Effect of magnetically dependent heating on the behaviour of magnetoacoustic waves in coronal plasma with thermal misbalance

ABSTRACT

The magnetic nature of coronal heating has been actively investigated within the framework of theoretical models and statistical analysis of observational data for decades. At present, a rather wide range of possible mechanisms has been proposed in the literature that requires additional verification. In this paper, we investigate the possibility of analysing the magnetic nature of coronal heating by means of magnetoacoustic (MA) waves propagating in coronal structures. To address this issue, we perform the analysis of fast and slow waves using a magnetic slab geometry. Applying the assumption of strong magnetic structuring, we derive the dispersion relation, which allows us to study the properties of MA waves. To analyse the dependence of phase velocity and wave decrement/increment on wavenumber, we numerically solved the obtained equations using the parameters corresponding to ‘warm’ coronal loop. It is shown that oscillations on the fundamental harmonic in a plasma with a weak magnetic field, where the effect of phase velocity dispersion is most pronounced, are best suited for diagnostics of magnetic heating using slow MA waves. In turn, the geometry remains the primary source for fast MA wave dispersion. Magnetic heating can either suppress or increase the damping of fast and slow MA waves. Moreover, the amplification of fast MA waves accompanied by damping of slow MA waves can be achieved. This issue is of interest in the context of the excitation of the decayless kink oscillations in the solar coronal loops.

instabilities, MHD, waves, Sun: corona, Sun: oscillations

1 INTRODUCTION

The solar atmosphere from the photosphere to the corona is highly structured and non-uniform. At the same time, it is also quite evident that one of the main structuring factors in the upper atmosphere is the strong magnetic field. It provides for the formation and long-term existence of various plasma structures with sizes from unobservably small to hundreds of megametres, such as coronal loops, prominences, plumes, etc. In fact, all these structures can play the role of waveguide for plasma perturbations. The latter are nowadays routinely observed with the help of space- and ground-based instruments (Roberts 2000; Nakariakov & Kolotkov 2020; Banerjee et al. 2021).

Magnetic fields not only introduce guiding for waves but also significantly affect their properties (Wiegelmann, Thalmann & Solanki 2014). In particular, the finite cross-section of coronal loops caused by magnetic structuring introduces the additional dispersion source for compression perturbations. Moreover, the presence of magnetic contribution to the total plasma pressure allows the existence of two compression mode types, the so-called fast and slow magnetoacoustic (MA) waves (see Priest 2014; Roberts 2019, for more information). The difference in these modes arises due to the fact that magnetic and gas-dynamic pressures act in phase and in opposite phases for fast and slow waves, respectively. For the first time, slow and fast MA waves were observed in the solar corona, thanks to data from high-resolution spacecraft, namely, Solar and Heliospheric Observatory (SOHO) and Transition Region and Coronal Explorer (TRACE) (e.g. Aschwanden et al. 1999; De Moortel, Walsh & Ireland 2000; Kliem et al. 2002; Ofman & Aschwanden 2002; Wang et al. 2002).

The manifestation of MA waves in observations can be quite different, as they can also have many additional features. In particular, MA perturbations can be trapped (trapped waves) in a magnetic structure (see Roberts 1981) or leak out of it (leaky waves). These issues were theoretically studied by Wilson (1981), Spruit (1982), and Cally (1986). At the same time, fast and slow waves can have different axial symmetry. The two main forms are sausage and kink; see pioneer works by Defouw (1976), Ryutov & Ryutova (1976), Roberts (1981), and Edwin & Roberts (1982, 1983), for details. The wave modes developing inside the structure and near the boundary are known as body and surface waves, respectively. Therefore, in the solar corona, there is a huge diversity of wave types corresponding to different types of compressional perturbations’ evolution.

The numerous observations associated with MA waves and oscillation and constructed magnetohydrodynamic (MHD) theories allowed us to indirectly measure the magnetic field strength in coronal structures (Jess et al. 2016; Cho et al. 2017). The technique of using MA waves for coronal plasma analysis is currently actively used in various applications and is known as MHD or coronal seismology. This method is not confined to the analysis of magnetic parameters of plasma. It can also be applied to the determination of thermodynamic parameters. In particular, the slow waves were used to measure the coronal temperature (Marsh & Walsh 2009) and transport coefficients (Wang et al. 2015).

Moreover, it turned out that the mentioned technique can be useful for solving the long-living problem of coronal heating. On closer inspection, the compression wave propagating through the coronal plasma disrupts not only the mechanical but also the thermal equilibrium of the corona. This is due to the known fact that optically thin radiation rate is the function of plasma density and temperature (Del Zanna et al. 2021). The unspecified coronal heating is also generally assumed to be a function of plasma parameters (Ibanez S. & Escalona T. 1993; Ibañez & Ballester 2022). In this case, the wave may experience feedback from unbalanced coronal heating and radiation cooling processes, which results in the energy transfer between the wave itself and the plasma. This phenomenon is known as a heating/cooling misbalance (or thermal misbalance; Field 1965; Molevich & Oraevsky 1988; Zavershinskii et al. 2019).

The effect of thermal misbalance has been intensively studied for several years (see the recent review by Kolotkov, Zavershinskii & Nakariakov 2021). It has been shown that the feedback between the waves and thermally active plasma leads to the dependence of the phase speed and growth/decay rate on the wave period, and also causes the growth or decay of the waves. Considering the heating as a function of temperature and density, these features of slow MA waves have been analysed using the infinite magnetic field approximation (Kolotkov, Nakariakov & Zavershinskii 2019; Zavershinskii et al. 2019) and ‘thin’ flux tube approximation (Belov, Molevich & Zavershinskii 2021). The analysis of both fast and slow MA waves affected by thermal misbalance has been conducted using the slab geometry by Agapova et al. (2022). It has been shown that the phase velocity behaviour of fast waves is less sensitive to non-adiabatic processes than slow waves. Thus, for a heating mechanism whose rate is mainly determined by the thermodynamic parameters of the plasma, slow waves look like a more promising seismological tool. Speaking of this, Kolotkov et al. (2019) introduced seismological constraints on the heating function, assuming the heating rate to be a power-law function [H(ρ, T) ∼ ρ^aT^b]. The limitations were set using observations of slow MA waves in various coronal structures and the requirement of attenuation of the slow MA and entropy modes through thermal misbalance. Another promising seismological technique is the analysis of the phase shifts between perturbations of different plasma parameters. This is due to the fact that heating/cooling and thermal conduction are also responsible for the variation of the phase shifts. Analytical studies of this issue are conducted in application to standing slow modes by Prasad et al. (2022) and to propagating slow modes by Prasad, Srivastava & Wang (2021) and Molevich et al. (2022). Furthermore, non-adiabatic processes affect the initial distribution of energy between modes; see the cases of thermal misbalance (Zavershinskii et al. 2021) and thermal conduction (Zavershinskii et al. 2023) for details.

It seems quite expected that the magnetic field should also play a significant role in the coronal heating on a par with density and/or temperature [e.g. H(ρ, T, B) ∼ ρ^aT^bB^c]. An important step in the study of this issue was the works of Rosner, Tucker & Vaiana (1978) and Hood & Priest (1979), who showed how the volumetric heating rate, averaged along the loop axis, could be related to pressure and loop length. Subsequently, this theory was applied by Porter & Klimchuk (1995) to scale the heating rate to the loop length using statistical relation between the pressures and lengths of the coronal loops observed by the Soft X-ray Telescope. This is quite a powerful result, which allows us to relate the coronal heating rate with the magnetic field by investigating statistical relationship between the magnetic field strength and length of magnetic flux tubes and applying some theoretical model meant to describe it (e.g. assuming heating by Alfvén wave turbulence; van Ballegooijen et al. 2011). An extensive analysis of more than 20 possible heating models (alternating currents (AC) and direct currents (DC)) was presented in Mandrini, Démoulin & Klimchuk (2000).This approach has been further developed by using the data analysis of extreme ultraviolet (Ugarte-Urra et al. 2019) and microwave gyroresonant (Fleishman et al. 2021) emission. However, the estimates constructed require verification by an additional approach.

We believe that such an approach could be the analysis of MA waves influenced by magnetically dependent heating through the effect of thermal misbalance. Previously, the coronal magnetic heating in the context of the thermal misbalance has been analysed using the ‘thin’ flux tube approximation (Duckenfield, Kolotkov & Nakariakov 2021; Kolotkov et al. 2021; Kolotkov, Nakariakov & Fihosy 2023). However, this approach allows only slow waves to be analysed, within the constraints of magnetic field strength and waveguide thickness. In this paper, we will use a more general approach and conduct the analysis using slab geometry. This will allow us to account more correctly for the effects of waveguide thickness, and to examine how magnetic heating affects fast MA waves through the effect of thermal misbalance.

Our manuscript is organized as follows. First, we present the main MHD equations describing thermally active plasma and obtain a differential equation that determines the dynamics of two-dimensional perturbations in Section 2. In Section 3, we derive the dispersion relation for waves in the magnetic slab under the assumption of strong magnetic structuring. We examine the obtained theoretical results for the solar corona in Section 4. Moreover, in Section 5, we present a comparison of the results obtained under the assumption of a ‘thin’ flux tube and a magnetic slab. The discussion and main conclusions are presented in Section 6.

2 MODEL AND BASIC EQUATIONS

The dynamics of MA oscillations and waves in the solar corona, which is composed of a fully ionized plasma where non-adiabatic processes such as heating and radiation cooling are present, can be described by the system of MHD equations presented in the work (see equations 1–6 in Agapova et al. 2022).

As we have mentioned previously, in current research, we have assumed that the heating processes depend not only on the temperature T and density ρ, but also on the magnitude of the magnetic field B. Therefore, the energy equation is represented as follows:

$$\begin{eqnarray} \frac{{\rho }^{\gamma }}{\gamma -1}\frac{{\mathrm{D}} }{{\mathrm{D}} t}\left(\frac{P}{{\rho }^{\gamma }}\right)=-\rho Q(\rho ,T,\left|\boldsymbol {B}\right|) . \end{eqnarray}$$

(1)

In equation (1), P is the plasma pressure, γ = C_P/C_V = 5/3 is the adiabatic index, where C_V = 3k_B/2m and C_P = C_V + k_B/m are the specific heat capacities at constant volume and pressure, respectively. The Boltzmann constant and the mean particle mass are denoted by k_B and m, respectively. |$\mathrm{ D}/{\mathrm{ D}{t}}=\partial /\partial t+\boldsymbol {v}\cdot \nabla$| – stands for the convective derivative.Heat-loss function Q(ρ, T, |B|) is defined as the difference between radiation cooling L(ρ, T) and heating H(ρ, T, |B|) functions, i.e. Q(ρ, T, |B|) = L(ρ, T) − H(ρ, T, |B|). In the stationary state, the heat-loss function Q(ρ₀, T₀, |B₀|) = 0, implying that non-adiabatic processes balance each other, i.e. L(ρ₀, T₀) = H(ρ₀, T₀, |B₀|).

In order to analyse the properties of the small amplitude perturbations, we have used the standard perturbation theory method. Given the undisturbed magnetic field B₀ directed along the z-axis, as well as, the stationary state ρ₀, and P₀, the complete set of linear MHD equations, including linearized equation (1), can be written as

$$\begin{eqnarray} \frac{\partial {B_1}_x}{\partial t}={B_0}\frac{\partial {v_1}_x}{\partial z}, \end{eqnarray}$$

(2)

$$\begin{eqnarray} \frac{\partial {B_1}_y}{\partial t}={B_0}\frac{\partial {v_1}_y}{\partial z}, \end{eqnarray}$$

(3)

$$\begin{eqnarray} \frac{\partial {B_1}_z}{\partial t} = {B_0} \frac{\partial {v_1}_z}{\partial z} -{B_0} \Theta , \end{eqnarray}$$

(4)

$$\begin{eqnarray} \nabla \cdot \boldsymbol B_1=0 , \end{eqnarray}$$

(5)

$$\begin{eqnarray} \rho _0 \frac{\partial {v_1}_x}{\partial t}=-\frac{\partial P_T}{\partial x} + \frac{B_0 }{4 \pi }\frac{\partial {B_1}_x}{\partial z} , \end{eqnarray}$$

(6)

$$\begin{eqnarray} \rho _0 \frac{\partial {v_1}_y}{\partial t}=-\frac{\partial P_T}{\partial y} + \frac{B_0 }{4 \pi }\frac{\partial {B_1}_y}{\partial z}, \end{eqnarray}$$

(7)

$$\begin{eqnarray} \rho _0 \frac{\partial {v_1}_z}{\partial t}=-\frac{\partial P_T}{\partial z} + \frac{B_0 }{4 \pi }\frac{\partial {B_1}_z}{\partial z}, \end{eqnarray}$$

(8)

$$\begin{eqnarray} \frac{\partial \rho _1}{\partial t} + \rho _0 \Theta = 0, \end{eqnarray}$$

(9)

$$\begin{eqnarray} \frac{1}{\gamma -1} \frac{\partial {P_1}}{\partial t}-\frac{\gamma }{\gamma -1} \frac{P_0}{\rho _0} \frac{\partial {\rho _1}}{\partial t} = -\rho _0 \left({Q_{0T}} T_1+{Q_{0\rho }} \rho _1+{Q_{0B}}{B_1}_z\right), \\ \end{eqnarray}$$

(10)

$$\begin{eqnarray} P_1=\frac{k_{\rm B}}{m}\left(\rho _0 T_1 +\rho _1 T_0 \right) , \end{eqnarray}$$

(11)

where the numeric indices ‘0’ and ‘1’ indicate the stationary state and linear perturbation of plasma parameters, respectively. Alphabetic subscripts (‘x’, ‘y’, and ‘z’) mean the corresponding component of magnetic |${\mathrm \bf {B}} $| or velocity |${\mathrm \bf {v}}$| vectors. We also introduce the following notation for the divergence of velocity perturbation |$\Theta =\nabla \cdot \boldsymbol {v}_1$|⁠, and derivatives of the heat-loss function |$Q_{0T}=\left.\partial {Q}/\partial {T}\right|_{\rho _0, T_0, B_0},Q_{0\rho }=\left.\partial {Q}/\partial {\rho }\right|_{\rho _0, T_0, B_0}, \, \text{and} \, Q_{0B}=\left.\partial {Q}/\partial {B}\right|_{\rho _0, T_0, B_0}$|⁠.

In the current research, we use the plane geometry and consider the magnetic slab of half-width x₀ locating between x = −x₀ and x = x₀. Furthermore, we have assumed the unperturbed state of the plasma with magnetic field B_0i, density ρ_0i, and the temperature T_0i inside the slab, and magnetic field B_0e, density ρ_0e, and temperature T_0e outside the slab (see Fig. 1). Thus, the equilibrium state may be described by

$$\begin{eqnarray} B_0\!\left(x\right), \rho _{0}\left(x\right), T_{0}\left(x\right)= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}B_{0i}, \rho _{0i}, T_{0i},\ \ \left|x\right|\le x_0, \\ B_{0e}, \rho _{0e}, T_{0e},\ \ \left|x\right|\gt x_0. \end{array}\right. \end{eqnarray}$$

(12)

Figure 1.

Configuration of a 2x₀-wide magnetic slab in a thermally active plasma. The black arrows show the vertical magnetic field |$B_0(x)\hat{\boldsymbol{ z}}$|⁠.

Separately for the plasma inside and outside the magnetic slab, the system of equations (2)–(11) can be combined to yield (we omit indices ‘i’ and ‘e’ for simplicity)

$$\begin{eqnarray} &&{\frac{\partial }{\partial t}\left(\frac{\partial P_T}{\partial t}-\rho _0c_{\rm A}^2\frac{\partial {\textit {v}_1}_z}{\partial z}+\rho _0\left(c_{\rm A}^2+c_{\rm S}^2\right)\!{\Theta }\right)\!\tau _V }\\ &&{\quad + \frac{\partial P_T}{\partial t}\! +\rho _0(c_{\rm SQB}^2-c_{\rm A}^2)\frac{\partial {\textit {v}_1}_z}{\partial z}}\\ &&{\quad+\rho _0\left(c_{\rm A}^2+c_{\rm SQ}^2-c_{\rm SQB}^2\right){\Theta }=0,} \end{eqnarray}$$

(13)

$$\begin{eqnarray} \frac{\partial ^{2} {\textit {v}_1}_x}{\partial t^{2}}=-\frac{1}{\rho _0}\frac{\partial ^{2} P_T}{\partial x^{}\partial t^{}}+c_{\rm A}^2\frac{\partial ^{2} {\textit {v}_1}_x}{\partial z^{2}}, \end{eqnarray}$$

(14)

$$\begin{eqnarray} \frac{\partial ^{2} {\textit {v}_1}_z}{\partial t^{2}}=-\frac{1}{\rho _0}\frac{\partial ^{2} P_T}{\partial z^{}\partial t^{}}+c_{\rm A}^2\left(\frac{\partial ^{2} {\textit {v}_1}_z}{\partial z^{2}}-\frac{\partial \Theta }{\partial z}\right). \end{eqnarray}$$

(15)

Here, |$P_T\!=\!P_1+({B_0}/{4\pi }){B_1}_z$| is the perturbation of the total pressure (gas-dynamic plus magnetic), and |$c_{\rm A}\!=\!\sqrt{B_0^2/{4\pi \rho _0}}$| and |$c_{\rm S}=\sqrt{\gamma k_{\rm B}T_0/m}$| are the Alfvén and sound speeds, respectively.

Before we go any further, let us introduce some characteristic speeds and time-scales. If the slow waves are affected by the thermal misbalance only, then the phase velocity c_ph = c_ph(ω) becomes a function of the wave frequency ω and varies from c_S to c_SQ with decreasing frequency. Here, c_SQ is the characteristic wave speed, which is determined by the dependence of heating/cooling processes on plasma temperature and density:

$$\begin{eqnarray} c_{\rm SQ} =\sqrt{\frac{\tau _V}{\tau _P } \gamma \frac{k_{\rm B}T_0}{m}}. \end{eqnarray}$$

(16)

In equation (16), we have used characteristic time-scales:

$$\begin{eqnarray} \tau _V={C_V}/{Q_{0T}}, \qquad \tau _P={C_P}T_0/\left({Q_{0T}} T_0-{Q_{0\rho }} \rho _0\right), \end{eqnarray}$$

(17)

which allows us to introduce ranges of weak (⁠|$\omega \,$||τ_{V, P}| ≫ 1) and strong (⁠|$\omega \,$||τ_{V, P}| ≪ 1) impact of thermal misbalance.

In addition, we have introduced velocity c_SQB that takes into account the effects of the magnetic field on the dispersion properties of MA waves:

$$\begin{eqnarray} c_{\rm SQB}^2=\frac{Q_{0B}B_0k_{\rm B}}{Q_{0T}m}. \end{eqnarray}$$

(18)

In practice, however, the physical meaning will be in the modified tube speed, which will take into account the aforementioned influence of the magnetic field via this term and which we will discuss further.

In order to describe the wave dynamics, we have substituted a solution in the form of a harmonic wave into the system of equations (13)–(15):

$$\begin{eqnarray} \textit {v}_{1x}={\tilde{\textit {v}}}_{1x}\left(x\right)\mathrm{e}^{\mathrm{i}\left(\omega t+k_yy+k_zz\right)},\\ \textit {v}_{1y}={\tilde{\textit {v}}}_{1y}\left(x\right)\mathrm{e}^{\mathrm{i}\left(\omega t+k_yy+k_zz\right)},\\ \textit {v}_{1z}={\tilde{\textit {v}}}_{1z}\left(x\right)\mathrm{e}^{\mathrm{i}\left(\omega t+k_yy+k_zz\right)}, \\ P_T={\tilde{P}}_T\left(x\right)\mathrm{e}^{\mathrm{i}\left(\omega t+k_yy+k_zz\right)}, \end{eqnarray}$$

where |${\tilde{v}}_{1x}$|⁠, |${\tilde{v}}_{1y}$|⁠, and |${\tilde{v}}_{1z}$| are the amplitude components of the velocity vector and |${\tilde{P}}_T$| is the total pressure perturbation; k_y and k_z are the wavenumbers in y- and z-directions, respectively.

After substituting the solution in the form of a harmonic wave and some mathematical transformations, the equations (13)–(15) can be reduced to the equation for the amplitude of the total pressure perturbation:

$$\begin{eqnarray} {\tilde{P}}_T=\frac{\mathrm{i}\rho _0(k_z^2c_{\rm A}^2-\omega ^2)(A^2_{\rm Q}+\mathrm{i}\omega \tau _VA^2)}{\omega \left(A^2_{\rm Q}\left(m^2_{\rm Q}+k_y^2\right)+\mathrm{i}\omega \tau _VA^2\left(m^2+k_y^2\right)\right)}\frac{\mathrm{ d}^{} \tilde{\textit {v}}_{1x}}{\mathrm{ d} x^{}}, \end{eqnarray}$$

(19)

where the following notations are introduced:

$$\begin{eqnarray} m^2=\frac{\left(k^2_zc^2_{\rm A}-\omega ^2\right)\left(k^2_zc^2_{\rm S}-\omega ^2\right)}{\left(c^2_{\rm A}+c^2_{\rm S}\right)\left(k^2_zc^2_{\rm T}-\omega ^2\right)}\!, \end{eqnarray}$$

$$\begin{eqnarray} m^2_{\rm Q}=\frac{\left(k^2_zc^2_{\rm A}-\omega ^2\right)\left(k^2_zc^2_{\rm SQ}-\omega ^2\right)}{\left(c^2_{\rm A}+c^2_{\rm SQ}-c^2_{\rm SQB}\right)\left(k^2_zc^2_{\rm TQ}-\omega ^2\right)}, \end{eqnarray}$$

$$\begin{eqnarray} A^2=\left(c^2_{\rm A}+c^2_{\rm S}\right)\left(k^2_zc^2_{\rm T}-\omega ^2\right) , \end{eqnarray}$$

$$\begin{eqnarray} A^2_{\rm Q}=\left(c^2_{\rm A}+c^2_{\rm SQ}-c^2_{\rm SQB}\right)\left(k^2_zc^2_{\rm TQ}-\omega ^2\right) . \end{eqnarray}$$

To describe the influence of the waveguiding dispersion in a thermally active magnetic structure, two additional characteristic phase speeds are to be introduced:

$$\begin{eqnarray} c_{\rm T}=\sqrt{ \frac{{c^2_{\rm S}c^2_{\rm A}}}{\left(c^2_{\rm A}+c^2_{\rm S}\right)}},\qquad c_{\rm TQ} =\sqrt{\frac{c^2_{\rm A}c^2_{\rm SQ}}{\left(c^2_{\rm A}+c^2_{\rm SQ}-c_{\rm SQB}^2\right)}}. \end{eqnarray}$$

(20)

Here, we show the well-known tube speed c_T for an ideal plasma, which is generally associated with slow waves and is the sound speed reduced by the effect of the magnetic field elasticity. In the ideal plasma, the phase velocity change from the speed of sound c_S in the short-period limit to the tube speed c_T in the long-period limit is related only with geometry dispersion effect. The combination of geometry dispersion and the dispersion caused by the thermal misbalance leads to the appearance of the modified tube speed c_TQ (see Belov et al. 2021 for details). In our consideration, the expression for c_TQ (equation 20) is further modified from the introduced (Belov et al. 2021) by using |$c_{\rm SQB}^2$| (equation 18) to take into account the influence of heating function dependence on magnetic field. The relation for c_TQ is consistent with the relation introduced in Kolotkov et al. (2021) using ‘thin’ flux tube approximation.

Let us derive the equation that will allow us to describe the dynamics of two-dimensional perturbations in the plasma inside and outside a magnetic slab with thermal misbalance. For this purpose, one should combine the Fourier transform of the equation (14) with the equation (19) differentiated by the x-coordinate. The resulting equation is as follows:

$$\begin{eqnarray} \frac{\left(\mathrm{i}\omega \tau _VA^2+A^2_{{\rm Q}}\right)}{\left(A^2_{{\rm Q}}\left(m^2_{{\rm Q}}+k_y^2\right)+\mathrm{i}\omega \tau _VA^2\left(m^2+k_y^2\right)\right)}\frac{\mathrm{ d}^{2} \tilde{\textit {v}}_{1x}}{\mathrm{ d} x^{2}}-\tilde{\textit {v}}_{1x}=0. \end{eqnarray}$$

(21)

3 THE DISPERSION RELATION

Using equation (21), we now can proceed to an analysis of the dispersion properties of MA waves. As well as a velocity component normal to the slab boundary v_x, the total pressure must be continuous at the boundaries of the magnetic slab (at x = ±x₀) to satisfy the condition of mechanical equilibrium:

$$\begin{eqnarray} P_e+\frac{B^2_e}{8\pi }=P_i+\frac{B^2_i}{8\pi }. \end{eqnarray}$$

(22)

In addition, for the stationary state to exist, the thermal equilibrium in the internal and external plasma should be established. This implies that

$$\begin{eqnarray} Q\left(\rho _{0_{i}}, T_{0_{i}}, B_{0_{i}}\right) = 0,\qquad Q\left(\rho _{0_{e}}, T_{0_{e}}, B_{0_{e}}\right) =0. \end{eqnarray}$$

(23)

We have limited ourselves to the analysis of two-dimensional perturbations, where the velocity component |${\textit {v}_y}$| and the wavenumber k_y in the y-direction equal zero. For such perturbations, the transverse component of the velocity vector is given as |$\textit {v}_x={\hat{\textit {v}}}_x\left(x\right)e^{i\left(\omega t+k_z z\right)}$|⁠. Additionally, it is assumed that the waves arising inside the slab disappear outside it (i.e. |$\textit {v}_x\rightarrow 0$| at x → ±∞), analogous assumptions applied in works Roberts (1981) and Arregui et al. (2007).

As a result, the differential equation for the velocity perturbation (equation 21) can be rewritten in the form shown below:

$$\begin{eqnarray} \frac{\mathrm{ d}^{2} {\hat{\textit {v}}}_{1x}}{\mathrm{ d} x^{2}}-k^2_{x_i}{\hat{\textit {v}}}_{1x}=0, \quad \left|x\right|\le x_0, \\ \frac{\mathrm{ d}^{2} {\hat{\textit {v}}}_{1x}}{\mathrm{ d} x^{2}}-k^2_{x_e}{\hat{\textit {v}}}_{1x}=0, \quad \left|x\right|\gt x_0, \end{eqnarray}$$

(24)

where we introduce |$k^2_{x_{i,e}}$| as follows:

$$\begin{eqnarray} k^2_{x_{i,e}}=\frac{\left(A^2_{{Q_{i,e}}}\!m^2_{{Q_{i,e}}}+i\omega {\tau _V}_{i,e} A^2_{i,e }m^2_{i,e}\right)}{\left(A^2_{{Q_{i,e}}}+i\omega {\tau _V}_{i,e}A^2_{i,e}\right)}. \end{eqnarray}$$

It has been shown previously (Agapova et al. 2022) that in a thermally active plasma, wavenumber k_x becomes a complex value. This differs from the case of an ideal plasma (see Edwin & Roberts 1982), where the value of k_x (or m in the author’s original notation) can be either real or purely imaginary, depending on the sign of |$k_x^2$|⁠. In the ideal plasma, the case |$k_x^2\gt 0$| is associated with surface modes, and the case |$k_x^2\lt 0$| is associated with body modes. Here, we follow the reasoning presented in Agapova et al. (2022), i.e. we assume that if the real part is positively defined by |$\mathrm{ Re}\!\!\left(k_x^2\right)\gt 0$|⁠, then Re(k_x) > Im(k_x) and the wave can be considered as a ‘surface rather than body’ mode. In the opposite case, |$\mathrm{ Re}\!\!\left(k_x^2\right)\lt 0$|⁠, the analogy assumes that Re(k_x) < Im(k_x), then such waves can be called ‘more a body than a surface’. Further, for brevity, we will refer to the waves simply as surface and body modes, meaning ‘rather a surface than a body’ and ‘rather a body than a surface’, respectively.

Next, we will solve the equations (24) for the positive real part of the wave vector Re(k_x) > 0. The mentioned absence of waves at infinity (⁠|${\hat{v}}_{1x}\!\rightarrow \!0$| at |x| → ∞) in terms of wavenumbers implies |$\mathrm{ Re}\!\!\left(k_{x_{e}}^2\right)\gt 0$| at |x| → ∞ (depending on the current conditions, inside the slab, the sign of |$\mathrm{ Re}\!\!\left(k^2_{x_{i}}\right)$| can be arbitrary). Thus, the solution can be written as

$$\begin{eqnarray} {\hat{v}}_{1x}=\left\lbrace \begin{array}{lr}\alpha _i{\cosh \left(k_{x_i}x\right)\ }+\beta _i{\sinh \left(k_{x_i}x\right)\ }, & \left|x\right|\le x_0, \\ \beta _e \exp \left({k_{x_e}\left(x+x_0\right)} \right), & x\lt {-x}_0,\\ \alpha _e \exp \left({-k_{x_e}\left(x-x_0\right)}\right), & x\gt x_0, \end{array} \right. \end{eqnarray}$$

(25)

where α_i, β_i, α_e, and β_e are arbitrary constants, which must be determined up to multiplicative constants. For this purpose, one has to choose the mode under consideration and use the boundary conditions on x = ±x₀ (for more information, see Agapova et al. 2022).

Finally, the requirement of the existence of a non-trivial solution to this system of equations gives a general dispersion relation for (surface/body) sausage/kink waves:

$$\begin{eqnarray} (k^2_z{c^2_{\rm A_{\mathit{ i}}}}\!-\omega ^2) \frac{k_{x_e}}{k_{x_i}} \! =\!-\frac{\rho _{0_e}}{\rho _{0_i}}\left(k^2_z{c^2_{\rm A_{\mathit{ e}}}}\!-\omega ^2\right)\!\!\begin{pmatrix} \!\mathrm{ tanh}\ (k_{x_i}x_0)\\ { \!\mathrm{ coth}\ (k_{x_i}x_0)} \end{pmatrix}\!. \end{eqnarray}$$

(26)

In order to describe kink and sausage modes, we use |$~\rm \!tanh()$| and |$~\rm \!coth()$| functions, respectively. It should be mentioned that in the absence of thermal misbalance (τ_{Vi, e} → 0), the equation (26) reduces to that obtained for ideal plasma by Edwin & Roberts (1982). If the heat-loss function does not depend on the magnetic field strength (i.e. Q_0B → 0), equation (26) is reduced to the case considered in Agapova et al. (2022).

Further, we will apply dispersion relation (26) to calculate the dependences of phase velocities and increment/decrement of fast and slow MA waves on wavenumbers using the solar corona conditions.

4 APPLICATION TO SOLAR CORONA

As we have noted above, the heat-loss function Q(ρ, T, |B|) is determined by the difference between heating H(ρ, T, |B|) and the radiative energy loss L(ρ, T). Let us now specify these functions for the solar corona conditions.

In these conditions, the main factor causing cooling is optically thin radiation. Therefore, the cooling rate L(ρ, T) can be modelled as a function of density and temperature as follows:

$$\begin{eqnarray} L(\rho , T)=\frac{\rho }{4m^2}\Lambda \!\left(T\right). \end{eqnarray}$$

(27)

Here, function Λ(T) denotes the temperature dependence of the radiation losses. In this paper, we determine it using the CHIANTI data base Version 10.2 (Del Zanna et al. 2015, 2021).

To study the effect of coronal heating, we assume that its rate is some power-law function of density, temperature, and magnetic field strength, namely

$$\begin{eqnarray} H(\rho , T, \left|B\right|)=h \rho ^aT^b\left|B\right|^c, \end{eqnarray}$$

(28)

where constant h is determined to balance the cooling process in steady-state conditions, such that H(ρ₀, T₀, |B₀|) = L(ρ₀, T₀). Assuming the coronal heating as the power-law function is a well-known practice used for the last few decades (see e.g. Rosner et al. 1978; Hood & Priest 1979; Porter & Klimchuk 1995; Mandrini et al. 2000; Duckenfield et al. 2021; Fleishman et al. 2021) to determine the possible mechanism by analysing observational data. It has been shown previously by Kolotkov, Duckenfield & Nakariakov (2020) that heating mechanisms originally proposed by Rosner et al. (1978) and widely used for calculations and analytics (Ibañez & Ballester 2022; Washinoue & Suzuki 2023) (Ohmic heating, constant heating per unit volume and mass, and heating by Alfvén waves via mode conversion and anomalous conduction damping), written in form of function of density and temperature H(ρ, T) ∼ ρ^aT^b imply the instabilities of compression modes in the magnetic structure. This implies that these mechanisms are inappropriate for describing long-lived coronal wave-guiding structures. However, two mechanisms were originally proposed to be also proportional to magnetic field ∼ρ^aT^b|B|^c, namely the anomalous conduction (a = 1, b = 1, and c = −3), and Alfvén mode conversion (a = 0, b = 1, and c = 1/3). Nevertheless, our analysis reveals that these forms also imply that compression waves remain unstable. In this regard, we followed the logic of previous works and used a seismologically introduced heating mechanism (Kolotkov et al. 2020) for our calculations and modified it to take into account the magnetic field influence. In other words, our basic heating model was H(ρ, T, |B|) ∼ ρ²T^−3.5|B|^c. This form will further allow us to compare the results of our study with similar results arising from the ‘thin’ flux tube model (see Section 5).

To calculate the values of phase velocity and MA-wave decrement, we numerically solved the equation (26) using the slab parameters from Table 1. These parameters correspond to ‘warm’ coronal loops, with one loop having a magnetic field of 10 G and a density contrast of 5, and another loop having a magnetic field of 50 G and a density contrast of 10. Figs 2 and 3 show plots of the phase velocity Re(ω)/k_z and P = 1/Im(ω) of MA waves on the dimensionless wavenumber k_zx₀, respectively. According to our calculations, all the roots found correspond to body waves.

Figure 2.

Phase velocity dependences Re(ω)/k on the dimensionless wavenumber k_zx₀, calculated for the coronal loop (see Table 1). Different spatial scales are used for fast and slow MA waves. The speed range on the vertical axis, where the scale changes, is indicated by the saw teeth. A logarithmic scale is used horizontally. The left column corresponds to the value of the magnetic field inside the slab equal to 10 G, and the right column shows the case of 50 G. The top and bottom panels correspond to different waves, namely sausage and kink, respectively. Different colours correspond to different heating scenarios. The grey line indicates the range in which it is impossible to find the roots.

Figure 3.

Decrement P = 1/Im(ω) as a function of the dimensionless wavenumber k_zx₀. The left column corresponds to the case when the magnetic field inside the slab is 10 G. The right column corresponds to the case of 50 G. The top and bottom panels correspond to sausage and kink waves, respectively. Different colours correspond to different heating scenarios.

Table 1.

Slab parameters used for calculations.

Parameter	Value
Magnetic field strength inside the slab, |$B_{0_{i}}$| (G)	10	50
Temperature inside the slab, |$T_{0_{i}}$| (MK)	1
Temperature outside the slab, |$T_{0_{e}}$| (MK)	1
Number density inside the slab, |$n_{0_{i}}$| (cm−3)	1010
Density contrast |$\left(n_{0_{i}}/n_{0_{e}}\right)$|	5	10
Slab half-width, x0 (Mm)	1

Speaking of the slow MA waves, one can notice (see Fig. 2) that the thermal misbalance causes significant changes in the phase velocity in the long-wave limit (low-frequency range of the spectrum). The phase speed of both sausage and kink waves now varies in the range from the modified tube speed |$c_{{\mathrm{ TQ}}_i}$| (equation 20) to sound speed inside the slab |$c_{{\rm S}_i}$| (rather than from the internal tube speed |$c_{{\mathrm{ T}}_i}$|⁠, as was the case in the weakly non-adiabatic plasma). In particular, for the case of B_0i = 10 G, the modified tube speed is |$c_{{\mathrm{ TQ}}_i} \approx 102 \pm 2$| km s⁻¹, while the internal tube speed is |$c_{{\mathrm{ T}}_i} \approx 133$| km s⁻¹. In turn, for the case of B_0i = 50 G, the modified tube speed is |$c_{{\mathrm{ TQ}}_i} \approx 108$| km s⁻¹, while the internal tube speed is |$c_{{\mathrm{ T}}_i} \approx 151$| km s⁻¹. In fact, such a difference between |$c_{{\mathrm{ TQ}}_i}$| and |$c_{{\mathrm{ T}}_i}$| can result in errors in seismological estimates of plasma parameters by slow waves (Belov et al. 2021). On the other hand, it may be used for seismological estimations of coronal heating function using found inconsistencies. The effect is most pronounced for oscillations at the fundamental harmonic.

It is quite noticeable that for the shown case B_0i = 10 G (left column in Fig. 2), the modified tube speed |$c_{{\mathrm{ TQ}}_i}$| has different values depending on the exponent c. One may also notice that an increase of exponent c leads to the decrease of modified tube speed |$c_{{\mathrm{ TQ}}_i}$|⁠. It turned out that such variation of the modified tube speed |$c_{{\mathrm{ TQ}}_i}$| related to magnetic heating formulation is more pronounced the weaker the magnetic field (the greater the plasma beta). In particular, for considered ‘strong’ magnetic field, where B_0i = 50 G (right column in Fig. 2), the difference in the values of |$c_{{\mathrm{ TQ}}_i}$| between the considered heating functions becomes neglectable. This result coincides with the one previously obtained by Duckenfield et al. (2021) using ‘thin’ flux tube approximation. Our calculations show that for considered loops, the difference between |$c_{{\mathrm{ TQ}}_i}$| for c = 1 and c = −1 is less |$1~{\%}$| starting with magnetic field strength 34 G, which corresponds to internal plasma β ≈ 0.02.

In turn, the thermal misbalance again has no noticeable effect on the phase speed dispersion of fast MA waves. Even variations in the magnetic heating that adhere to the seismological constraints (Kolotkov et al. 2023) do not produce significant effects. The dependences of fast wave phase velocity on wavenumber, calculated for a plasma with thermal activity, are nearly identical to those for an ideal plasma. In other words, the geometry of the slab remains the primary source of phase velocity dispersion for fast waves. As in the case of ideal plasma, the phase velocity of fast waves oscillates between |$c_{\rm A_\mathit{ e}}$| and |$c_{\rm A_\mathit{ i}}$|⁠.

Thermal misbalance is known to cause not only wave dispersion, but also the growth/decay of compression perturbation depending on the regime of thermal misbalance (see e.g. Zavershinskii et al. 2021; Kolotkov et al. 2023). The chosen heating mechanism implies that all compression modes should be damped for c = 0. Fig. 3 shows the decay periods P = 1/Im(ω) of fast and slow kink/sausage MA waves calculated for selected parameters and heating mechanisms c = −1, 0, and 1. It was shown previously that slow MA waves grow/decay faster than fast waves in the uniform plasma due to the thermal misbalance in the plasma with the beta less than 1 (see e.g. Zavershinskii et al. 2020 for details). The same is true for magnetically structured and heated plasma.

Our calculations also reveal that magnetic heating affects the attenuation (or possible amplification) of slow and fast MA waves in different ways. In particular, the increase in the magnetic field strength generally leads to an increase in slow MA wave damping and a decrease in fast MA wave damping due to the misbalance effect. These issues are clearly seen by comparison of calculated decay periods P shown in Fig. 3 for |$B_{0i}=10\, \mathrm{ G}$| (left column) and |$B_{0i}=50\, \mathrm{ G}$| (right column). All considered decay periods of slow MA waves calculated for the case of |$B_{0i}=10\, \mathrm{ G}$| are greater than decay periods calculated for the case of |$B_{0i}=50\, \mathrm{ G}$|⁠. Conversely, in the case of fast MA waves, all considered decay periods of fast MA waves calculated for the case of |$B_{0i}=10\, \mathrm{ G}$| are lower than decay periods calculated for the case of |$B_{0i}=50\, \mathrm{ G}$|⁠. In the course of the comparison, one may also notice that the influence of magnetic heating formulation (power index c) becomes weaker for slow MA waves with the increase of the magnetic field strength. This effect is shown in Fig. 3 using the case |$B_{0i}=50\, \mathrm{ G}$| (right column). One may notice that dependences of the decay period of slow MA waves in this case are practically identical on the considered range of wavenumbers. The same effect has been shown by Duckenfield et al. (2021) using the ‘thin’ flux tube approximation. In turn, the decay period of the fast MA wave has a noticeable dependence on the heating mechanism formulation.

Continuing to discuss the effect of the heating mechanism formulation, we should mention that it also affects the fast and slow MA waves in different ways. To be more precise, the increase of power index c leads to the growth of the slow MA wave damping. This effect is more pronounced for low magnetic field cases (see the case of |$B_{0i}=10\, \mathrm{ G}$| in Fig. 3). In contrast, the increase of power index c leads to the decrease of the fast MA wave damping. In other words, heating function dependence on magnetic field can suppress or enhance the attenuation of fast and slow MA waves due to the thermal misbalance effect related to the temperature and density dependences of heating and radiation cooling.

Moreover, by continuing to increase the power index c one can achieve not only compensation of attenuation but also amplification of fast MA waves. Meanwhile, the slow MA waves remain damped waves. In order to show such an intriguing issue, we considered heating with power indices c = 3, 4, and 5 for magnetic field. Our calculations are shown in Fig. 4. The theses previously stated in the attenuation conjecture are also applicable to the case of amplification. In particular, the effect of amplification is more pronounced in the case of low magnetic field (high beta plasma). The effect is visible for both sausage and kink modes. When it comes to the amplification at the fundamental harmonic, it is worth noting that there can be both an increase and decrease in amplification rate with increasing loop length (corresponding to a decreasing of wave vector k_zx₀). We believe that such an issue is of great interest in the context of decayless kink oscillation and requires special analysis in the future.

Figure 4.

Absolute value of increment |P| = 1/Im(ω) as a function of the dimensionless wavenumber k_zx₀. The left column corresponds to sausage waves. The right column corresponds to kink waves. The solid line corresponds to the case where the magnetic field inside the plate is 10 G. The dashed line corresponds to the case of 50 G. Different colours correspond to different heating scenarios.

5 ‘THIN’ FLUX TUBES AND MAGNETIC SLAB: COMPARISON

Despite the generality of the approach used within the framework of this work, its application to helioseismological problems is more complicated than the application of the ‘thin’ flux tube approximation. The question may therefore be asked when the application of the simplified approach may lead to sufficient estimation errors. In order to answer this question, further, we compare the results of the ‘thin’ flux tube model previously obtained by Kolotkov et al. (2021) with the model presented in this article. Since the ‘thin’ flux tube model describes only waves of axial symmetry (sausage) waves and does not take into account the parameters of the external environment, which is important for fast waves, the comparison is made only for slow sausage waves.

The comparison between the models is shown in Fig. 5. For the ‘weak’ magnetic field (left panel), it can be noticed that the phase speed dependences described by the magnetic slab and ‘thin’ flux tube models are quite close to the long-wavelength limit. In this limit, the phase speed varies between the introduced modified tube speed |$c_{{\mathrm{ TQ}}_i}$| (equation 20) and the internal tube speed |$c_{{\mathrm{ T}}_i}$|⁠. The small discrepancy between the models in the mentioned range is due to the fact that phase velocity described by the ‘thin’ flux tube models increases faster with the wavelength decrease because the tube becomes no longer ‘thin’ for smaller wavelengths. However, the curves coincide again when phase speeds approach |$c_{{\mathrm{ T}}_i}$| and begin to diverge up to 10 per cent with the increase of a dimensionless wavenumber. The reason for this divergence is the same as for the long-wavelength limit.

Figure 5.

Phase velocity dependences Re(ω)/k on the dimensionless wavenumber k_zx₀, calculated for the coronal loop (see Table 1) obtained with the slab (straight lines) and ‘thin’ flux tube (dashed lines) models. The left column corresponds to the value of the magnetic field inside the slab equal to 10 G, and the right column shows the case of 50 G. The grey line indicates the range in which it is impossible to find the roots.

For the case of a ‘strong’ magnetic field (right panel of Fig. 5), there is a slight deviation in the range from 0.01 to 0.2 for k_zx₀ values, which is about 4 per cent of the difference. Nevertheless, it can be assumed that phase curves are almost identical because a ‘strong’ magnetic field leads to the fact that |$c_{{\mathrm{ T}}_i}\approx c_{{\mathrm{ S}}_i}$| and |$c_{{\mathrm{ TQ}}_i}\approx c_{{\mathrm{ SQ}}_i}$|⁠; thus, geometrical dispersion can be neglected.

Therefore, the shown difference in the results of the models underlines the limitations of the applicability of the ‘thin’ flux tube model. This model can describe the dispersion of slow waves quite well except for the short-wavelength limit of the ‘weak’ magnetic field case. Nevertheless, it may be a good first approximation for the application to real observations.

6 SUMMARY AND CONCLUSION

The routinely observed long-lived magnetic structures in the solar corona, in fact, owe their existence to the presence of mechanical and thermal equilibrium. Speaking of the latter, it is generally presumed that losses of radiation are compensated by the inflow of magnetic energy stored in the plasma. Despite the fact that the magnetic nature of the coronal heating rate seems to be quite expected, its formulation remains enigmatic. However, some statistical relation between plasma magnetic field and coronal loop parameters has been found, allowing to seismologically specify the theoretically proposed mechanisms (see e.g. Porter & Klimchuk 1995; Mandrini et al. 2000; Ugarte-Urra et al. 2019; Fleishman et al. 2021). Nevertheless, to verify the proposed models, some additional diagnostic tool is required. Within the framework of this work, we assumed that such an instrument can be MA waves affected by the thermal misbalance effect.

To be more specific, we aimed to investigate the effect of magnetic heating on the dispersion properties of MA waves in the coronal plasma. Such a problem has been previously investigated by means of first-order ‘thin’ flux tube approximation (see e.g. Duckenfield et al. 2021; Kolotkov et al. 2021, 2023). However, this approximation has limitations on the spectrum of described waves and is unable to describe the properties of fast MA waves. Therefore, in order to enhance the capability of seismological assessments, a more general approach is to be applied.

In the current research, we analyse the wave properties using slab geometry (see Fig. 1). Applying the perturbation theory methods, we derived equation (21) allowing us to describe the dynamics of two-dimensional perturbations in a magnetically structured plasma. Considering the case of strong magnetic structuring, we obtained the dispersion relation (26), which can describe the properties (surface/body) of sausage/kink fast and slow MA waves. In order to show the influence of magnetic heating, we consider the magnetic slab with parameters corresponding to ‘warm’ coronal loops. Below, we briefly describe the revealed features:

• The combination of geometry dispersion and the dispersion caused by the thermal misbalance accounting for heating function dependence on the magnetic field leads to the appearance of the modified tube speed c_TQ (equation 20), which is now a long-wavelength limit of slow MA wave phase velocity. The obtained expression coincides with the one previously obtained using ‘thin’ flux tube approximation (Kolotkov et al. 2021). The presence of a difference with respect to the case of no dependence on magnetic field (Belov et al. 2021; Agapova et al. 2022) gives the principal opportunity to use the slow waves for the seismological study of magnetic field contribution to the heating function. According to our results, oscillations on the fundamental harmonic in regions with weak magnetic fields, where the differences are more significant, are best suited for this purpose. The trend showing an increasing effect of the magnetic field on the observed phase velocity difference with the decrease of magnetic field strength was demonstrated in Fig. 2.

• The presence of heating dependence on the magnetic field has no noticeable effect on the behaviour of fast MA wave velocity. Indeed, the calculated dependences of the phase velocity of fast MA waves on the wavenumber almost completely coincide with the case of the absence of thermal misbalance and vary between external |$c_{\mathrm{ A}_e}$| and internal |$c_{\mathrm{ A}_i}$| Alfvén speeds. Thus, even in magnetically heated plasma, the geometry dispersion remains primary for fast MA waves.

• According to the results obtained, heating function dependence on the magnetic field can influence the attenuation/amplification of MA waves. In particular, it can either suppress or increase the damping of slow waves (see Fig. 3). Assuming the coronal heating to be a power-law function of magnetic field strength (equation 28), we show that the increase of power index c leads to the growth of the slow MA wave damping. However, as the magnetic field increases, this effect becomes weaker (compare the results shown in the left and right columns in Fig. 3).

• Special attention should be given to the demonstrated possibility of fast MA wave amplification in the structured plasma. The peculiarity in this case is that the slow waves remain damped (see Fig. 4). The effect is visible for both sausage and kink modes. It was shown in Fig. 3 that the fast MA wave decay period decreases with the increase of power index c. By continuing to increase the power index c, we achieved not only compensation of attenuation but also the mentioned amplification of fast MA waves. Our calculations show that amplification becomes more pronounced with the decrease in magnetic field strength. From our point of view, the amplification of fast MA waves accompanied by damped slow MA waves is of interest in the context of observed decayless kink oscillation (Anfinogentov, Nakariakov & Nisticò 2015; Zhong et al. 2022a, b; Li & Long 2023). In fact, the feature found suggests an alternative cause for the excitation (Ruderman & Petrukhin 2021; Mandal et al. 2022) of the above-mentioned oscillations due to the feedback between the coronal plasma and the wave, without any input of energy from lower atmospheric layers. One of the arguments in favour of such an approach is the observation of 30-min decayless kink oscillation (Zhong et al. 2023), providing important evidence for their non-resonant origin. In fact, the possibility of fast MA wave amplification in the case of damped slow MA waves requires additional analysis, which we will conduct in our future work.

• Last but not least are the results following from the conducted comparison of phase velocity dependences of wavenumber using ‘thin’ flux tube approximation and magnetic slab geometry. Our calculation reveals that the ‘thin’ flux tube approximation describes the wave dispersion quite well for the limit of long wavelengths in the case of a ‘weak’ magnetic field, while, for the ‘strong’ magnetic field, in the opposite, the discrepancy is noticeable for long wavelengths and disappears for the shorter wavelengths.

In summary, it is worth noting that the constructed models expand the knowledge about the influence of magnetic heating on the dynamics of the solar plasma. From our point of view, they can be used both to verify the available heating models (Mandrini et al. 2000; Ugarte-Urra et al. 2019; Fleishman et al. 2021) and as a development of theories for constructing seismological constraints (Kolotkov et al. 2023) on the heating functions.

ACKNOWLEDGEMENTS

The study was supported in part by the Ministry of Education and Science of Russia by State assignment to educational and research institutions under project nos. FSSS-2023-0009 and 0023-2019-0003. CHIANTI is a collaborative project involving George Mason University, the University of Michigan (USA), University of Cambridge (UK), and NASA Goddard Space Flight Center (USA).

DATA AVAILABILITY

The data underlying this article will be shared at the reasonable request of the corresponding author.