# Stability of Slow Magnetoacoustic and Entropy Waves in the Solar Coronal Plasma with Thermal Misbalance

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## Abstract

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## 1. Introduction

## 2. Governing Equations and Dispersion Relation

## 3. Stabilization by Thermal Conduction

## 4. Effect of Finite Plasma-β

## 5. Discussion and Conclusions

- The field-aligned thermal conduction tends to stabilize both slow and entropy modes, thus effectively broadening the parametric domain in the $(a,b,c)$ space, within which the coronal plasma remains stable. The efficiency of this stabilization by thermal conduction, however, strongly depends on the equilibrium plasma conditions and perturbation wavelength, determined by the thermal and acoustic Field’s lengths given by Equations (11) and (12), respectively. While for slow waves in quiescent loops this effect is found to be rather minor, hot loops are shown to be predominantly stable to slow and entropy modes due to highly effective thermal conduction. In other words, the considered functional form of the heating model and the field-aligned Spitzer conductivity cannot account for the development of slow magnetoacoustic overstability and/or rapid coronal condensations, associated with isentropic and isobaric thermal instabilities in hot active region loops (see, e.g., [24], for the most recent review).
- The stability of the entropy mode is found to be insensitive to the dependence of the coronal heating function on the local magnetic field, i.e., the parameter c, and plasma-$\beta $. In other words, this result suggests that catastrophic cooling and condensations of the coronal plasma, caused by the entropy mode instability, are independent of the magnetic properties of host active regions and are fully driven by the loss of balance between optically thin radiation, field-aligned conduction, and heating, which is consistent with numerical findings of, e.g., Ref. [54]. On the other hand, there could be an indirect link through the local plasma heating by magnetic reconnection (see, e.g., [55]), which is out of the scope of the present study.
- The stability of slow magnetoacoustic waves, by contrast, is found to depend on the product of two magnetic field parameters, $\beta c$. Thus, in finite-$\beta $ plasma, one may expect to probe the functional dependence of the coronal heating function on the magnetic field, i.e., the parameter c, with the theory presented here. This result effectively extends the seismological diagnostics of the coronal heating function developed in Ref. [37]. In particular, it allows us to consider both existing AC and DC theories of coronal heating (see, e.g., Table 5 in Ref. [56]) for comparison and validation. For example, following Rosner et al. [45], Ibanez S. and Escalona T. [47] considered coronal heating mechanisms by electric current dissipation, mode conversion, and anomalous conduction damping of Alfvén waves in a similar power-law form. Together with the diagnostic potential of microwave observations, see [57], such a seismological technique offers a unique opportunity to constrain the link between the coronal magnetic field and thermodynamic properties of the corona, which is not directly available in extreme ultraviolet or soft X-ray observations traditionally used for coronal heating studies.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Thermal (red) and acoustic (blue) Field’s lengths ${\lambda}_{\mathrm{F}}^{\mathrm{thermal}}$ (11) and ${\lambda}_{\mathrm{F}}^{\mathrm{acoustic}}$ (12) in Mm, evaluated for (

**left**) ${\rho}_{0}=3\times {10}^{-12}$ kg m${}^{-3}$ and ${T}_{0}=1$ MK (typical for quiescent coronal loops hosting propagating slow waves), and (

**right**) ${\rho}_{0}=6\times {10}^{-12}$ kg/m${}^{3}$ and ${T}_{0}=6$ MK (typical for hot active region loops hosting standing slow magnetoacoustic oscillations, also known as “SUMER” oscillations). In the green-shaded regions, both ${\lambda}_{\mathrm{F}}^{\mathrm{thermal}}$ and ${\lambda}_{\mathrm{F}}^{\mathrm{acoustic}}$ tend to infinity, i.e., the effect of the field-aligned thermal conduction on the wave stability vanishes.

**Figure 2.**Values of the power-indices $(a,b,c)$ in the parametrization of the coronal heating function, $\mathcal{H}(\rho ,T,B)\propto {\rho}^{a}{T}^{b}{B}^{c}$, below which the coronal plasma is stable to slow (gradient of blue) and entropy (gradient of red) wave modes. The stability conditions are obtained from the numerical solution of dispersion relation (8) for plasma parameters typical of quiescent coronal loops with low (

**top**) and finite (

**bottom**) values of the plasma parameter $\beta $, hosting propagating slow waves: ${\rho}_{0}=3\times {10}^{-12}$ kg/m${}^{3}$; ${T}_{0}=1$ MK providing the adiabatic sound speed, ${c}_{\mathrm{s}}\approx 152\times \sqrt{{T}_{0}\left[\mathrm{MK}\right]}\approx 152$ km/s; ${B}_{0}=20$ G (

**top**) and ${B}_{0}=4$ G (

**bottom**) providing plasma-$\beta \approx 0.01$ and $\beta \approx 0.3$, respectively; $\lambda =70$ Mm providing the adiabatic slow wave period, $\lambda /{c}_{\mathrm{s}}\approx 7$ min. The red and blue dots (

**top left**) illustrate the grid size used in the numerical solution here. The contour labels (

**bottom left**and

**right**) indicate values of the power indices c and b, respectively. The white lines (

**right**) indicate the intersection of the $abc$-planes determined by the slow and entropy wave stability conditions.

**Figure 3.**Same as Figure 2 but for SUMER-type loops hosting standing slow oscillations with low (

**top**) and finite (

**bottom**) values of the plasma parameter $\beta $: ${\rho}_{0}=6\times {10}^{-12}$ kg/m${}^{3}$; ${T}_{0}=6$ MK providing the adiabatic sound speed ${C}_{\mathrm{s}}\approx 152\times \sqrt{{T}_{0}\left[\mathrm{MK}\right]}\approx 372$ km/s; ${B}_{0}=80$ G (

**top**) and ${B}_{0}=15$ G (

**bottom**) providing plasma-$\beta \approx 0.01$ and $\beta \approx 0.3$, respectively; $\lambda =500$ Mm providing the adiabatic slow wave period $\lambda /{C}_{\mathrm{s}}\approx 22$ min.

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**MDPI and ACS Style**

Kolotkov, D.Y.; Nakariakov, V.M.; Fihosy, J.B.
Stability of Slow Magnetoacoustic and Entropy Waves in the Solar Coronal Plasma with Thermal Misbalance. *Physics* **2023**, *5*, 193-204.
https://doi.org/10.3390/physics5010015

**AMA Style**

Kolotkov DY, Nakariakov VM, Fihosy JB.
Stability of Slow Magnetoacoustic and Entropy Waves in the Solar Coronal Plasma with Thermal Misbalance. *Physics*. 2023; 5(1):193-204.
https://doi.org/10.3390/physics5010015

**Chicago/Turabian Style**

Kolotkov, Dmitrii Y., Valery M. Nakariakov, and Joseph B. Fihosy.
2023. "Stability of Slow Magnetoacoustic and Entropy Waves in the Solar Coronal Plasma with Thermal Misbalance" *Physics* 5, no. 1: 193-204.
https://doi.org/10.3390/physics5010015