# Damping and Dispersion of Non-Adiabatic Acoustic Waves in a High-Temperature Plasma: A Radiative-Loss Function

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method and Basic Equations

#### 2.1. Heating/Cooling Function

#### 2.2. Interpolation of a Radiative-Loss Function

#### 2.3. Basic Equations

## 3. Results

#### 3.1. Wave Instability

#### 3.2. Wave Damping

#### 3.3. Wave Dispersion

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Srivastava, A.K.; Kuridze, D.; Zaqarashvili, T.V.; Dwivedi, B.N. Intensity oscillations observed with Hinode near the south pole of the Sun: Leakage of low frequency magneto-acoustic waves into the solar corona. Astron. Astrophys.
**2008**, 481, L95–L98. [Google Scholar] [CrossRef] - De Moortel, I. Longitudinal waves in coronal loops. Space Sci. Rev.
**2009**, 149, 65–81. [Google Scholar] [CrossRef] - Banerjee, D.; Gupta, G.R.; Teriaca, L. Propagating MHD waves in coronal holes. Space Sci. Rev.
**2011**, 158, 267–288. [Google Scholar] [CrossRef] - Banerjee, D.; Krishna Prasad, S. MHD waves in coronal holes. In Low-Frequency Waves in Space Plasmas; Keiling, A., Lee, D.-H., Nakariakov, V.M., Eds.; John Wiley & Sons, Inc.: New York, NY, USA, 2016; pp. 419–430. [Google Scholar] [CrossRef] [Green Version]
- Krishna Prasad, S.; Van Doorsselaere, T. Compressive oscillations in hot coronal loops: Are sloshing oscillations and standing slow waves independent? Astrophys. J.
**2021**, 914, 81. [Google Scholar] [CrossRef] - Wang, T.J. Standing slow-mode waves in hot coronal loops: Observations, modeling, and coronal seismology. Space Sci. Rev.
**2011**, 158, 397–419. [Google Scholar] [CrossRef] [Green Version] - De Moortel, I.; Nakariakov, V.M. Magnetohydrodynamic waves and coronal seismology: An overview of recent results. Phil. Trans. Roy. Soc. A
**2012**, 370, 3193–3216. [Google Scholar] [CrossRef] [Green Version] - Ofman, L.; Wang, T. Hot coronal loop oscillations observed by SUMER: Slow magnetosonic wave damping by thermal conduction. Astrophys. J.
**2002**, 580, L85–L88. [Google Scholar] [CrossRef] - De Moortel, I.; Hood, A.W. The damping of slow MHD waves in solar coronal magnetic fields. Astron. Astrophys.
**2003**, 408, 755–765. [Google Scholar] [CrossRef] - Priest, E.R.; Foley, C.R.; Heyvaerts, J.; Arber, T.D.; Culhane, J.L.; Acton, L.W. Nature of the heating mechanism for the diffuse solar corona. Nature
**1998**, 393, 545–547. [Google Scholar] [CrossRef] - Aschwanden, M.J.; Terradas, J. The effect of radiative cooling on coronal loop oscillations. Astrophys. J.
**2008**, 686, L127–L130. [Google Scholar] [CrossRef] - Mikhalyaev, B.B.; Veselovskii, I.S.; Khongorova, O.V. Radiation effects on the MHD wave behavior in the solar corona. Sol. Syst. Res.
**2013**, 47, 50–57. [Google Scholar] [CrossRef] - Kolotkov, D.Y.; Nakariakov, V.M.; Zavershinskii, D.I. Damping of slow magnetoacoustic oscillations by the misbalance between heating and cooling processes in the solar corona. Astron. Astrophys.
**2019**, 628, A133. [Google Scholar] [CrossRef] [Green Version] - Zavershinskii, D.I.; Kolotkov, D.Y.; Nakariakov, V.M.; Molevich, N.E.; Ryashchikov, D.S. Formation of quasi-periodic slow magnetoacoustic wave trains by the heating/cooling misbalance. Phys. Plasmas
**2019**, 26, 082113. [Google Scholar] [CrossRef] [Green Version] - Belov, S.A.; Molevich, N.E.; Zavershinskii, D.L. Dispersion of slow magnetoacoustic waves in the active region fan loops introduced by thermal misbalance. Sol. Phys.
**2021**, 296, 122. [Google Scholar] [CrossRef] - Ginzburg, V.L. Concerning the general relationship between absorption and dispersion of sound waves. Sov. Phys. - Acoustics
**1955**, 1, 32–41. [Google Scholar] - Dere, K.P.; Landi, E.; Young, P.R.; Del Zanna, G.; Landini, M.; Mason, H.E. CHIANTI—An atomic database for emission lines IX. Ionization rates, recombination rates, ionization equilibria for the elements hydrogen through zinc and updated atomic data. Astron. Astrophys.
**2009**, 498, 915–929. [Google Scholar] [CrossRef] - Dudík, J.; Dzifčáková, E.; Karlický, M.; Kulinová, A. The bound-bound and free-free radiative losses for the nonthermal distributions in solar and stellar coronae. Astron. Astrophys.
**2011**, 529, A103. [Google Scholar] [CrossRef] - Del Zanna, G.; Dere, K.P.; Young, P.R.; Landi, E. CHIANTI—An atomic database for emission lines. XVI. Version 10, further extensions. Astrophys. J.
**2021**, 909, 38. [Google Scholar] [CrossRef] - Field, G.B. Thermal instability. Astrophys. J.
**1965**, 142, 531–567. [Google Scholar] [CrossRef] - Priest, E.R. Magnetohydrodynamics of the Sun; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar] [CrossRef]
- Dere, K.P.; Landi, E.; Mason, H.E.; Monsignori Fossi, B.C.; Young, P.R. CHIANTI—An atomic database for emission lines. I. Wavelengths greater than 50 Å. Astron. Astrophys. Suppl. Ser.
**1997**, 125, 149–173. [Google Scholar] [CrossRef] [Green Version] - Landi, E.; Landini, M. Radiative losses of optically thin coronal plasmas. Astron. Astrophys.
**1999**, 347, 401–408. Available online: https://ui.adsabs.harvard.edu/abs/1999A%26A...347..401L/ (accessed on 25 January 2023). - Rosner, R.; Tucker, W.H.; Vaiana, G.S. Dynamics of the quiescent solar corona. Astrophys. J.
**1978**, 220, 643–665. [Google Scholar] [CrossRef] - Carbonell, M.; Oliver, R.; Ballester, J.L. Time damping of linear non-adiabatic magnetohydrodynamic waves in an unbounded plasma with solar coronal properties. Astron. Astrophys.
**2004**, 45, 739–750. [Google Scholar] [CrossRef] [Green Version] - Klimchuk, J.A.; Patsourakos, S.; Cargill, P.J. Highly efficient modeling of dynamic coronal loops. Astrophys. J.
**2008**, 682, 1351–1362. [Google Scholar] [CrossRef] [Green Version] - Landi, E.; Del Zanna, G.; Young, P.R.; Dere, K.P.; Mason, H.E.; Landini, M. CHIANTI—An atomic database for emission lines. VII. New data for X-rays and other improvements. Astrophys. J. Supp. Ser.
**2006**, 162, 261–280. [Google Scholar] [CrossRef] [Green Version] - Parker, E.N. Instability of thermal fields. Astrophys. J.
**1953**, 117, 431–436. [Google Scholar] [CrossRef] - Weymann, R. Heating of stellar chromospheres by shock waves. Astrophys. J.
**1960**, 132, 452–460. [Google Scholar] [CrossRef] - Kolotkov, D.Y.; Duckenfield, T.J.; Nakariakov, V.M. Seismological constraints on the solar coronal heating function. Astron. Astrophys.
**2020**, 644, A33. [Google Scholar] [CrossRef] - Kolotkov, D.Y.; Zavershinskii, D.I.; Nakariakov, V.M. The solar corona as an active medium for magnetoacoustic waves. Plasma Phys. Control. Fusion
**2021**, 63, 124008. [Google Scholar] [CrossRef] - CHIANTI: An Atomic Database for Spectroscopic Diagnostics of Astrophysical Plasmas. Available online: http://www.chiantidatabase.org (accessed on 25 January 2023).
- Reale, F. Coronal loops: Observations and modeling of onfined plasma. Living Rev. Sol. Phys.
**2014**, 11, 4. [Google Scholar] [CrossRef] - Spitzer, L., Jr. Physics of Fully Ionized Gases; Dover Publications, Inc.: Mineola, NY, USA, 2006. [Google Scholar]
- Nakariakov, V.M.; Afanasyev, A.N.; Kumar, S.; Moon, Y.-J. Effect of local thermal equilibrium misbalance on long-wavelength slow magnetoacoustic waves. Astrophys. J.
**2017**, 849, 62. [Google Scholar] [CrossRef]

**Figure 3.**The frequency real part (

**left**), the phase speed, ${V}_{\mathrm{ph}}=\omega /k$ (

**middle**) and the damping coefficient, $\delta =-\mathrm{Im}\tilde{\omega}$ (

**right**) for ${T}_{0}={10}^{6}\phantom{\rule{4pt}{0ex}}\mathrm{K}$, ${n}_{0}=1\xb7{10}^{15}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{-3}$. See text for more details.

**Figure 4.**The frequency real part (

**left**), the phase speed, ${V}_{\mathrm{ph}}=\omega /k$ (

**middle**) and the damping coefficient, $\delta =-\mathrm{Im}\tilde{\omega}$ (

**right**) for ${T}_{0}={10}^{6}\phantom{\rule{4pt}{0ex}}\mathrm{K}$, ${n}_{0}=5\xb7{10}^{15}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{-3}$. See text for more details.

**Figure 5.**Localized pulse dispersion due to thermal conductivity and heating/cooling (

**left**), thermal conductivity only (

**middle**), and heating/cooling only (

**right**) for ${T}_{0}={10}^{6}\phantom{\rule{4pt}{0ex}}\mathrm{K}$, ${n}_{0}=1\xb7{10}^{15}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{-3}$.

**Figure 6.**Localized pulse dispersion due to thermal conductivity and heating/cooling (

**left**), thermal conductivity only (

**middle**), and heating/cooling only (

**right**) for ${T}_{0}={10}^{6}\phantom{\rule{4pt}{0ex}}\mathrm{K}$, ${n}_{0}=5\xb7{10}^{15}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{-3}$.

**Table 1.**The values of the radiative-loss function ${\Lambda}_{i}$, as a function of temperature, ${T}_{i}$. $\rho $ and n denote the plasma and particle densities, respectively.

i | ${\mathit{T}}_{\mathit{i}}$ (K), $\times {10}^{6}$ | ${\mathit{\rho}}^{2}{\mathit{\Lambda}}_{\mathit{i}}/{\mathit{n}}^{2}\phantom{\rule{4pt}{0ex}}(\mathrm{erg}\xb7{\mathrm{cm}}^{3}\xb7{s}^{-1})$, $\times {10}^{-22}$ | ${\mathit{\Lambda}}_{\mathit{i}}(\mathrm{erg}\xb7{g}^{-2}\xb7{\mathrm{cm}}^{3}\xb7{s}^{-1})$, $\times {10}^{26}$ |
---|---|---|---|

0 | 0.5011872 | 2.267829 | 2.108884 |

1 | 0.562341 | 2.367434 | 2.201509 |

2 | 0.630957 | 2.418156 | 2.248675 |

3 | 0.707946 | 2.469151 | 2.296096 |

4 | 0.794328 | 2.547926 | 2.369351 |

5 | 0.891251 | 2.622882 | 2.439053 |

6 | 1 | 2.646271 | 2.460803 |

7 | 1.122018 | 2.602311 | 2.419923 |

8 | 1.258925 | 2.523537 | 2.346671 |

9 | 1.412538 | 2.421656 | 2.25193 |

10 | 1.584893 | 2.266581 | 2.107724 |

11 | 1.778279 | 2.019601 | 1.878054 |

12 | 1.995262 | 1.665344 | 1.548626 |

13 | 2.238721 | 1.269071 | 1.180126 |

14 | 2.511886 | 0.946541 | 0.880201 |

15 | 2.818383 | 0.729857 | 0.678704 |

16 | 3.162278 | 0.59583 | 0.55407 |

17 | 3.548134 | 0.52004 | 0.483592 |

18 | 3.981072 | 0.483017 | 0.449164 |

19 | 4.466836 | 0.471106 | 0.438087 |

20 | 5.011872 | 0.474839 | 0.441559 |

21 | 5.623413 | 0.487475 | 0.45331 |

22 | 6.309573 | 0.504156 | 0.468822 |

23 | 7.079458 | 0.520485 | 0.484006 |

24 | 7.943282 | 0.531125 | 0.493901 |

25 | 8.912509 | 0.530389 | 0.493216 |

26 | 10 | 0.513176 | 0.47721 |

27 | 11.220185 | 0.476465 | 0.443071 |

28 | 12.589254 | 0.421238 | 0.391715 |

29 | 14.125375 | 0.357961 | 0.332873 |

**Table 2.**Coefficients of the dimensionless cubic interpolation (7).

i | Ai | Bi | Ci | Di |
---|---|---|---|---|

0 | −8.164325 | −6.655046 | 1.952125 | 2.108884 |

1 | 42.533874 | −8.152892 | 1.046559 | 2.201509 |

2 | 6.927944 | 0.602624 | 0.52849 | 2.248675 |

3 | −11.622274 | 2.202738 | 0.74447 | 2.296096 |

4 | −7.161141 | -0.809143 | 0.864853 | 2.36935 |

5 | 0.696996 | −2.891375 | 0.506189 | 2.439053 |

6 | 5.9093052 | −2.6639816 | −0.097951 | 2.4608027 |

7 | 0.940888 | −0.500849 | −0.484119 | 2.419923 |

8 | −1.306459 | −0.114406 | −0.568352 | 2.3466709 |

9 | −0.579185 | −0.71647 | −0.695985 | 2.25193 |

10 | 0.091538 | −1.015948 | −0.994577 | 2.107724 |

11 | 1.443157 | −0.962841 | −1.377247 | 1.878054 |

12 | 1.4062024 | −0.02342 | −1.591249 | 1.548626 |

13 | −0.261502 | 1.003637 | −1.352606 | 1.180126 |

14 | −0.388805 | 0.789338 | −0.862828 | 0.880201 |

15 | −0.189246 | 0.431835 | −0.488543 | 0.678704 |

16 | −0.102562 | 0.236593 | −0.258674 | 0.55407 |

17 | −0.046161 | 0.117871 | −0.121902 | 0.483592 |

18 | −0.021779 | 0.057916 | −0.045797 | 0.449164 |

19 | −0.009937 | 0.026178 | −0.004947 | 0.438087 |

20 | −0.004251 | 0.009929 | 0.0147325 | 0.044156 |

21 | −0.002043 | 0.002131 | 0.0221072 | 0.453309 |

22 | −0.001395 | −0.20738 | 0.022146 | 0.468822 |

23 | −0.000595 | −0.005296 | 0.0164728 | 0.484006 |

24 | −0.000074 | −0.006839 | 0.005991 | 0.4939 |

25 | 0.000361 | −0.007054 | −0.007474 | 0.493216 |

26 | 0.00049 | −0.005878 | −0.021536 | 0.47721 |

27 | 0.000944 | −0.004084 | −0.033691 | 0.443071 |

28 | 0.000667 | −0.000021 | −0.039563 | 0.391715 |

**Table 3.**The values of the dimensionless thermal conductivity, $\tilde{\varkappa}$, the radiative-loss function, $\tilde{\Lambda}$, and the its derivative, ${\tilde{\Lambda}}^{\prime}$. $\gamma =5/3$ is the adiabatic index.

i | ${\tilde{\mathit{T}}}_{\mathit{i}}$ | $\tilde{\mathit{\varkappa}}\left({\tilde{\mathit{T}}}_{\mathit{i}}\right)$ | ${\tilde{\mathit{\Lambda}}}^{\prime}\left({\tilde{\mathit{T}}}_{\mathit{i}}\right)$ | $\tilde{\mathit{\Lambda}}\left({\tilde{\mathit{T}}}_{\mathit{i}}\right)/{\tilde{\mathit{T}}}_{\mathit{i}}$ | $(\mathit{\gamma}-1){\tilde{\mathit{\Lambda}}}^{\prime}\left({\tilde{\mathit{T}}}_{\mathit{i}}\right)+\tilde{\mathit{\Lambda}}\left({\tilde{\mathit{T}}}_{\mathit{i}}\right)/{\tilde{\mathit{T}}}_{\mathit{i}}$ |
---|---|---|---|---|---|

0 | 0.501187 | 0.405448 | 1.952125 | 4.207778 | 5.509194 |

1 | 0.562341 | 0.540673 | 1.04656 | 3.9149 | 4.6126 |

2 | 0.63096 | 0.721 | 0.52849 | 3.5639 | 3.916237 |

3 | 0.707946 | 0.961468 | 0.744471 | 3.243322 | 3.739636 |

4 | 0.794328 | 1.282138 | 0.864853 | 2.982836 | 3.559404 |

5 | 0.891251 | 1.709759 | 0.506189 | 2.736663 | 3.074122 |

6 | 1 | 2.28 | −0.097951 | 2.460803 | 2.395502 |

7 | 1.122019 | 3.040429 | −0.484119 | 2.156759 | 1.834013 |

8 | 1.258925 | 4.054477 | −0.568352 | 1.864027 | 1.485126 |

9 | 1.412538 | 5.406732 | −0.695985 | 1.594244 | 1.130255 |

10 | 1.584893 | 7.209993 | −0.994577 | 1.329884 | 0.666833 |

11 | 1.778279 | 9.61468 | −1.377247 | 1.056107 | 0.137943 |

12 | 1.995262 | 12.821382 | −1.591249 | 0.776152 | −0.284681 |

13 | 2.238721 | 17.097588 | −1.352606 | 0.527143 | −0.374595 |

14 | 2.5118864 | 22.8 | −0.862828 | 0.350415 | −0.224804 |

15 | 2.818383 | 30.404289 | −0.488543 | 0.240813 | −0.084882 |

16 | 3.162278 | 40.544771 | −0.258674 | 0.175212 | 0.002763 |

17 | 3.548134 | 54.06732 | −0.121902 | 0.136295 | 0.055027 |

18 | 3.981072 | 72.099931 | −0.045797 | 0.112825 | 0.008229 |

19 | 4.466836 | 96.146803 | −0.004947 | 0.098076 | 0.094778 |

20 | 5.011872 | 128.21382 | 0.014733 | 0.088103 | 0.097924 |

21 | 5.623413 | 170.97588 | 0.022107 | 0.080611 | 0.095349 |

22 | 6.309573 | 228 | 0.022146 | 0.074303 | 0.089068 |

23 | 7.079458 | 304.04289 | 0.016473 | 0.068368 | 0.07935 |

24 | 7.943282 | 405.44771 | 0.005991 | 0.062178 | 0.066172 |

25 | 8.912509 | 540.6732 | −0.007474 | 0.05534 | 0.050357 |

26 | 10 | 720.9993 | −0.021536 | 0.047721 | 0.033363 |

27 | 11.220185 | 961.46803 | −0.033691 | 0.039489 | 0.017028 |

28 | 12.589254 | 1 282.1382 | −0.039563 | 0.031115 | 0.00474 |

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**MDPI and ACS Style**

Derteev, S.; Shividov, N.; Bembitov, D.; Mikhalyaev, B.
Damping and Dispersion of Non-Adiabatic Acoustic Waves in a High-Temperature Plasma: A Radiative-Loss Function. *Physics* **2023**, *5*, 215-228.
https://doi.org/10.3390/physics5010017

**AMA Style**

Derteev S, Shividov N, Bembitov D, Mikhalyaev B.
Damping and Dispersion of Non-Adiabatic Acoustic Waves in a High-Temperature Plasma: A Radiative-Loss Function. *Physics*. 2023; 5(1):215-228.
https://doi.org/10.3390/physics5010017

**Chicago/Turabian Style**

Derteev, Sergei, Nikolai Shividov, Dzhirgal Bembitov, and Badma Mikhalyaev.
2023. "Damping and Dispersion of Non-Adiabatic Acoustic Waves in a High-Temperature Plasma: A Radiative-Loss Function" *Physics* 5, no. 1: 215-228.
https://doi.org/10.3390/physics5010017