# p-Mode Oscillations in Highly Gravitationally Stratified Magnetic Solar Atmospheres

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## Abstract

**:**

## 1. Introduction

## 2. Motivation

## 3. Numerical Computation Methods

## 4. Structures in the Solar Atmosphere

## 5. Computational Model

## 6. Numerical Drivers for p-Mode Oscillations

## 7. Global Magnetoacoustic Waves in Uniform Vertical Magnetic Field Configurations

## 8. Frequency Analysis

## 9. Discussion

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AIA | Atmospheric Imaging Assembly |

DKIST | the Daniel K. Inouye Solar Telescope |

FFT | fast Fourier transform |

GPU | graphical processing unit |

MHD | magnetohydrodynamics |

SDO | Solar Dynamics Observatory |

SMAUG | Sheffield magnetohydrodynamics accelerated using GPUs |

VAC | Versatile Advection Code |

VALIIIc | Vernazza, Avrett and Loeser |

1D, 2D, 3D | one-, two-, three-dimensional |

## References

- Erdélyi, R. Magnetic coupling of waves and oscillations in the lower solar atmosphere: Can the tail wag the dog? Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2005**, 364, 351–381. [Google Scholar] [CrossRef] - Taroyan, Y.; Erdélyi, R. Global acoustic resonance in a stratified solar atmosphere. Sol. Phys.
**2008**, 251, 523–531. [Google Scholar] [CrossRef] - Pintér, B.; Erdélyi, R.; Goossens, M. Global oscillations in a magnetic solar model. Astron. Astrophys.
**2007**, 466, 377–388. [Google Scholar] [CrossRef] - Mein, N.; Mein, P. Velocity waves in the quiet solar chromosphere. Sol. Phys.
**1976**, 49, 231–248. [Google Scholar] [CrossRef] - Schmieder, B.; Mein, N. Mechanical flux in the solar chromosphere. II. Determination of the mechanical flux. Astron. Astrophys.
**1980**, 84, 99–105. Available online: https://ui.adsabs.harvard.edu/abs/1980A%26A....84...99S/ (accessed on 18 March 2023). - 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.
**2010**, 158, 267–288. [Google Scholar] [CrossRef] - Mathioudakis, M.; Jess, D.; Erdélyi, R. Alfvén waves in the solar atmosphere. Space Sci. Rev.
**2012**, 175, 1–27. [Google Scholar] [CrossRef] - Ruderman, M.S.; Erdélyi, R. Transverse oscillations of coronal loops. Space Sci. Rev.
**2009**, 149, 199–228. [Google Scholar] [CrossRef] - Wang, T. Standing slow-mode waves in hot coronal loops: Observations, modeling, and coronal seismology. Space Sci. Rev.
**2011**, 158, 397–419. [Google Scholar] [CrossRef] - Auchère, F.; Bocchialini, K.; Solomon, J.; Tison, E. Long-period intensity pulsations in the solar corona during activity cycle 23. Astron. Astrophys.
**2014**, 563, A8. [Google Scholar] [CrossRef] - Jensen, E.; Orrall, F.Q. Observational study of macroscopic inhomogeneities in the solar atmosphere. IV. Velocity and intensity fluctuations observed in the K line. Astrophys. J.
**1963**, 138, 252–270. [Google Scholar] [CrossRef] - Banerjee, D.; Erdélyi, R.; Oliver, R.; O’Shea, E. Present and future observing trends in atmospheric magnetoseismology. Sol. Phys.
**2007**, 246, 3–29. [Google Scholar] [CrossRef] - Erdélyi, R.; Taroyan, Y. Hinode EUV spectroscopic observations of coronal oscillations. Astron. Astrophys.
**2008**, 489, L49–L52. [Google Scholar] [CrossRef] - Roberts, B.; Edwin, P.M.; Benz, A. On coronal oscillations. Astrophys. J.
**1984**, 279, 857–865. [Google Scholar] [CrossRef] - Verth, G.; Erdélyi, R.; Goossens, M. Magnetoseismology: Eigenmodes of torsional Alfvén waves in stratified solar waveguides. Astrophys. J.
**2010**, 714, 1637–1648. [Google Scholar] [CrossRef] - Zaqarashvili, T.V.; Murawski, K. Torsional oscillations of longitudinally inhomogeneous coronal loops. Astron. Astrophys.
**2007**, 470, 353–357. [Google Scholar] [CrossRef] - Gyenge, N.; Erdélyi, R. Periodic recurrence patterns in X-Ray solar flare appearances. Astrophys. J.
**2018**, 859, 169. [Google Scholar] [CrossRef] - Gudiksen, B.V.; Carlsson, M.; Hansteen, V.H.; Hayek, W.; Leenaarts, J.; Martínez-Sykora, J. The stellar atmosphere simulation codeBifrost. Astron. Astrophys.
**2011**, 531, A154. [Google Scholar] [CrossRef] - Vögler, A.; Shelyag, S.; Schüssler, M.; Cattaneo, F.; Emonet, T.; Linde, T. Simulations of magneto-convection in the solar photosphere. Equations, methods, and results of the MURaM code. Astron. Astrophys.
**2005**, 429, 335–351. [Google Scholar] [CrossRef] - Zhang, F.; Poedts, S.; Lani, A.; Kuźma, B.; Murawski, K. Two-fluid modeling of acoustic wave propagation in gravitationally stratified isothermal media. Astrophys. J.
**2021**, 911, 119. [Google Scholar] [CrossRef] - Fedun, V.; Erdélyi, R.; Shelyag, S. Oscillatory response of the 3D solar atmosphere to the leakage of photospheric motion. Sol. Phys.
**2009**, 258, 219–241. [Google Scholar] [CrossRef] - Fedun, V.; Shelyag, S.; Verth, G.; Mathioudakis, M.; Erdélyi, R. MHD waves generated by high-frequency photospheric vortex motions. Ann. Geophys.
**2011**, 29, 1029–1035. [Google Scholar] [CrossRef] - Vigeesh, G.; Fedun, V.; Hasan, S.S.; Erdélyi, R. Three-dimensional simulations of magnetohydrodynamic waves in magnetized solar atmosphere. Astrophys. J.
**2012**, 755, A18. [Google Scholar] [CrossRef] - Griffiths, M.; Fedun, V.; Erdélyi, R.; Zheng, R. Solar atmosphere wave dynamics generated by solar global oscillating eigenmodes. Adv. Space Res.
**2018**, 61, 720–737. [Google Scholar] [CrossRef] - Santamaria, I.C.; Khomenko, E.; Collados, M. Magnetohydrodynamic wave propagation from the subphotosphere to the corona in an arcade-shaped magnetic field with a null point. Astron. Astrophys.
**2015**, 577, A70. [Google Scholar] [CrossRef] - Khomenko, E.; Calvo Santamaria, I. Magnetohydrodynamic waves driven by p-modes. J. Phys. Conf. Ser.
**2013**, 440, 012048. [Google Scholar] [CrossRef] - Shelyag, S.; Fedun, V.; Erdélyi, R. Magnetohydrodynamic code for gravitationally-stratified media. Astron. Astrophys.
**2008**, 486, 655–662. [Google Scholar] [CrossRef] - Griffiths, M.K.; Fedun, V.; Erdélyi, R. A Fast MHD code for gravitationally stratified media using graphical processing units: SMAUG. J. Astrophys. Astron.
**2015**, 36, 197–223. [Google Scholar] [CrossRef] - Tóth, G. A General code for modeling MHD flows on parallel computers: Versatile Advection Code. Astrophys. Lett. Commun.
**1996**, 34, 245–250. Available online: https://ui.adsabs.harvard.edu/abs/1996ApL%26C..34..245T (accessed on 18 March 2023). - Caunt, S.E.; Korpi, M.J. A 3D MHD model of astrophysical flows: Algorithms, tests and parallelisation. Astron. Astrophys.
**2001**, 369, 706–728. [Google Scholar] [CrossRef] - Simon, G.W.; Weiss, N.O. On the magnetic field in pores. Sol. Phys.
**1970**, 13, 85–103. [Google Scholar] [CrossRef] - Cameron, R.; Schüssler, M.; Vögler, A.; Zakharov, V. Radiative magnetohydrodynamic simulations of solar pores. Astron. Astrophys.
**2007**, 474, 261–272. [Google Scholar] [CrossRef] - Vernazza, J.E.; Avrett, E.H.; Loeser, R. Structure of the solar chromosphere. III. Models of the EUV brightness components of the quiet-sun. Astrophys. J. Suppl. Ser.
**1981**, 45, 635–725. [Google Scholar] [CrossRef] - McWhirter, R.W.P.; Thonemann, P.C.; Wilson, R. The heating of the solar corona. II. A model based on energy balance. Astron. Astrophys.
**1975**, 40, 63–73. Available online: https://ui.adsabs.harvard.edu/abs/1975A%26A....40...63M/ (accessed on 18 March 2023). - Murawski, K.; Zaqarashvili, T.V. Numerical simulations of spicule formation in the solar atmosphere. Astron. Astrophys.
**2010**, 519, A8. [Google Scholar] [CrossRef] - Kalkofen, W. The validity of dynamical models of the solar atmosphere. Sol. Phys.
**2011**, 276, 75–95. [Google Scholar] [CrossRef] - Carlsson, M.; Stein, R.F. Does a nonmagnetic solar chromosphere exist? Astrophys. J. Lett.
**1995**, 440, L29–L32. [Google Scholar] [CrossRef] - Leenaarts, J.; Carlsson, M.; Hansteen, V.; Gudiksen, B.V. On the minimum temperature of the quiet solar chromosphere. Astron. Astrophys.
**2011**, 530, A124. [Google Scholar] [CrossRef] - Gent, F.A.; Fedun, V.; Mumford, S.J.; Erdélyi, R. Magnetohydrostatic equilibrium—I. Three-dimensional open magnetic flux tube in the stratified solar atmosphere. Mon. Not. R. Astron. Soc.
**2013**, 435, 689–697. [Google Scholar] [CrossRef] - Schüssler, M.; Rempel, M. The dynamical disconnection of sunspots from their magnetic roots. Astron. Astrophys.
**2005**, 441, 337–346. [Google Scholar] [CrossRef] - Griffiths, M.; von Fay-Siebenburgen, R.; Erdélyi. Videos of p-Mode Oscillations in Highly Gravitationally Stratified Magnetic Solar Atmospheres. The University of Sheffield. 2018. Available online: https://figshare.shef.ac.uk/articles/dataset/Videos_of_p-Mode_Oscillations_in_Highly_Gravitationally_Stratified_Magnetic_Solar_Atmospheres/7378046 (accessed on 18 March 2023).
- Bogdan, T.J.; Hansteen, M.C.V.; McMurry, A.; Rosenthal, C.S.; Johnson, M.; Petty, S.; Zita, E.J.; Stein, R.F.; McIntosh, S.W.; Nordlund, P. Waves in the magnetized solar atmosphere. II. Waves from localized sources in magnetic flux concentrations. Astrophys. J.
**2003**, 599, 626–660. [Google Scholar] [CrossRef] - Pesnell, W.D.; Thompson, B.J.; Chamberlin, P.C. The Solar Dynamics Observatory (SDO). Sol. Phys.
**2012**, 275, 3–15. [Google Scholar] [CrossRef] - Lemen, J.R.; Title, A.M.; Akin, D.J.; Boerner, P.F.; Chou, C.; Drake, J.F.; Duncan, D.W.; Edwards, C.G.; Friedlaender, F.M.; Heyman, G.F.; et al. The Atmospheric Imaging Assembly (AIA) on the Solar Dynamics Observatory (SDO). Sol. Phys.
**2012**, 275, 17–40. [Google Scholar] [CrossRef] - Roberts, B. Wave propagation in a magnetically structured atmosphere. I: Surface waves at a magnetic interface. Sol. Phys.
**1981**, 169, 27–38. [Google Scholar] [CrossRef] - Roberts, B. Wave propagation in a magnetically structured atmosphere. II: Waves in a magnetic slab. Sol. Phys.
**1981**, 69, 39–56. [Google Scholar] [CrossRef] - Edwin, P.M.; Roberts, B. Wave propagation in a magnetic cylinder. Sol. Phys.
**1983**, 88, 179–191. [Google Scholar] [CrossRef] - Hindman, B.; Zweibel, E.G.; Cally, P. Driven acoustic oscillations within a vertical magnetic field. Astrophys. J.
**1996**, 459, 760–772. [Google Scholar] [CrossRef] - Hasan, S.S.; Christensen-Dalsgaard, J. The influence of a vertical magnetic field on oscillations in an isothermal stratified atmosphere. Astrophys. J.
**1992**, 396, 311–332. [Google Scholar] [CrossRef] - Kontogiannis, I.; Tsiropoula, G.; Tziotziou, K. Power halo and magnetic shadow in a solar quiet region observed in the Hα line. Astron. Astrophys.
**2010**, 510, A41. [Google Scholar] [CrossRef] - Malins, C. On transition region convection cells in simulations of p-mode propagation. Astron. Notes/Astron. Nachr.
**2007**, 328, 752–755. [Google Scholar] [CrossRef] - Didkovsky, L.; Kosovichev, A.; Judge, D.; Wieman, S.; Woods, T. Variability of solar five-minute oscillations in the corona as observed by the Extreme Ultraviolet Spectrophotometer (ESP) on the Solar Dynamics Observatory/Extreme Ultraviolet Variability Experiment (SDO/EVE). Sol. Phys.
**2012**, 287, 171–184. [Google Scholar] [CrossRef] - Ireland, J.; McAteer, R.T.J.; Inglis, A. Coronal fourier power spectra: Implications for coronal seismology and coronal heating. Astrophys. J.
**2015**, 798, 1. [Google Scholar] [CrossRef] - Campbell, W.R.; Roberts, B. The influence of a chromospheric magnetic field on the solar p- and f-modes. Astrophys. J.
**1989**, 338, 538–556. [Google Scholar] [CrossRef] - Kostogryz, N.M.; Fournier, D.; Gizon, L. Modelling continuum intensity perturbations caused by solar acoustic oscillations. Astron. Astrophys.
**2021**, 654, A1. [Google Scholar] [CrossRef] - Christensen-Dalsgaard, J. ADIPLS—The Aarhus adiabatic oscillation package. Astrophys. Space Sci.
**2008**, 316, 113–120. [Google Scholar] [CrossRef] - Rast, M.; Martinez Pillet, V. Resolving the Source of the Solar Acoustic Oscillations: What Will Be Possible with DKIST? AAS/Solar Physics Division Abstracts #47. 2016. Available online: https://ui.adsabs.harvard.edu/abs/2016SPD....4720105R%2F (accessed on 18 March 2023).

**Figure 2.**Atmospheric profiles for the density and temperature for the model solar atmosphere based on the VALIIIc model [34].

**Figure 3.**Profiles of the computed speed of sound and the frequency cut-off for a solar model atmosphere based on the VALIIIc model [34].

**Figure 4.**The initial magnetic field configuration showing the radial field distribution that is cylindrical and uniform in the vertical direction with a maximum value of 100 G. The color bar shows the magnetic field strength in Gauss.

**Figure 5.**The vertical component of the velocity for different sections of the simulation after different times 76 s, 150 s and 225 s for a vertical field with maximum field of 100 G.

**Figure 6.**The vertical component of the velocity for different sections of the simulation after different times 76 s, 150 s and 225 s for a magnetic field of 0 G.

**Figure 7.**Time–distance plot of the vertical component of the velocity in the mid-chromosphere, for the magnetic field of 100 G. The central red line is the line for plasma-$\beta $ equal to 1.

**Figure 8.**Time–distance plot of the vertical component of the velocity in the mid-chromosphere, for a magnetic field of 0 G.

**Figure 9.**The ratio of the integrated energy flux ratio for different values of the field: 0 G (blue), 50 G (orange), 75 G (purple), and 100 G (red).

**Figure 10.**Temporal analysis of ${B}_{z}$ vertical slices at 2 Mm: (

**top**) the selected vertical slices, indicated by gray colours; (

**middle**) the obtained signal after applying a Hanning window function; and (

**bottom**) the result of the fast Fourier transform (FFT) analysis based on the 5 selected vertical ${B}_{z}$ slices.

**Figure 11.**Temporal analysis of ${V}_{z}$ vertical slices at 2 Mm: (

**top**) the selected vertical slices, indicated by gray colours; (

**middle**) the obtained signal after applying a Hanning window function; and (

**bottom**) the result of the FFT analysis based on the 5 selected vertical ${V}_{z}$ slices.

**Figure 12.**Temporal analysis of the intensity of a 50-pixel large area based on the Atmospheric Imaging Assembly 1600 Å, 1700 Å, 304 Å, 171 Å and 193 Å data between 18:00 UT to 20:00 UT on 22 August 2010 [45]. The variation of the Z-score (de-trended by first difference and normalised pixel intensity data) and the FFT of the analysed observational data are shown as indicated. The dashed line indicates the significance level of 3 standard deviation calculated using a Monte Carlo method.

**Table 1.**The speeds obtained from the time–distance plots for the 300 s period driver with magnetic fields of 0 G, 50 G, 75 G and 100 G.

Wave Speed (km/s) | 0 G | 50 G | 75 G | 100 G |
---|---|---|---|---|

2 Mm | 12.6 | 96.5 | 47.7 | 25.2 |

1 Mm | 10.1 | 64.1 | 44.4 | 45.4 |

0.5 Mm | 8.7 | 45.4 | 37.8 | 32.3 |

**Table 2.**The time-averaged and integrated energy flux ratio obtained for the 300 s period driver with magnetic fields of 0 G, 50 G, 75 G and 100 G.

Magnetic Field (G) | 1 Mm | 2 Mm | 4 Mm | 5.5 Mm |
---|---|---|---|---|

0 G | 0.155 | −1.771 ×${10}^{-5}$ | 1.227 × ${10}^{-6}$ | 8.194 × ${10}^{-7}$ |

50 G | 0.270 | −0.399 | 0.040 | 0.021 |

75 G | −0.507 | −0.126 | 0.015 | 0.007 |

100 G | −0.255 | −0.226 | 0.019 | 0.006 |

**Table 3.**The table shows observed frequencies for the quiet Sun between 18:00 UT to 20:00 UT on 22 August 2010 [45], the frequencies have been identified from the temporal analysis of a 50-pixel large area based on different AIA bands. Significant peaks around 5 min (0.0033 Hz) appear in bold type.

1700 Å | 1600 Å | 304 Å | 171 Å | 193 Å |
---|---|---|---|---|

0.00332 | 0.00311 | 0.00269 | 0.00141 | 0.00168 |

0.00391 | 0.00352 | 0.00382 | 0.00207 | 0.00227 |

0.00423 | 0.00432 | 0.00451 | 0.00238 | 0.00318 |

0.00470 | 0.00488 | 0.00480 | 0.00375 | 0.00369 |

0.00512 | 0.00557 | 0.00619 | 0.00456 | - |

0.00643 | 0.00599 | 0.00742 | 0.00762 | - |

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**MDPI and ACS Style**

Griffiths, M.; Gyenge, N.; Zheng, R.; Korsós, M.; Erdélyi, R.
*p*-Mode Oscillations in Highly Gravitationally Stratified Magnetic Solar Atmospheres. *Physics* **2023**, *5*, 461-482.
https://doi.org/10.3390/physics5020032

**AMA Style**

Griffiths M, Gyenge N, Zheng R, Korsós M, Erdélyi R.
*p*-Mode Oscillations in Highly Gravitationally Stratified Magnetic Solar Atmospheres. *Physics*. 2023; 5(2):461-482.
https://doi.org/10.3390/physics5020032

**Chicago/Turabian Style**

Griffiths, Michael, Norbert Gyenge, Ruisheng Zheng, Marianna Korsós, and Robertus Erdélyi.
2023. "*p*-Mode Oscillations in Highly Gravitationally Stratified Magnetic Solar Atmospheres" *Physics* 5, no. 2: 461-482.
https://doi.org/10.3390/physics5020032