Abstract
Characterized by cyclic axisymmetric perturbations to both the magnetic and fluid parameters, magnetohydrodynamic fast sausage modes (FSMs) have proven useful for solar coronal seismology given their strong dispersion. This review starts by summarizing the dispersive properties of the FSMs in the canonical configuration where the equilibrium quantities are transversely structured in a step fashion. With this preparation we then review the recent theoretical studies on coronal FSMs, showing that the canonical dispersion features have been better understood physically, and further exploited seismologically. In addition, we show that departures from the canonical equilibrium configuration have led to qualitatively different dispersion features, thereby substantially broadening the range of observations that FSMs can be invoked to account for. We also summarize the advances in forward modeling studies, emphasizing the intricacies in interpreting observed oscillatory signals in terms of FSMs. All these advances notwithstanding, we offer a list of aspects that remain to be better addressed, with the physical connection of coronal FSMs to the quasi-periodic pulsations in solar flares particularly noteworthy.
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Notes
The ambient is taken to be always static in this review.
Kink modes are addressed only when necessary in this review. For more details, see the companion reviews by Nakariakov et al. (in preparation, this issue) and Van Doorsselaere et al. (2020, this issue).
In the ER83 setup, the equilibrium quantities are transversely structured in a piece-wise constant manner, meaning that the subscript \(\mathrm{i}\) (\(\mathrm{e}\)) applies to the entire interior (exterior). We choose to introduce the subscripts this way such that we can avoid re-introducing them when discussing the equilibria with continuous transverse structuring.
The first several zeros are \(j_{0, 1} = 2.4048\), \(j_{0, 2} = 5.5201\), \(j_{1, 1} = 3.8317\), and \(j_{1, 2} = 7.0156\).
As detailed in Sect. 2.2, much can be learned for the dispersive properties of FSMs by examining the slab counterpart of the ER83 equilibrium. In the slab case, the analytical study by Karamimehr et al. (2019) showed that leaky FSMs of different transverse orders (\(l\)) behave differently with decreasing \(k\). Relative to its transverse harmonics (\(l \ge 2\)), the transverse fundamental (\(l = 1\)) is characterized by a faster decrease in \(\omega _{\mathrm{R}}\) accompanied by a more rapid increase of \(\omega _{\mathrm{I}}\).
In the slab case, it is possible to analytically establish that the damping time experiences its shortest value at the long wave-length limit when the density ratio of the internal and external medium is close to unity, while experiencing its highest value just below the cut-off value when the density ratio possesses its highest value (Karamimehr et al. 2019).
These two papers found that the damping time \(\tau \) (or equivalently \(\omega _{\mathrm{I}}\)) at \(kR \to 0\) does not depend on the transverse order. In addition, \(\tau /P\) is proportional to \(\sqrt{\rho _{\mathrm{i}}/ \rho _{\mathrm{e}}}\) rather than \(\rho _{\mathrm{i}}/\rho _{\mathrm{e}}\) for sufficiently large density contrasts. These differences from the cylindrical case are not regarded as “primary”, though.
This is not to say that Fig. 3 contradicts Figure 3a in Roberts et al. (1984), which sketches the three-phase scenario for impulsively generated wavetrains. It is just that there are certain requirements for the sketch to apply, with the need to sample a wavetrain at a distance sufficiently far from the exciter already implied in Equation (7). On top of that, some further time-dependent simulations have suggested that the duration of the exciter needs to be \(\lesssim R/v_{\mathrm{Ai}}\) (Goddard et al. 2019), and the spatial extent of the exciter needs to be comparable to the waveguide radius (Yu et al. 2017). We take these requirements as encouraging rather than discouraging for one to further examine impulsive wavetrains, because the deviations of the signatures of observed wavetrains from the sketch actually encode a rich set of seismic information on, say, the exciters. For more details, see Sect. 5.2.
Crazy tadpoles were found in nonlinear sausage wavetrains in density-enhanced slabs as well (Pascoe et al. 2017c). Nonlinear waves, however, are beyond the scope of this review.
If the width of the current sheet (CS) is taken to vanish, then the slab results directly apply (e.g. Feng et al. 2011, appendix). If the CS width is taken to be finite, then the Alfvén speed vanishes somewhere in the CS, and the characteristic timescales of FSMs tend to be comparable to the transverse sound time evaluated with the internal sound speed (Edwin et al. 1986, Fig. 2). Impulsive wavetrains, however, remain qualitatively the same in the sense of MHD (e.g., Jelínek and Karlický 2012; Mészárosová et al. 2014), which is understandable because Features 4 and 5 persist therein. We note that the Morlet spectra of the modulated radio fluxes further depend on the radiation mechanism that bridges the MHD results and measurables. The spectra may be similar across a substantial range of radio frequencies, as in the first identification of tadpoles in decimetric type IV bursts (Mészárosová et al. 2009b). Their morphology may be frequency-dependent at lower radio frequencies, resulting in the “drifting tadpoles” named by Mészárosová et al. (2009a).
Implied here is that the magnetic trap forms first, and FSMs are excited in this preset host by some continuation of the energy release. The energy release is nonetheless “impulsive” or thrust-like, by which Zaitsev and Stepanov (1982) meant that its duration is shorter than the periods of the FSMs. Intuitively speaking, the expanded segment may shrink again, provided that the magnetic field therein is not that weak. FSMs may therefore develop around some new quasi-equilibrium, in conjunction with the response of the segment to the “thrust” imparted by the energy release. Thrust-excited FSMs were numerically explored by Pascoe et al. (2009) for preset expanded loops that are in force balance with the environment. However, to our knowledge, one has yet to quantitatively examine the excitation of FSMs during the restoration processes of expanded loops that are not in force balance with their ambient. At this point, we note also that the subsecond-period oscillations recently detected in decimetric radio bursts by Yu and Chen (2019) actually agree with the interpretation of thrust-excited FSMs in post-reconnection loops (see Fig. 11 therein). However, different from the Zaitsev and Stepanov (1982) scenario is that the modulation to the radio flux was attributed to the trapping and/or acceleration of energetic electrons within the time-varying wavepackets rather than the modulation of the mirror ratio of the local trap by the time-varying FSMs.
Recall that the period \(P\) is of the order of the transverse Alfvén time (\(R/v_{\mathrm{Ai}}\), Equation (5)). Adopting the radiative loss function given by Rosner et al. (1978), one finds that the associated damping timescale \(\tau _{\mathrm{rad}}\) is given by \(3600 T_{6}/n_{9}\) sec when \(0.56< T_{6} <2\) and \(2200 T_{6}^{5/3}/n_{9}\) sec when \(2 < T_{6} <10\), where \(T_{6}\) (\(n_{9}\)) is the loop temperature (electron density) in \(10^{6}\) K (\(10^{9}\) cm−3). Typically \(\tau _{\mathrm{rad}} \gg R/v_{\mathrm{Ai}}\).
In this context, we note that impulsively generated wavetrains have also been modeled for curved slabs by Nisticò et al. (2014) with applications to AR loops in mind. If we extend the idea to non-straight equilibria rather than specifically curved loops, then magnetic funnels were modeled by Pascoe et al. (2013b) and coronal holes by Pascoe et al. (2014). These studies demonstrate that leaky components can form quasi-periodic wavetrains in the external medium. Unlike the simple straight slab geometry, the propagation of these wavetrains is no longer necessarily perpendicular to the waveguide. For example, the refraction due to the nonuniform external medium can allow the wavetrain to propagate in the direction of the structure, despite that the wavetrain itself is outside the structure and not structure-guided.
The fluctuating plasma velocity was not considered because it does not affect the gyrosynchrotron emission.
For trapped FSMs, Fig. 1 indicates that when \(\rho _{\mathrm{i}}/\rho _{\mathrm{e}}\) is given, \(v_{\mathrm{ph}}/v_{\mathrm{Ai}}\) decreases with increasing \(kR\). This is not to be confused with that \(v_{\mathrm{ph}}/v_{\mathrm{Ai}}\) for a relatively small \(kR\) shows little dependence on \(\rho _{\mathrm{i}}/\rho _{\mathrm{e}}\). When evaluating \(v_{\mathrm{ph}}= 2L/P\), KMR12 somehow adopted twice the measured period. KMR12 further supposed that \(v_{\mathrm{ph}}\) is close to the external Alfvén speed \(v_{\mathrm{Ae}}\), and employed the smallness of \(v_{\mathrm{Ae}}\) to argue that it is difficult to reconcile the measured period with an FSM in an ER83 equilibrium. We note that, the insensitivity of \(v_{\mathrm{ph}}/v_{\mathrm{Ai}}\) to \(\rho _{\mathrm{i}}/\rho _{\mathrm{e}}\) for a relatively small \(kR\) means that \(v_{\mathrm{ph}}/v_{\mathrm{Ae}}\) is approximately \(\propto \sqrt{\rho _{\mathrm{e}}/ \rho _{\mathrm{i}}}\), and hence \(v_{\mathrm{Ae}}\) can be much larger than the measured \(v_{\mathrm{ph}}\) for large \(\rho _{\mathrm{i}}/\rho _{\mathrm{e}}\). Despite that, their argument remains valid.
Practical seismological diagnostics using kink oscillations therefore includes using additional observables such as the shape of the damping profile (Hood et al. 2013; Pascoe et al. 2013a, 2016, 2019), independent sources of information such as the EUV intensity (Pascoe et al. 2017b, 2018), and Bayesian analysis (Arregui et al. 2013b; Pascoe et al. 2017a,d) including MCMC sampling (Anfinogentov et al. 2020) which permits constraints to be calculated without the need for a unique solution.
Without imaging information, this study assumed that \(L/R \gg 1\), and derived \(R/v_{\mathrm{Ai}}\) rather than \(v_{\mathrm{Ai}}\).
References
M. Allcock, R. Erdélyi, Magnetohydrodynamic waves in an asymmetric magnetic slab. Sol. Phys. 292(2), 35 (2017). https://doi.org/10.1007/s11207-017-1054-y
J. Andries, M. Goossens, On the continuous spectrum of leaky magnetohydrodynamic modes and the associated quasimodes. Phys. Plasmas 14(5), 052101 (2007). https://doi.org/10.1063/1.2714513
J. Andries, I. Arregui, M. Goossens, Determination of the coronal density stratification from the observation of harmonic coronal loop oscillations. Astrophys. J. 624(1), L57–L60 (2005). https://doi.org/10.1086/430347
J. Andries, T. Van Doorsselaere, B. Roberts, G. Verth, E. Verwichte, R. Erdélyi, Coronal seismology by means of kink oscillation overtones. Space Sci. Rev. 149(1–4), 3–29 (2009). https://doi.org/10.1007/s11214-009-9561-2
S.A. Anfinogentov, V.M. Nakariakov, D.J. Pascoe, C.R. Goddard, Solar Bayesian analysis toolkit – a new Markov chain Monte Carlo IDL code for Bayesian parameter inference. Arxiv e-prints (2020). arXiv:2005.05365
P. Antolin, Thermal instability and non-equilibrium in solar coronal loops: from coronal rain to long-period intensity pulsations. Comments Plasma Phys. Control. Fusion 62(1), 014016 (2020). https://doi.org/10.1088/1361-6587/ab5406
P. Antolin, L. Rouppe van der Voort, Observing the fine structure of loops through high-resolution spectroscopic observations of coronal rain with the CRISP instrument at the Swedish solar telescope. Astrophys. J. 745(2), 152 (2012). https://doi.org/10.1088/0004-637X/745/2/152. arXiv:1112.0656
P. Antolin, T. Van Doorsselaere, Line-of-sight geometrical and instrumental resolution effects on intensity perturbations by sausage modes. Astron. Astrophys. 555, A74 (2013). https://doi.org/10.1051/0004-6361/201220784. arXiv:1303.6147
P. Antolin, P. Pagano, I. De Moortel, V.M. Nakariakov, I. Situ, Generation of transverse magnetohydrodynamic waves from colliding flows in the solar corona. J. Astrophys. 861(2), L15 (2018). https://doi.org/10.3847/2041-8213/aacf98. arXiv:1807.00395
T.D. Arber, A.W. Longbottom, C.L. Gerrard, A.M. Milne, A staggered grid, Lagrangian-Eulerian remap code for 3-D MHD simulations. J. Comput. Phys. 171(1), 151–181 (2001). https://doi.org/10.1006/jcph.2001.6780
I. Arregui, Bayesian coronal seismology. Adv. Space Res. 61(2), 655–672 (2018). https://doi.org/10.1016/j.asr.2017.09.031. arXiv:1709.08372
I. Arregui, M. Goossens, No unique solution to the seismological problem of standing kink magnetohydrodynamic waves. Astron. Astrophys. 622, A44 (2019). https://doi.org/10.1051/0004-6361/201833813. arXiv:1812.07266
I. Arregui, A. Asensio Ramos, A.J. Díaz, Bayesian analysis of multiple harmonic oscillations in the solar corona. Astrophys. J. 765(1), L23 (2013a). https://doi.org/10.1088/2041-8205/765/1/L23. arXiv:1302.3393
I. Arregui, A. Asensio Ramos, D.J. Pascoe, Determination of transverse density structuring from propagating magnetohydrodynamic waves in the solar atmosphere. Astrophys. J. 769(2), L34 (2013b).https://doi.org/10.1088/2041-8205/769/2/L34. arXiv:1304.6869
M.J. Aschwanden, Physics of the Solar Corona. An Introduction (2004)
M.J. Aschwanden, H. Peter, The width distribution of loops and strands in the solar corona—are we hitting rock bottom? Astrophys. J. 840(1), 4 (2017). https://doi.org/10.3847/1538-4357/aa6b01. arXiv:1701.01177
M.J. Aschwanden, V.M. Nakariakov, V.F. Melnikov, Magnetohydrodynamic sausage-mode oscillations in coronal loops. Astrophys. J. 600(1), 458–463 (2004). https://doi.org/10.1086/379789. arXiv:astro-ph/0309493
A.O. Benz, G. Mann, Intermediate drift bursts and the coronal magnetic field. Astron. Astrophys. 333, 1034–1042 (1998)
D. Berghmans, P. de Bruyne, M. Goossens, The footpoint-driven coronal sausage wave. Astrophys. J. 472, 398 (1996). https://doi.org/10.1086/178072
T.E.X. Bernold, R.A. Treumann, The fiber fine structure during solar type IV radio bursts - observations and theory of radiation in presence of localized whistler turbulence. Astrophys. J. 264, 677–688 (1983). https://doi.org/10.1086/160641
P.S. Cally, Leaky and non-leaky oscillations in magnetic flux tubes. Sol. Phys. 103(2), 277–298 (1986). https://doi.org/10.1007/BF00147830
P.S. Cally, M. Xiong, Sensitivity of coronal loop sausage mode frequencies and decay rates to radial and longitudinal density inhomogeneities: a spectral approach. J. Phys. A, Math. Gen. 51(2), 025501 (2018). https://doi.org/10.1088/1751-8121/aa9755. arXiv:1711.00256
P.J. Cargill, Coronal magnetism: difficulties and prospects. Space Sci. Rev. 144(1–4), 413–421 (2009). https://doi.org/10.1007/s11214-008-9446-9
S.X. Chen, B. Li, L.D. Xia, Y.J. Chen, H. Yu, Effects of field-aligned flows on standing kink and sausage modes supported by coronal loops. Sol. Phys. 289(5), 1663–1681 (2014). https://doi.org/10.1007/s11207-013-0420-7. arXiv:1309.3484
S.X. Chen, B. Li, L.D. Xia, H. Yu, Periods and damping rates of fast sausage oscillations in multishelled coronal loops. Sol. Phys. 290(8), 2231–2243 (2015a). https://doi.org/10.1007/s11207-015-0751-7. arXiv:1507.02169
S.X. Chen, B. Li, M. Xiong, H. Yu, M.Z. Guo, Standing sausage modes in nonuniform magnetic tubes: an inversion scheme for inferring flare loop parameters. Astrophys. J. 812(1), 22 (2015b).https://doi.org/10.1088/0004-637X/812/1/22. arXiv:1509.01442
S.X. Chen, B. Li, M. Xiong, H. Yu, M.Z. Guo, Fast sausage modes in magnetic tubes with continuous transverse profiles: effects of a finite plasma beta. Astrophys. J. 833(1), 114 (2016). https://doi.org/10.3847/1538-4357/833/1/114. arXiv:1610.03254
S.X. Chen, B. Li, S. Kumar, H. Yu, M. Shi, Fast standing modes in transversley nonuniform solar coronal slabs: the effects of a finite plasma beta. Astrophys. J. 855(1), 47 (2018a). https://doi.org/10.3847/1538-4357/aaab5d. arXiv:1801.09204
X. Chen, E.P. Kontar, S. Yu, Y. Yan, J. Huang, B. Tan, Fine structures of solar radio type III bursts and their possible relationship with coronal density turbulence. Astrophys. J. 856(1), 73 (2018b). https://doi.org/10.3847/1538-4357/aaa9bf. arXiv:1801.07545
I. De Moortel, S.J. Bradshaw, Forward modelling of coronal intensity perturbations. Sol. Phys. 252(1), 101–119 (2008).https://doi.org/10.1007/s11207-008-9238-0
B. De Pontieu, M. Carlsson, L.H.M. Rouppe van der Voort, R.J. Rutten, V.H. Hansteen, H. Watanabe, Ubiquitous torsional motions in type II spicules. Astrophys. J. 752(1), L12 (2012). https://doi.org/10.1088/2041-8205/752/1/L12. arXiv:1205.5006
G. Del Zanna, H.E. Mason, Solar UV and X-ray spectral diagnostics. Living Rev. Sol. Phys. 15(1), 5 (2018). https://doi.org/10.1007/s41116-018-0015-3. arXiv:1809.01618
A.J. Díaz, Fast magnetohydrodynamic oscillations in an elliptical coronal arcade. Astron. Astrophys. 456(2), 737–746 (2006). https://doi.org/10.1051/0004-6361:20065184
A.J. Díaz, T. Zaqarashvili, B. Roberts, Fast magnetohydrodynamic oscillations in a force-free line-tied coronal arcade. Astron. Astrophys. 455(2), 709–717 (2006). https://doi.org/10.1051/0004-6361:20054430
A.J. Díaz, G.R. Donnelly, B. Roberts, Effect of coronal structure on loop oscillations: exponential profiles. Astron. Astrophys. 476(1), 359–368 (2007). https://doi.org/10.1051/0004-6361:20078385
G.R. Donnelly, A.J. Díaz, B. Roberts, Effect of longitudinal density structure on a straight magnetic field modelling coronal arcade oscillations. Astron. Astrophys. 471(3), 999–1009 (2007). https://doi.org/10.1051/0004-6361:20066094
I. Dorotovič, R. Erdélyi, V. Karlovský, Identification of linear slow sausage waves in magnetic pores, in Waves & Oscillations in the Solar Atmosphere: Heating and Magneto-Seismology, IAU Symposium, vol. 247, ed. by R. Erdélyi, C.A. Mendoza-Briceno (2008), pp. 351–354. https://doi.org/10.1017/S174392130801507X
I. Dorotovič, R. Erdélyi, N. Freij, V. Karlovský, I. Márquez, Standing sausage waves in photospheric magnetic waveguides. Astron. Astrophys. 563, A12 (2014). https://doi.org/10.1051/0004-6361/201220542. arXiv:1210.6476
P.M. Edwin, B. Roberts, Wave propagation in a magnetically structured atmosphere - Part three - the slab in a magnetic environment. Sol. Phys. 76(2), 239–259 (1982). https://doi.org/10.1007/BF00170986
P.M. Edwin, B. Roberts, Wave propagation in a magnetic cylinder. Sol. Phys. 88(1–2), 179–191 (1983). https://doi.org/10.1007/BF00196186
P.M. Edwin, B. Roberts, Impulsively generated fast coronal pulsations, in NASA Conference Publication, NASA Conference Publication, vol. 2449 (1986), pp. 347–357
P.M. Edwin, B. Roberts, W.J. Hughes, Dispersive ducting of MHD waves in the plasma sheet: a source of Pi2 wave bursts. Geophys. Res. Lett. 13(4), 373–376 (1986). https://doi.org/10.1029/GL013i004p00373
R. Erdélyi, V. Fedun, Linear MHD sausage waves in compressible magnetically twisted flux tubes. Sol. Phys. 246(1), 101–118 (2007). https://doi.org/10.1007/s11207-007-9022-6
R. Erdélyi, R.J. Morton, Magnetohydrodynamic waves in a compressible magnetic flux tube with elliptical cross-section. Astron. Astrophys. 494(1), 295–309 (2009). https://doi.org/10.1051/0004-6361:200810318
X. Fang, D. Yuan, C. Xia, T. Van Doorsselaere, R. Keppens, The role of Kelvin-Helmholtz instability for producing loop-top hard X-ray sources in solar flares. Astrophys. J. 833(1), 36 (2016). https://doi.org/10.3847/1538-4357/833/1/36
S.W. Feng, Y. Chen, B. Li, H.Q. Song, X.L. Kong, L.D. Xia, X.S. Feng, Streamer wave events observed in solar cycle 23. Sol. Phys. 272(1), 119–136 (2011). https://doi.org/10.1007/s11207-011-9814-6. arXiv:1105.4331
G.D. Fleishman, A.A. Kuznetsov, Fast gyrosynchrotron codes. Astrophys. J. 721(2), 1127–1141 (2010). https://doi.org/10.1088/0004-637X/721/2/1127
G.D. Fleishman, T.S. Bastian, D.E. Gary, Broadband quasi-periodic radio and X-ray pulsations in a solar flare. Astrophys. J. 684(2), 1433–1447 (2008). https://doi.org/10.1086/589821. arXiv:0804.4037
L. Fletcher, B.R. Dennis, H.S. Hudson, S. Krucker, K. Phillips, A. Veronig, M. Battaglia, L. Bone, A. Caspi, Q. Chen, P. Gallagher, P.T. Grigis, H. Ji, W. Liu, R.O. Milligan, M. Temmer, An observational overview of solar flares. Space Sci. Rev. 159(1–4), 19–106 (2011). https://doi.org/10.1007/s11214-010-9701-8. arXiv:1109.5932
G.A. Gary, Plasma beta above a solar active region: rethinking the paradigm. Sol. Phys. 203(1), 71–86 (2001). https://doi.org/10.1023/A:1012722021820
I. Giagkiozis, V. Fedun, R. Erdélyi, G. Verth, Axisymmetric modes in magnetic flux tubes with internal and external magnetic twist. Astrophys. J. 810(1), 53 (2015). https://doi.org/10.1088/0004-637X/810/1/53. arXiv:1706.09669
I. Giagkiozis, M. Goossens, G. Verth, V. Fedun, T. Van Doorsselaere, Resonant absorption of axisymmetric modes in twisted magnetic flux tubes. Astrophys. J. 823(2), 71 (2016). https://doi.org/10.3847/0004-637X/823/2/71. arXiv:1706.09665
C.A. Gilchrist-Millar, D.B. Jess, S.D.T. Grant, P.H. Keys, C. Beck, S. Jafarzadeh, R. JM Van Doorselaere T, B. Ruiz Cobo, Magnetoacoustic wave energy dissipation in the atmosphere of solar pores. Arxiv:e-prints (2020). arXiv:2007.11594
C.R. Goddard, G. Nisticò, V.M. Nakariakov, I.V. Zimovets, S.M. White, Observation of quasi-periodic solar radio bursts associated with propagating fast-mode waves. Astron. Astrophys. 594, A96 (2016). https://doi.org/10.1051/0004-6361/201628478. arXiv:1608.04232
C.R. Goddard, V.M. Nakariakov, D.J. Pascoe, Fast magnetoacoustic wave trains with time-dependent drivers. Astron. Astrophys. 624, L4 (2019). https://doi.org/10.1051/0004-6361/201935401
M. Goossens, R. Erdélyi, M.S. Ruderman, Resonant MHD waves in the solar atmosphere. Space Sci. Rev. 158(2–4), 289–338 (2011). https://doi.org/10.1007/s11214-010-9702-7
M. Goossens, T. Van Doorsselaere, R. Soler, G. Verth, Energy content and propagation in transverse solar atmospheric waves. Astrophys. J. 768(2), 191 (2013). https://doi.org/10.1088/0004-637X/768/2/191
S.D.T. Grant, D.B. Jess, M.G. Moreels, R.J. Morton, D.J. Christian, I. Giagkiozis, G. Verth, V. Fedun, P.H. Keys, T. Van Doorsselaere, R. Erdélyi, Wave damping observed in upwardly propagating sausage-mode oscillations contained within a magnetic pore. Astrophys. J. 806(1), 132 (2015). https://doi.org/10.1088/0004-637X/806/1/132. arXiv:1505.01484
M. Gruszecki, K. Murawski, L. Ofman, Standing fast magnetoacoustic kink waves of solar coronal loops with field-aligned flow. Astron. Astrophys. 488(2), 757–761 (2008). https://doi.org/10.1051/0004-6361:200809873
M. Gruszecki, V.M. Nakariakov, T. Van Doorsselaere, Intensity variations associated with fast sausage modes. Astron. Astrophys. 543, A12 (2012). https://doi.org/10.1051/0004-6361/201118168
J.C. Guevara Gómez, S. Jafarzadeh, S. Wedemeyer, M. Szydlarski, M. Stangalini, B. Fleck, P. Keys, High-frequency oscillations in small chromospheric bright features observed with ALMA. Arxiv:e-prints (2020). arXiv:2008.04179
M.Z. Guo, S.X. Chen, B. Li, L.D. Xia, H. Yu, Inferring flare loop parameters with measurements of standing sausage modes. Sol. Phys. 291(3), 877–896 (2016). https://doi.org/10.1007/s11207-016-0868-3. arXiv:1512.03692
B.W. Hindman, R. Jain, Eigenmodes of three-dimensional magnetic arcades in the sun’s corona. Astrophys. J. 814(2), 105 (2015). https://doi.org/10.1088/0004-637X/814/2/105. arXiv:1511.09411
Hinode Review Team, K. Al-Janabi, P. Antolin, D. Baker, L.R. Bellot Rubio, L. Bradley, D.H. Brooks, R. Centeno, J.L. Culhane, G. Del Zanna, G.A. Doschek, L. Fletcher, H. Hara, L.K. Harra, A.S. Hillier, S. Imada, J.A. Klimchuk, J.T. Mariska, T.M.D. Pereira, K.K. Reeves, T. Sakao, T. Sakurai, T. Shimizu, M. Shimojo, D. Shiota, S.K. Solanki, A.C. Sterling, Y. Su, Y. Suematsu, T.D. Tarbell, S.K. Tiwari, S. Toriumi, I. Ugarte-Urra, H.P. Warren, T. Watanabe, P.R. Young, Achievements of Hinode in the first eleven years. Publ. Astron. Soc. Jpn. 71(5), R1 (2019). https://doi.org/10.1093/pasj/psz084
A.W. Hood, M. Ruderman, D.J. Pascoe, I. De Moortel, J. Terradas, A.N. Wright, Damping of kink waves by mode coupling. I. Analytical treatment. Astron. Astrophys. 551, A39 (2013). https://doi.org/10.1051/0004-6361/201220617
C. Hornsey, V.M. Nakariakov, A. Fludra, Sausage oscillations of coronal plasma slabs. Astron. Astrophys. 567, A24 (2014). https://doi.org/10.1051/0004-6361/201423524
A.R. Inglis, V.M. Nakariakov, V.F. Melnikov, Multi-wavelength spatially resolved analysis of quasi-periodic pulsations in a solar flare. Astron. Astrophys. 487(3), 1147–1153 (2008). https://doi.org/10.1051/0004-6361:20079323
A.R. Inglis, T. Van Doorsselaere, C.S. Brady, V.M. Nakariakov, Characteristics of magnetoacoustic sausage modes. Astron. Astrophys. 503(2), 569–575 (2009). https://doi.org/10.1051/0004-6361/200912088. arXiv:1303.6301
P. Jelínek, M. Karlický, Magnetoacoustic waves in diagnostics of the flare current sheets. Astron. Astrophys. 537, A46 (2012). https://doi.org/10.1051/0004-6361/201117883
K. Kaneda, H. Misawa, K. Iwai, S. Masuda, F. Tsuchiya, Y. Katoh, T. Obara, Detection of propagating fast sausage waves through detailed analysis of a zebra-pattern fine structure in a solar radio burst. Astrophys. J. 855(2), L29 (2018). https://doi.org/10.3847/2041-8213/aab2a5
M.R. Karamimehr, S. Vasheghani Farahani, H. Ebadi, Characteristics of magnetoacoustic waves and coronal seismology. Astrophys. J. 886(2), 112 (2019). https://doi.org/10.3847/1538-4357/ab4b4e
M. Karlický, L. Yasnov, Estimating density and magnetic field turbulence in solar flares using radio zebra observations. Astron. Astrophys. 638, A22 (2020). https://doi.org/10.1051/0004-6361/202037936
M. Karlický, H. Mészárosová, P. Jelínek, Radio fiber bursts and fast magnetoacoustic wave trains. Astron. Astrophys. 550, A1 (2013). https://doi.org/10.1051/0004-6361/201220296. arXiv:1212.2421
A.C. Katsiyannis, D.R. Williams, R.T.J. McAteer, P.T. Gallagher, F.P. Keenan, F. Murtagh, Eclipse observations of high-frequency oscillations in active region coronal loops. Astron. Astrophys. 406, 709–714 (2003). https://doi.org/10.1051/0004-6361:20030458. arXiv:astro-ph/0305225
P.H. Keys, R.J. Morton, D.B. Jess, G. Verth, S.D.T. Grant, M. Mathioudakis, D.H. Mackay, J.G. Doyle, D.J. Christian, F.P. Keenan, R. Erdélyi, Photospheric observations of surface and body modes in solar magnetic pores. Astrophys. J. 857(1), 28 (2018). https://doi.org/10.3847/1538-4357/aab432. arXiv:1803.01859
O.V. Khongorova, B.B. Mikhalyaev, M.S. Ruderman, Fast sausage waves in current-carrying coronal loops. Sol. Phys. 280(1), 153–163 (2012). https://doi.org/10.1007/s11207-012-0056-z
D.Y. Kolotkov, V.M. Nakariakov, E.G. Kupriyanova, H. Ratcliffe, K. Shibasaki, Multi-mode quasi-periodic pulsations in a solar flare. Astron. Astrophys. 574, A53 (2015).https://doi.org/10.1051/0004-6361/201424988
D.Y. Kolotkov, V.M. Nakariakov, E.P. Kontar, Origin of the modulation of the radio emission from the solar corona by a fast magnetoacoustic wave. Astrophys. J. 861(1), 33 (2018). https://doi.org/10.3847/1538-4357/aac77e. arXiv:1805.08282
E.P. Kontar, S. Yu, A.A. Kuznetsov, A.G. Emslie, B. Alcock, N.L.S. Jeffrey, V.N. Melnik, N.H. Bian, P. Subramanian, Imaging spectroscopy of solar radio burst fine structures. Nat. Commun. 8, 1515 (2017). https://doi.org/10.1038/s41467-017-01307-8. arXiv:1708.06505
Y.G. Kopylova, A.V. Stepanov, Y.T. Tsap, Radial oscillations of coronal loops and microwave radiation from solar flares. Astron. Lett. 28(11), 783–791 (2002). https://doi.org/10.1134/1.1518717
Y.G. Kopylova, A.V. Melnikov, A.V. Stepanov, Y.T. Tsap, T.B. Goldvarg, Oscillations of coronal loops and second pulsations of solar radio emission. Astron. Lett. 33(10), 706–713 (2007). https://doi.org/10.1134/S1063773707100088
P. Kumar, V.M. Nakariakov, K.S. Cho, Quasi-periodic radio bursts associated with fast-mode waves near a magnetic null point. Astrophys. J. 844(2), 149 (2017). https://doi.org/10.3847/1538-4357/aa7d53. arXiv:1706.09988
E.G. Kupriyanova, V.F. Melnikov, V.M. Nakariakov, K. Shibasaki, Types of microwave quasi-periodic pulsations in single flaring loops. Sol. Phys. 267(2), 329–342 (2010). https://doi.org/10.1007/s11207-010-9642-0
E.G. Kupriyanova, V.F. Melnikov, K. Shibasaki, Spatially resolved microwave observations of multiple periodicities in a flaring loop. Sol. Phys. 284(2), 559–578 (2013). https://doi.org/10.1007/s11207-012-0141-3
E. Kupriyanova, D. Kolotkov, V. Nakariakov, A. Kaufman, Quasi-periodic pulsations in solar and stellar flares. Rev. Sol.-Terr. Phys. 6(1), 3–23 (2020). https://doi.org/10.12737/stp-61202001
A.A. Kuznetsov, Generation of intermediate drift bursts by magnetohydrodynamic waves in the solar corona. Sol. Phys. 237(1), 153–171 (2006). https://doi.org/10.1007/s11207-006-0122-5
A.A. Kuznetsov, Y.T. Tsap, Loss-cone instability and formation of zebra patterns in type IV solar radio bursts. Sol. Phys. 241(1), 127–143 (2007). https://doi.org/10.1007/s11207-006-0351-7
A.A. Kuznetsov, T. Van Doorsselaere, V.E. Reznikova, Simulations of gyrosynchrotron microwave emission from an oscillating 3D magnetic loop. Sol. Phys. 290(4), 1173–1194 (2015). https://doi.org/10.1007/s11207-015-0662-7. arXiv:1502.06716
X. Li, H. Morgan, D. Leonard, L. Jeska, A solar tornado observed by AIA/SDO: rotational flow and evolution of magnetic helicity in a prominence and cavity. Astrophys. J. 752(2), L22 (2012). https://doi.org/10.1088/2041-8205/752/2/L22. arXiv:1205.3819
B. Li, S.R. Habbal, Y. Chen, The period ratio for standing kink and sausage modes in solar structures with siphon flow. I. Magnetized slabs. Astrophys. J. 767(2), 169 (2013). https://doi.org/10.1088/0004-637X/767/2/169. arXiv:1303.1689
B. Li, S.X. Chen, L.D. Xia, H. Yu, Standing sausage modes in coronal loops with plasma flow. Astron. Astrophys. 568, A31 (2014). https://doi.org/10.1051/0004-6361/201323352. arXiv:1406.4688
B. Li, M.Z. Guo, H. Yu, S.X. Chen, Impulsively generated wave trains in coronal structures. II. Effects of transverse structuring on sausage waves in pressurelesss slabs. Astrophys. J. 855(1), 53 (2018). https://doi.org/10.3847/1538-4357/aaaf19. arXiv:1802.04651
D. Lim, V.M. Nakariakov, Y.J. Moon, Sausage oscillations in a plasma cylinder with a surface current. J. Atmos. Sol.-Terr. Phys. 175, 49–55 (2018). https://doi.org/10.1016/j.jastp.2018.04.013
D. Lim, V.M. Nakariakov, D.J. Yu, I.H. Cho, Y.J. Moon, Higher radial harmonics of sausage oscillations in coronal loops. Astrophys. J. 893(1), 62 (2020). https://doi.org/10.3847/1538-4357/ab7d3d
W. Liu, L. Ofman, Advances in observing various coronal EUV waves in the SDO era and their seismological applications (invited review). Sol. Phys. 289(9), 3233–3277 (2014). https://doi.org/10.1007/s11207-014-0528-4. arXiv:1404.0670
W. Liu, A.M. Title, J. Zhao, L. Ofman, C.J. Schrijver, M.J. Aschwanden, B. De Pontieu, T.D. Tarbell, Direct imaging of quasi-periodic fast propagating waves of ∼2000 km s−1 in the low solar corona by the solar dynamics observatory atmospheric imaging assembly. Astrophys. J. 736(1), L13 (2011). https://doi.org/10.1088/2041-8205/736/1/L13. arXiv:1106.3150
J. Liu, C.J. Nelson, B. Snow, Y. Wang, R. Erdélyi, Evidence of ubiquitous Alfvén pulses transporting energy from the photosphere to the upper chromosphere. Nat. Commun. 10, 3504 (2019). https://doi.org/10.1038/s41467-019-11495-0
I. Lopin, I. Nagorny, Fast-sausage oscillations in coronal loops with smooth boundary. Astron. Astrophys. 572, A60 (2014). https://doi.org/10.1051/0004-6361/201424546
I. Lopin, I. Nagorny, Fast waves in smooth coronal slab. Astrophys. J. 801(1), 23 (2015a). https://doi.org/10.1088/0004-637X/801/1/23
I. Lopin, I. Nagorny, Sausage waves in transversely nonuniform monolithic coronal tubes. Astrophys. J. 810(2), 87 (2015b). https://doi.org/10.1088/0004-637X/810/2/87
I. Lopin, I. Nagorny, Dispersion of sausage waves in coronal waveguides with transverse density structuring. Mon. Not. R. Astron. Soc. 488(1), 660–675 (2019a). https://doi.org/10.1093/mnras/stz1737
I. Lopin, I. Nagorny, MHD pulsations of cylindrical coronal loops with tangential magnetic discontinuity. Astrophys. J. 882(2), 134 (2019b). https://doi.org/10.3847/1538-4357/ab32dc
I. Lopin, I. Nagorny, Oscillations of a coronal plasma slab excited by an external source. Mon. Not. R. Astron. Soc. 496(3), 3035–3042 (2020). https://doi.org/10.1093/mnras/staa1654
M.L. Luoni, P. Démoulin, C.H. Mandrini, L. van Driel-Gesztelyi, Twisted flux tube emergence evidenced in longitudinal magnetograms: magnetic tongues. Sol. Phys. 270(1), 45–74 (2011). https://doi.org/10.1007/s11207-011-9731-8
N. Magyar, V.M. Nakariakov, Standing kink waves in sigmoid solar coronal loops: implications for coronal seismology. Astrophys. J. 894(2), L23 (2020). https://doi.org/10.3847/2041-8213/ab8e36. arXiv:2004.14083
J.A. McLaughlin, V.M. Nakariakov, M. Dominique, P. Jelínek, S. Takasao, Modelling quasi-periodic pulsations in solar and stellar flares. Space Sci. Rev. 214(1), 45 (2018). https://doi.org/10.1007/s11214-018-0478-5. arXiv:1802.04180
D.J. McLean, K.V. Sheridan, A damped train of regular metre-wave pulses from the sun. Sol. Phys. 32(2), 485–489 (1973). https://doi.org/10.1007/BF00154961
D.J. McLean, K.V. Sheridan, R.T. Stewart, J.P. Wild, Regular pulses from the sun and a possible clue to the origin of solar cosmic rays. Nature 234(5325), 140–142 (1971). https://doi.org/10.1038/234140a0
B.I. Meerson, P.V. Sasorov, A.V. Stepanov, Pulsations of type IV solar radio emission: the bounce-resonance effects. Sol. Phys. 58(1), 165–179 (1978). https://doi.org/10.1007/BF00152562
V.F. Melnikov, V.E. Reznikova, K. Shibasaki, V.M. Nakariakov, Spatially resolved microwave pulsations of a flare loop. Astron. Astrophys. 439(2), 727–736 (2005). https://doi.org/10.1051/0004-6361:20052774
D.B. Melrose, The emission and absorption of waves by charged particles in magnetized plasmas. Astrophys. Space Sci. 2(2), 171–235 (1968). https://doi.org/10.1007/BF00651567
H. Mészárosová, M. Karlický, J. Rybák, K. Jiřička, “Drifting tadpoles” in wavelet spectra of decimetric radio emission of fiber bursts. Astron. Astrophys. 502(2), L13–L15 (2009a). https://doi.org/10.1051/0004-6361/200912487
H. Mészárosová, M. Karlický, J. Rybák, K. Jiřička, Tadpoles in wavelet spectra of a solar decimetric radio burst. Astrophys. J. 697(2), L108–L110 (2009b). https://doi.org/10.1088/0004-637X/697/2/L108
H. Mészárosová, M. Karlický, J. Rybák, Magnetoacoustic wave trains in the 11 July 2005 radio event with fiber bursts. Sol. Phys. 273(2), 393–402 (2011). https://doi.org/10.1007/s11207-011-9794-6
H. Mészárosová, M. Karlický, P. Jelínek, J. Rybák, Magnetoacoustic waves propagating along a dense slab and Harris current sheet and their wavelet spectra. Astrophys. J. 788(1), 44 (2014). https://doi.org/10.1088/0004-637X/788/1/44
M.G. Moreels, T. Van Doorsselaere S.D.T. Grant, D.B. Jess, M. Goossens, Energy and energy flux in axisymmetric slow and fast waves. Astron. Astrophys. 578, A60 (2015). https://doi.org/10.1051/0004-6361/201425468
R.J. Morton, M.S. Ruderman, Kink and fluting modes of stratified coronal magnetic loops with elliptical cross-sections. Astron. Astrophys. 527, A53 (2011). https://doi.org/10.1051/0004-6361/201016028. arXiv:1011.2377
R.J. Morton, G. Verth, D.B. Jess, D. Kuridze, M.S. Ruderman, M. Mathioudakis, R. Erdélyi, Observations of ubiquitous compressive waves in the Sun’s chromosphere. Nat. Commun. 3, 1315 (2012). https://doi.org/10.1038/ncomms2324. arXiv:1306.4124
G. Mossessian, G.D. Fleishman, Modeling of gyrosynchrotron radio emission pulsations produced by magnetohydrodynamic loop oscillations in solar flares. Astrophys. J. 748(2), 140 (2012). https://doi.org/10.1088/0004-637X/748/2/140. arXiv:1112.0609
V.M. Nakariakov, D.Y. Kolotkov, Magnetohydrodynamic waves in the solar corona. Annu. Rev. Astron. Astrophys. 58(1), 441–481 (2020). https://doi.org/10.1146/annurev-astro-032320-042940
V.M. Nakariakov, V.F. Melnikov, Modulation of gyrosynchrotron emission in solar and stellar flares by slow magnetoacoustic oscillations. Astron. Astrophys. 446(3), 1151–1156 (2006). https://doi.org/10.1051/0004-6361:20053944
V.M. Nakariakov, V.F. Melnikov, Quasi-periodic pulsations in solar flares. Space Sci. Rev. 149(1–4), 119–151 (2009). https://doi.org/10.1007/s11214-009-9536-3
V.M. Nakariakov, B. Roberts, Magnetosonic waves in structured atmospheres with steady flows, I. Sol. Phys. 159(2), 213–228 (1995a). https://doi.org/10.1007/BF00686530
V.M. Nakariakov, B. Roberts, On fast magnetosonic coronal pulsations. Sol. Phys. 159(2), 399–402 (1995b). https://doi.org/10.1007/BF00686541
V.M. Nakariakov, E. Verwichte, Coronal waves and oscillations. Living Rev. Sol. Phys. 2(1), 3 (2005). https://doi.org/10.12942/lrsp-2005-3
V.M. Nakariakov, V.F. Melnikov, V.E. Reznikova, Global sausage modes of coronal loops. Astron. Astrophys. 412, L7–L10 (2003). https://doi.org/10.1051/0004-6361:20031660
V.M. Nakariakov, T.D. Arber, C.E. Ault, A.C. Katsiyannis, D.R. Williams, F.P. Keenan, Time signatures of impulsively generated coronal fast wave trains. Mon. Not. R. Astron. Soc. 349(2), 705–709 (2004). https://doi.org/10.1111/j.1365-2966.2004.07537.x
V.M. Nakariakov, C. Hornsey, V.F. Melnikov, Sausage oscillations of coronal plasma structures. Astrophys. J. 761(2), 134 (2012). https://doi.org/10.1088/0004-637X/761/2/134
G. Nisticò, D.J. Pascoe, V.M. Nakariakov, Observation of a high-quality quasi-periodic rapidly propagating wave train using SDO/AIA. Astron. Astrophys. 569, A12 (2014). https://doi.org/10.1051/0004-6361/201423763
L. Ofman, W. Liu, Quasi-periodic counter-propagating fast magnetosonic wave trains from neighboring flares: SDO/AIA observations and 3D MHD modeling. Astrophys. J. 860(1), 54 (2018). https://doi.org/10.3847/1538-4357/aac2e8. arXiv:1805.00365
L. Ofman, T.J. Wang, Hinode observations of transverse waves with flows in coronal loops. Astron. Astrophys. 482(2), L9–L12 (2008). https://doi.org/10.1051/0004-6361:20079340
L. Ofman, W. Liu, A. Title, M. Aschwanden, Modeling super-fast magnetosonic waves observed by SDO in active region funnels. Astrophys. J. 740(2), L33 (2011). https://doi.org/10.1088/2041-8205/740/2/L33
R. Oliver, M.S. Ruderman, J. Terradas, Propagation and dispersion of sausage wave trains in magnetic flux tubes. Astrophys. J. 806(1), 56 (2015). https://doi.org/10.1088/0004-637X/806/1/56. arXiv:1502.01330
N.R. Owen, I. De Moortel, A.W. Hood, Forward modelling to determine the observational signatures of propagating slow waves for TRACE, SoHO/CDS, and Hinode/EIS. Astron. Astrophys. 494(1), 339–353 (2009). https://doi.org/10.1051/0004-6361:200810828
W. Oxley, N.K. Zsámberger, R. Erdélyi, Standing MHD waves in a magnetic slab embedded in an asymmetric plasma environment: slow surface waves. Astrophys. J. 890(2), 109 (2020). https://doi.org/10.3847/1538-4357/ab67b3
P. Pagano, H.J. Van Damme P. Antolin, I. De Moortel, MHD simulations of the in situ generation of kink and sausage waves in the solar corona by collision of dense plasma clumps. Astron. Astrophys. 626, A53 (2019). https://doi.org/10.1051/0004-6361/201935539. arXiv:1905.03749
J.M. Pasachoff, E.F. Ladd, High-frequency oscillations in the corona observed at the 1983 eclipse. Sol. Phys. 109(2), 365–372 (1987). https://doi.org/10.1007/BF00160659
J.M. Pasachoff, D.A. Landman, High frequency coronal oscillations and coronal heating. Sol. Phys. 90(2), 325–330 (1984). https://doi.org/10.1007/BF00173960
D.J. Pascoe, V.M. Nakariakov, Standing sausage modes in curved coronal slabs. Astron. Astrophys. 593, A52 (2016). https://doi.org/10.1051/0004-6361/201526546
D.J. Pascoe, V.M. Nakariakov, T.D. Arber, Sausage oscillations in multishell coronal structures. Sol. Phys. 246(1), 165–175 (2007). https://doi.org/10.1007/s11207-007-9055-x
D.J. Pascoe, V.M. Nakariakov, T.D. Arber, K. Murawski, Sausage oscillations in loops with a non-uniform cross-section. Astron. Astrophys. 494(3), 1119–1125 (2009). https://doi.org/10.1051/0004-6361:200810541
D.J. Pascoe, A.W. Hood, I. De Moortel, A.N. Wright, Damping of kink waves by mode coupling. II. Parametric study and seismology. Astron. Astrophys. 551, A40 (2013a). https://doi.org/10.1051/0004-6361/201220620
D.J. Pascoe, V.M. Nakariakov, E.G. Kupriyanova, Fast magnetoacoustic wave trains in magnetic funnels of the solar corona. Astron. Astrophys. 560, A97 (2013b). https://doi.org/10.1051/0004-6361/201322678
D.J. Pascoe, V.M. Nakariakov, E.G. Kupriyanova, Fast magnetoacoustic wave trains in coronal holes. Astron. Astrophys. 568, A20 (2014). https://doi.org/10.1051/0004-6361/201423931
D.J. Pascoe, C.R. Goddard, G. Nisticò, S. Anfinogentov, V.M. Nakariakov, Coronal loop seismology using damping of standing kink oscillations by mode coupling. Astron. Astrophys. 589, A136 (2016). https://doi.org/10.1051/0004-6361/201628255
D.J. Pascoe, S. Anfinogentov, G. Nisticò, C.R. Goddard, V.M. Nakariakov, Coronal loop seismology using damping of standing kink oscillations by mode coupling. II. Additional physical effects and Bayesian analysis. Astron. Astrophys. 600, A78 (2017a). https://doi.org/10.1051/0004-6361/201629702
D.J. Pascoe, C.R. Goddard, S. Anfinogentov, V.M. Nakariakov, Coronal loop density profile estimated by forward modelling of EUV intensity. Astron. Astrophys. 600, L7 (2017b). https://doi.org/10.1051/0004-6361/201730458
D.J. Pascoe, C.R. Goddard, V.M. Nakariakov, Dispersive evolution of nonlinear fast magnetoacoustic wave trains. Astrophys. J. 847(2), L21 (2017c). https://doi.org/10.3847/2041-8213/aa8db8
D.J. Pascoe, A.J.B. Russell, S.A. Anfinogentov, P.J.A. Simões, C.R. Goddard, V.M. Nakariakov, L. Fletcher, Seismology of contracting and expanding coronal loops using damping of kink oscillations by mode coupling. Astron. Astrophys. 607, A8 (2017d). https://doi.org/10.1051/0004-6361/201730915
D.J. Pascoe, S.A. Anfinogentov, C.R. Goddard, V.M. Nakariakov, Spatiotemporal analysis of coronal loops using seismology of damped kink oscillations and forward modeling of EUV intensity profiles. Astrophys. J. 860(1), 31 (2018). https://doi.org/10.3847/1538-4357/aac2bc
D.J. Pascoe, A.W. Hood, T. Van Doorsselaere, Coronal loop seismology using standing kink oscillations with a lookup table. Front. Astron. Space Sci. 6, 22 (2019). https://doi.org/10.3389/fspas.2019.00022
R. Ramaty, Gyrosynchrotron emission and absorption in a magnetoactive plasma. Astrophys. J. 158, 753 (1969). https://doi.org/10.1086/150235
V.E. Reznikova, V.F. Melnikov, Y. Su, G. Huang, Pulsations of microwave flaring emission at low and high frequencies. Astron. Rep. 51(7), 588–596 (2007). https://doi.org/10.1134/S1063772907070086
V.E. Reznikova, P. Antolin, T. Van Doorsselaere, Forward modeling of gyrosynchrotron intensity perturbations by sausage modes. Astrophys. J. 785(2), 86 (2014). https://doi.org/10.1088/0004-637X/785/2/86
V.E. Reznikova, T. Van Doorsselaere, A.A. Kuznetsov, Perturbations of gyrosynchrotron emission polarization from solar flares by sausage modes: forward modeling. Astron. Astrophys. 575, A47 (2015). https://doi.org/10.1051/0004-6361/201424548
B. Roberts, Waves and oscillations in the corona. Sol. Phys. 193, 139–152 (2000). https://doi.org/10.1023/A:1005237109398
B. Roberts, Progress in coronal seismology, in Waves & Oscillations in the Solar Atmosphere: Heating and Magneto-Seismology, IAU Symposium, vol. 247, ed. by R. Erdélyi, C.A. Mendoza-Briceno (2008), pp. 3–19. https://doi.org/10.1017/S1743921308014609
B. Roberts, P.M. Edwin, A.O. Benz, Fast pulsations in the solar corona. Nature 305(5936), 688–690 (1983). https://doi.org/10.1038/305688a0
B. Roberts, P.M. Edwin, A.O. Benz, On coronal oscillations. Astrophys. J. 279, 857–865 (1984). https://doi.org/10.1086/161956
H. Rosenberg, Evidence for MHD pulsations in the solar corona. Astron. Astrophys. 9, 159 (1970)
R. Rosner, W.H. Tucker, G.S. Vaiana, Dynamics of the quiescent solar corona. Astrophys. J. 220, 643–645 (1978). https://doi.org/10.1086/155949
W. Ruan, C. Xia, R. Keppens, Extreme-ultraviolet and X-ray emission of turbulent solar flare loops. Astrophys. J. 877(1), L11 (2019). https://doi.org/10.3847/2041-8213/ab1f78
M.S. Ruderman, The resonant damping of oscillations of coronal loops with elliptic cross-sections. Astron. Astrophys. 409, 287–297 (2003). https://doi.org/10.1051/0004-6361:20031079
M.S. Ruderman, On the vertical and horizontal transverse oscillations of curved coronal loops. Astron. Astrophys. 506(2), 885–893 (2009). https://doi.org/10.1051/0004-6361/200912652
T. Samanta, J. Singh, G. Sindhuja, D. Banerjee, Detection of high-frequency oscillations and damping from multi-slit spectroscopic observations of the corona. Sol. Phys. 291(1), 155–174 (2016). https://doi.org/10.1007/s11207-015-0821-x. arXiv:1511.07160
C.J. Schrijver, Braiding-induced interchange reconnection of the magnetic field and the width of solar coronal loops. Astrophys. J. 662(2), L119–L122 (2007). https://doi.org/10.1086/519455
M. Selwa, S.K. Solanki, L. Ofman, The role of active region loop geometry. II. Symmetry breaking in three-dimensional active region: why are vertical kink oscillations observed so rarely? Astrophys. J. 728(2), 87 (2011). https://doi.org/10.1088/0004-637X/728/2/87
I.N. Sharykin, E.P. Kontar, A.A. Kuznetsov, LOFAR observations of fine spectral structure dynamics in type IIIb radio bursts. Sol. Phys. 293(8), 115 (2018). https://doi.org/10.1007/s11207-018-1333-2. arXiv:1806.01046
Y. Shen, Y. Liu, Observational study of the quasi-periodic fast-propagating magnetosonic waves and the associated flare on 2011 May 30. Astrophys. J. 753(1), 53 (2012). https://doi.org/10.1088/0004-637X/753/1/53. arXiv:1204.6649
Y.D. Shen, Y. Liu, J.T. Su, H. Li, X.F. Zhang, Z.J. Tian, R.J. Zhao, A. Elmhamdi, Observations of a quasi-periodic, fast-propagating magnetosonic wave in multiple wavelengths and its interaction with other magnetic structures. Sol. Phys. 288(2), 585–602 (2013). https://doi.org/10.1007/s11207-013-0395-4. arXiv:1307.6099
Y. Shen, P.F. Chen, Y.D. Liu, K. Shibata, Z. Tang, Y. Liu, First unambiguous imaging of large-scale quasi-periodic extreme-ultraviolet wave or shock. Astrophys. J. 873(1), 22 (2019). https://doi.org/10.3847/1538-4357/ab01dd. arXiv:1901.08199
S. Shestov, V.M. Nakariakov, S. Kuzin, Fast magnetoacoustic wave trains of sausage symmetry in cylindrical waveguides of the solar corona. Astrophys. J. 814(2), 135 (2015). https://doi.org/10.1088/0004-637X/814/2/135. arXiv:1510.07908
S.V. Shestov, V.M. Nakariakov, A.S. Ulyanov, A.A. Reva, S.V. Kuzin, Nonlinear evolution of short-wavelength torsional Alfvén waves. Astrophys. J. 840(2), 64 (2017). https://doi.org/10.3847/1538-4357/aa6c65. arXiv:1705.02790
M. Shi, B. Li, Z. Huang, S.X. Chen, Synthetic emissions of the Fe XXI 1354 Å line from flare loops experiencing fundamental fast sausage oscillations. Astrophys. J. 874(1), 87 (2019a). https://doi.org/10.3847/1538-4357/ab07b8. arXiv:1902.06087
M. Shi, B. Li, Z. Huang, S.X. Chen, Synthetic extreme-ultraviolet emissions modulated by leaky fast sausage modes in solar active region loops. Astrophys. J. 883(2), 196 (2019b). https://doi.org/10.3847/1538-4357/ab3d42. arXiv:1908.07131
M. Shi, B. Li, T. Van Doorsselaere, S.X. Chen, Z. Huang, Non-equilibrium ionization effects on extreme-ultraviolet emissions modulated by standing sausage modes in coronal loops. Astrophys. J. 870(2), 99 (2019c). https://doi.org/10.3847/1538-4357/aaf393. arXiv:1811.08571
J.M. Smith, B. Roberts, R. Oliver, Magnetoacoustic wave propagation in current sheets. Astron. Astrophys. 327, 377–387 (1997)
R. Soler, M. Goossens, J. Terradas, R. Oliver, The behavior of transverse waves in nonuniform solar flux tubes. I. Comparison of ideal and resistive results. Astrophys. J. 777(2), 158 (2013). https://doi.org/10.1088/0004-637X/777/2/158. arXiv:1309.3423
H.C. Spruit, Propagation speeds and acoustic damping of waves in magnetic flux tubes. Sol. Phys. 75(1–2), 3–17 (1982). https://doi.org/10.1007/BF00153456
A.K. Srivastava, T.V. Zaqarashvili, W. Uddin, B.N. Dwivedi, P. Kumar, Observation of multiple sausage oscillations in cool post-flare loop. Mon. Not. R. Astron. Soc. 388(4), 1899–1903 (2008). https://doi.org/10.1111/j.1365-2966.2008.13532.x. arXiv:0806.0897
J.T. Su, Y.D. Shen, Y. Liu, Y. Liu, X.J. Mao, Imaging observations of quasi-periodic pulsations in solar flare loops with SDO/AIA. Astrophys. J. 755(2), 113 (2012). https://doi.org/10.1088/0004-637X/755/2/113
T. Takano, H. Nakajima, S. Enome, K. Shibasaki, M. Nishio, Y. Hanaoka, Y. Shiomi, H. Sekiguchi, S. Kawashima, T. Bushimata, N. Shinohara, C. Torii, K. Fujiki, Y. Irimajiri, An Upgrade of Nobeyama Radioheliograph to a dual-Frequency (17 and 34 GHz). System, vol. 483 (1997). 183 p. https://doi.org/10.1007/BFb0106457
J. Terradas, R. Oliver, J.L. Ballester, On the excitation of trapped and leaky modes in coronal slabs. Astron. Astrophys. 441(1), 371–378 (2005). https://doi.org/10.1051/0004-6361:20053198
J. Terradas, J. Andries, M. Goossens, On the excitation of leaky modes in cylindrical loops. Sol. Phys. 246(1), 231–242 (2007). https://doi.org/10.1007/s11207-007-9067-6
H. Thackray, R. Jain, Fast magnetohydrodynamic waves in a solar coronal arcade. Astron. Astrophys. 608, A108 (2017). https://doi.org/10.1051/0004-6361/201731193
H. Tian, P.R. Young, K.K. Reeves, T. Wang, P. Antolin, B. Chen, J. He, Global sausage oscillation of solar flare loops detected by the interface region imaging spectrograph. Astrophys. J. 823(1), L16 (2016). https://doi.org/10.3847/2041-8205/823/1/L16. arXiv:1605.01963
T. Van Doorsselaere, E. Verwichte, J. Terradas, The effect of loop curvature on coronal loop kink oscillations. Space Sci. Rev. 149(1–4), 299–324 (2009). https://doi.org/10.1007/s11214-009-9530-9
T. Van Doorsselaere, A. De Groof, J. Zender, D. Berghmans, M. Goossens, LYRA observations of two oscillation modes in a single flare. Astrophys. J. 740(2), 90 (2011). https://doi.org/10.1088/0004-637X/740/2/90
T. Van Doorsselaere, S.E. Gijsen, J. Andries, G. Verth, Energy propagation by transverse waves in multiple flux tube systems using filling factors. Astrophys. J. 795(1), 18 (2014). https://doi.org/10.1088/0004-637X/795/1/18
T. Van Doorsselaere, E.G. Kupriyanova, D. Yuan, Quasi-periodic pulsations in solar and stellar flares: an overview of recent results (invited review). Sol. Phys. 291(11), 3143–3164 (2016). https://doi.org/10.1007/s11207-016-0977-z. arXiv:1609.02689
T. Van Doorsselaere, A.K. Srivastava, P. Antolin, N. Magyar, S. Vasheghani Farahani, H. Tian, D. Kolotkov, L. Ofman, M.-Z. Guo, I. Arregui, I. De Moortel, D. Pascoe, Coronal heating by MHD waves. Space Sci. Rev. (2020, submitted)
S. Vasheghani Farahani, C. Hornsey, T. Van Doorsselaere, M. Goossens, Frequency and damping rate of fast sausage waves. Astrophys. J. 781(2), 92 (2014). https://doi.org/10.1088/0004-637X/781/2/92
G. Verth, R. Erdélyi, Effect of longitudinal magnetic and density inhomogeneity on transversal coronal loop oscillations. Astron. Astrophys. 486(3), 1015–1022 (2008). https://doi.org/10.1051/0004-6361:200809626
E. Verwichte, C. Foullon, V.M. Nakariakov, Fast magnetoacoustic waves in curved coronal loops. Astron. Astrophys. 446(3), 1139–1149 (2006a). https://doi.org/10.1051/0004-6361:20053955
E. Verwichte, C. Foullon, V.M. Nakariakov, Fast magnetoacoustic waves in curved coronal loops. II. Tunneling modes. Astron. Astrophys. 449(2), 769–779 (2006b). https://doi.org/10.1051/0004-6361:20054398
T. Wang, L. Ofman, Determination of transport coefficients by coronal seismology of flare-induced slow-mode waves: numerical parametric study of a 1D loop model. Astrophys. J. 886(1), 2 (2019). https://doi.org/10.3847/1538-4357/ab478f. arXiv:1909.10910
F. Wang, H. Deng, B. Li, S. Feng, X. Bai, L. Deng, Y. Yang, Z. Xue, R. Wang, High-frequency oscillations in the atmosphere above a sunspot umbra. Astrophys. J. 856(1), L16 (2018). https://doi.org/10.3847/2041-8213/aab4f3. arXiv:1803.09046
T.-J. Wang, L. Ofman, D. Yuan, F. Reale, D.Y. Kolotkov, A.K. Srivastava, Slow-mode magnetoacoustic waves in coronal loops. Space Sci. Rev. (2020, submitted)
S. Wedemeyer-Böhm, E. Scullion, O. Steiner, L. Rouppe van der Voort, J. de La Cruz Rodriguez, V. Fedun, R. Erdélyi, Magnetic tornadoes as energy channels into the solar corona. Nature 486(7404), 505–508 (2012). https://doi.org/10.1038/nature11202
D.R. Williams, K.J.H. Phillips, P. Rudawy, M. Mathioudakis, P.T. Gallagher, E. O’Shea, F.P. Keenan, P. Read, B. Rompolt, High-frequency oscillations in a solar active region coronal loop. Mon. Not. R. Astron. Soc. 326(2), 428–436 (2001). https://doi.org/10.1046/j.1365-8711.2001.04491.x
D.R. Williams, M. Mathioudakis, P.T. Gallagher, K.J.H. Phillips, R.T.J. McAteer, F.P. Keenan, P. Rudawy, A.C. Katsiyannis, An observational study of a magneto-acoustic wave in the solar corona. Mon. Not. R. Astron. Soc. 336(3), 747–752 (2002). https://doi.org/10.1046/j.1365-8711.2002.05764.x
S. Yu, B. Chen, Possible detection of subsecond-period propagating magnetohydrodynamics waves in post-reconnection magnetic loops during a two-ribbon solar flare. Astrophys. J. 872(1), 71 (2019). https://doi.org/10.3847/1538-4357/aaff6d. arXiv:1901.05379
S. Yu, V.M. Nakariakov, L.A. Selzer, B. Tan, Y. Yan, Quasi-periodic wiggles of microwave zebra structures in a solar flare. Astrophys. J. 777(2), 159 (2013). https://doi.org/10.1088/0004-637X/777/2/159. arXiv:1309.5777
H. Yu, B. Li, S.X. Chen, M.Z. Guo, Kink and sausage modes in nonuniform magnetic slabs with continuous transverse density distributions. Astrophys. J. 814(1), 60 (2015). https://doi.org/10.1088/0004-637X/814/1/60. arXiv:1510.05747
H. Yu, S.X. Chen, B. Li, L.D. Xia, Period ratios for standing kink and sausage modes in magnetized structures with siphon flow on the Sun. Res. Astron. Astrophys. 16(6), 92 (2016a). https://doi.org/10.1088/1674-4527/16/6/092. arXiv:1601.01803
H. Yu, B. Li, S.X. Chen, M. Xiong, M.Z. Guo, Impulsively generated sausage waves in coronal tubes with transversally continuous structuring. Astrophys. J. 833(1), 51 (2016b). https://doi.org/10.3847/1538-4357/833/1/51. arXiv:1610.04316
S. Yu, V.M. Nakariakov, Y. Yan, Effect of a sausage oscillation on radio zebra-pattern structures in a solar flare. Astrophys. J. 826(1), 78 (2016c). https://doi.org/10.3847/0004-637X/826/1/78. arXiv:1608.04289
H. Yu, B. Li, S.X. Chen, M. Xiong, M.Z. Guo, Impulsively generated wave trains in coronal structures. I. Effects of transverse structuring on sausage waves in pressureless tubes. Astrophys. J. 836(1), 1 (2017). https://doi.org/10.3847/1538-4357/836/1/1. arXiv:1612.09479
D. Yuan, Y. Shen, Y. Liu, V.M. Nakariakov, B. Tan, J. Huang, Distinct propagating fast wave trains associated with flaring energy releases. Astron. Astrophys. 554, A144 (2013). https://doi.org/10.1051/0004-6361/201321435
D. Yuan, T. Van Doorsselaere, D. Banerjee, P. Antolin, Forward modeling of standing slow modes in flaring coronal loops. Astrophys. J. 807(1), 98 (2015). https://doi.org/10.1088/0004-637X/807/1/98. arXiv:1504.07475
V.V. Zaitsev, A.V. Stepanov, On the origin of the hard X-ray pulsations during solar flares. Sov. Astron. Lett. 8, 132–134 (1982)
V.V. Zajtsev, A.V. Stepanov, On the origin of pulsations of type IV solar radio emission. Plasma cylinder oscillations (I). Issled. Geomagn. Aeron. Fiz. Solnca 37, 3–10 (1975)
J. Zhang, Y. Liu, Ubiquitous rotating network magnetic fields and extreme-ultraviolet cyclones in the Quiet Sun. Astrophys. J. 741(1), L7 (2011). https://doi.org/10.1088/2041-8205/741/1/L7. arXiv:1109.5003
V.V. Zheleznyakov, E.Y. Zlotnik, Cyclotron wave instability in the corona and origin of solar radio emission with fine structure. III. Origin of zebra-pattern. Sol. Phys. 44(2), 461–470 (1975). https://doi.org/10.1007/BF00153225
I.V. Zimovets, S.A. Kuznetsov, A.B. Struminsky, Fine structure of the sources of quasi-periodic pulsations in “single-loop” solar flares. Astron. Lett. 39(4), 267–278 (2013). https://doi.org/10.1134/S1063773713040063
I.V. Zimovets, J.A. McLaughlin, A.K. Srivastava, D.Y. Kolotkov, A.A. Kuznetsov, E.G. Kupriyanova, I.-H. Cho, A.R. Inglis, F. Reale, D.J. Pascoe, H. Tian, D. Yuan, D. Li, Q.M. Zhang, Establishing the analogy between quasi-periodic pulsations in solar and stellar flares. Space Sci. Rev. (2020, submitted)
N.K. Zsámberger, M. Allcock, R. Erdélyi, Magneto-acoustic waves in a magnetic slab embedded in an asymmetric magnetic environment: the effects of asymmetry. Astrophys. J. 853(2), 136 (2018). https://doi.org/10.3847/1538-4357/aa9ffe
Acknowledgements
We thank the reviewers for their constructive comments and suggestions, which helped improve this manuscript substantially. We gratefully acknowledge ISSI-BJ for supporting the workshop on “Oscillatory Processes in Solar and Stellar Coronae”, during which this review was initiated. BL was supported by the National Natural Science Foundation of China (41674172, 11761141002, 41974200). PA acknowledges funding from his STFC Ernest Rutherford Fellowship (No. ST/R004285/2). AAK was supported by budgetary funding of Basic Research program II.16. DJP and TVD were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 724326) and the C1 grant TRACEspace of Internal Funds KU Leuven.
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Oscillatory Processes in Solar and Stellar Coronae
Edited by Valery M. Nakariakov, Dipankar Banerjee, Bo Li, Tongjiang Wang, Ivan Zimovets and Maurizio Falanga
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Li, B., Antolin, P., Guo, MZ. et al. Magnetohydrodynamic Fast Sausage Waves in the Solar Corona. Space Sci Rev 216, 136 (2020). https://doi.org/10.1007/s11214-020-00761-z
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DOI: https://doi.org/10.1007/s11214-020-00761-z