Alfvén wave

In plasma physics, an Alfvén wave, named after Hannes Alfvén, is a type of magnetohydrodynamic wave in which ions oscillate in response to a restoring force provided by an effective tension on the magnetic field lines.[1]

A cluster of double layers forming in an Alfvén wave, about a sixth of the distance from the left. Red = electrons, Green = ions, Yellow = electric potential, Orange = parallel electric field, Pink = charge density, Blue = magnetic field

Definition

An Alfvén wave in a plasma is a low-frequency (compared to the ion cyclotron frequency) travelling oscillation of the ions and the magnetic field. The ion mass density provides the inertia and the magnetic field line tension provides the restoring force.

The wave propagates in the direction of the magnetic field, although waves exist at oblique incidence and smoothly change into the magnetosonic wave when the propagation is perpendicular to the magnetic field.

The motion of the ions and the perturbation of the magnetic field are in the same direction and transverse to the direction of propagation. The wave is dispersionless.

Alfvén velocity

The low-frequency relative permittivity $ε$ of a magnetized plasma is given by[2]

\epsilon =1+{\frac {c^{2}\mu _{0}\rho }{B^{2}}}

where $B$ is the magnetic field strength, $c$ is the speed of light, $μ 0$ is the permeability of the vacuum, and $ρ = \sum n s m s$ is the total mass density of the charged plasma particles. Here, $s$ goes over all plasma species, both electrons and (few types of) ions.

Therefore, the phase velocity of an electromagnetic wave in such a medium is

v={\frac {c}{\sqrt {\epsilon }}}={\frac {c}{\sqrt {1+{\dfrac {c^{2}\mu _{0}\rho }{B^{2}}}}}}

or

v={\frac {v_{A}}{\sqrt {1+{\dfrac {v_{A}^{2}}{c^{2}}}}}}

where

v_{A}={\frac {B}{\sqrt {\mu _{0}\rho }}}

is the Alfvén velocity. If $v A ≪ c$ , then $v \approx v A$ . On the other hand, when $v A \to \infty$ , then $v \to c$ . That is, at high field or low density, the velocity of the Alfvén wave approaches the speed of light, and the Alfvén wave becomes an ordinary electromagnetic wave.

Neglecting the contribution of the electrons to the mass density and assuming that there is a single ion species, we get

v_{A}={\frac {B}{\sqrt {\mu _{0}n_{i}m_{i}}}}~~

in SI

v_{A}={\frac {B}{\sqrt {4\pi n_{i}m_{i}}}}~~

in Gauss

v_{A}\approx \left(2.18\times 10^{11}\,{\mbox{cm s}}^{-1}\right)\left({\frac {m_{i}}{m_{p}}}\right)^{-{\frac {1}{2}}}\left({\frac {n_{i}}{{\rm {cm}}^{-3}}}\right)^{-{\frac {1}{2}}}\left({\frac {B}{\rm {gauss}}}\right)

where $n i$ is the ion number density and $m i$ is the ion mass.

Alfvén time

In plasma physics, the Alfvén time $τ A$ is an important timescale for wave phenomena. It is related to the Alfvén velocity by:

\tau _{A}={\frac {a}{v_{A}}}

where $a$ denotes the characteristic scale of the system. For example, $a$ could be the minor radius of the torus in a tokamak.

Relativistic case

In 1993, Gedalin derived the Alfvén wave velocity using relativistic magnetohydrodynamics[3] to be

v={\frac {c}{\sqrt {1+{\dfrac {e+P}{2P_{m}}}}}}

where $e$ is the total energy density of plasma particles, $P$ is the total plasma pressure, and $P m = .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap} B 2 / 2 μ 0$ is the magnetic pressure. In the non-relativistic limit $P ≪ e \approx ρc 2$ , and we immediately recover the expression from the previous section.

History

The coronal heating problem

The study of Alfvén waves began from the coronal heating problem, a longstanding question in heliophysics. It was unclear why the temperature of the solar corona is hot (about one million kelvins) compared to its surface (the photosphere), which is only a few thousand kelvins. Intuitively, it would make sense to see a decrease in temperature when moving away from a heat source, but this does not seem to be the case even though the photosphere is denser and would generate more heat than the corona.

In 1942, Hannes Alfvén proposed in Nature the existence of an electromagnetic-hydrodynamic wave which would carry energy from the photosphere to heat up the corona and the solar wind. He claimed that the sun had all the necessary criteria to support these waves and they may in turn be responsible for sun spots. He stated:

Magnetic waves, called Alfvén S-waves, flow from the base of black hole jets.

If a conducting liquid is placed in a constant magnetic field, every motion of the liquid gives rise to an E.M.F. which produces electric currents. Owing to the magnetic field, these currents give mechanical forces which change the state of motion of the liquid. Thus a kind of combined electromagnetic–hydrodynamic wave is produced.
— Hannes Alfvén, Existence of Electromagnetic-Hydrodynamic Waves, [4]

This would eventually turn out to be Alfvén waves. He received the 1970 Nobel Prize in Physics for this discovery.

Experimental studies and observations

The convection zone of the sun, the region beneath the photosphere in which energy is transported primarily by convection, is sensitive to the motion of the core due to the rotation of the sun. Together with varying pressure gradients beneath the surface, electromagnetic fluctuations produced in the convection zone induce random motion on the photospheric surface and produce Alfvén waves. The waves then leave the surface, travel through the chromosphere and transition zone, and interact with the ionized plasma. The wave itself carries energy and some of the electrically charged plasma.

In the early 1990s, De Pontieu[5] and Haerendel[6] suggested that Alfvén waves may also be associated with the plasma jets known as spicules. It was theorized these brief spurts of superheated gas were carried by the combined energy and momentum of their own upward velocity, as well as the oscillating transverse motion of the Alfvén waves. In 2007, Alfvén waves were reportedly observed for the first time traveling towards the corona by Tomcyzk et al., but their predictions could not conclude that the energy carried by the Alfvén waves was sufficient to heat the corona to its enormous temperatures, for the observed amplitudes of the waves were not high enough.[7] However, in 2011, McIntosh et al. reported the observation of highly energetic Alfvén waves combined with energetic spicules which could sustain heating the corona to its million-kelvin temperature. These observed amplitudes (20.0 km/s against 2007's observed 0.5 km/s) contained over one hundred times more energy than the ones observed in 2007.[8] The short period of the waves also allowed more energy transfer into the coronal atmosphere. The 50,000-km-long spicules may also play a part in accelerating the solar wind past the corona.[9] However, the above-mentioned discoveries of Alfvén waves in the complex Sun's atmosphere starting from Hinode era in 2007 for next 10 years mostly fall in the realm of Alfvénic waves essentially generated as a mixed mode due to transverse structuring of the magnetic and plasma properties in the localized fluxtubes. In 2009, Jess et al.[10] reported the periodic variation of H-alpha line-width as observed by Swedish Solar Telescope (SST) above chromospheric bright-points. They claimed first direct detection of the long-period (126–700 s) incompressible torsional Alfvén waves in the lower solar atmosphere. After the seminal work of Jess et al. (2009), in 2017 Srivastava et al.[11] detected the existence of high-frequency torsional Alfvén waves in the Sun's chromospheric fine-structured flux tubes. They discovered that these high-frequency waves carry substantial energy capable of heating the Sun's corona and also in originating the supersonic solar wind. In 2018, using spectral imaging observations, non-LTE inversions and magnetic field extrapolations of sunspot atmospheres, Grant et al.[12] found evidence for elliptically-polarized Alfvén waves forming fast-mode shocks in the outer regions of the chromospheric umbral atmosphere. They provided quantification of the degree of physical heat provided by the dissipation of such Alfvén wave modes above active region spots.

Historical timeline

1942: Alfvén suggests the existence of electromagnetic-hydromagnetic waves in a paper published in Nature 150, 405–406 (1942).
1949: Laboratory experiments by S. Lundquist produce such waves in magnetized mercury, with a velocity that approximated Alfvén's formula.
1949: Enrico Fermi uses Alfvén waves in his theory of cosmic rays. According to Alexander J. Dessler in a 1970 Science journal article, Fermi had heard a lecture at the University of Chicago, Fermi nodded his head exclaiming "of course" and the next day, the physics world said "of course".
1950: Alfvén publishes the first edition of his book, Cosmical Electrodynamics, detailing hydromagnetic waves, and discussing their application to both laboratory and space plasmas.
1952: Additional confirmation appears in experiments by Winston Bostick and Morton Levine with ionized helium.
1954: Bo Lehnert produces Alfvén waves in liquid sodium.[13]
1958: Eugene Parker suggests hydromagnetic waves in the interstellar medium.
1958: Berthold, Harris, and Hope detect Alfvén waves in the ionosphere after the Argus nuclear test, generated by the explosion, and traveling at speeds predicted by Alfvén formula.
1958: Eugene Parker suggests hydromagnetic waves in the Solar corona extending into the Solar wind.
1959: D. F. Jephcott produces Alfvén waves in a gas discharge.[14]
1959: C. H. Kelley and J. Yenser produce Alfvén waves in the ambient atmosphere.
1960: Coleman et al. report the measurement of Alfvén waves by the magnetometer aboard the Pioneer and Explorer satellites.[15]
1961: Sugiura suggests evidence of hydromagnetic waves in the Earth's magnetic field.[16]
1961: Normal Alfvén modes and resonances in liquid sodium are studied by Jameson.
1966: R. O. Motz generates and observes Alfven waves in mercury.[17]
1970: Hannes Alfvén wins the 1970 Nobel Prize in physics for "fundamental work and discoveries in magneto-hydrodynamics with fruitful applications in different parts of plasma physics".
1973: Eugene Parker suggests hydromagnetic waves in the intergalactic medium.
1974: J. V. Hollweg suggests the existence of hydromagnetic waves in interplanetary space.[18]
1977: Mendis and Ip suggest the existence of hydromagnetic waves in the coma of Comet Kohoutek.[19]
1984: Roberts et al. predict the presence of standing MHD waves in the solar corona[20] and opens the field of coronal seismology.
1999: Aschwanden et al.[21] and Nakariakov et al. report the detection of damped transverse oscillations of solar coronal loops observed with the EUV imager on board the Transition Region And Coronal Explorer (TRACE), interpreted as standing kink (or "Alfvénic") oscillations of the loops. This confirms the theoretical prediction of Roberts et al. (1984).
2007: Tomczyk et al. reported the detection of Alfvénic waves in images of the solar corona with the Coronal Multi-Channel Polarimeter (CoMP) instrument at the National Solar Observatory, New Mexico.[22] However, these observations turned out to be kink waves of coronal plasma structures.[23]
2007: A special issue on the Hinode space observatory was released in the journal Science.[24] Alfvén wave signatures in the coronal atmosphere were observed by Cirtain et al.,[25] Okamoto et al.,[26] and De Pontieu et al.[27] An estimation of the observed waves' energy density by De Pontieu et al. have show that the energy associated with the waves is sufficient to heat the corona and accelerate the solar wind.
2008: Kaghashvili et al. uses driven wave fluctuations as a diagnostic tool to detect Alfvén waves in the solar corona.[28]
2009: Jess et al. detect torsional Alfvén waves in the structured Sun's chromosphere using the Swedish Solar Telescope.[10]
2011: Alfvén waves are shown to propagate in a liquid metal alloy made of Gallium.[29]
2017: 3D numerical modelling performed by Srivastava et al. show that the high-frequency (12–42 mHz) Alfvén waves detected by the Swedish Solar Telescope can carry substantial energy to heat the Sun's inner corona.[11]
2018: Using spectral imaging observations, non-LTE inversions and magnetic field extrapolations of sunspot atmospheres, Grant et al. found evidence for elliptically-polarized Alfvén waves forming fast-mode shocks in the outer regions of the chromospheric umbral atmosphere. For the first time, these authors provided quantification of the degree of physical heat provided by the dissipation of such Alfvén wave modes.[12]

References

Iwai, K; Shinya, K,; Takashi, K. and Moreau, R. (2003) "Pressure change accompanying Alfvén waves in a liquid metal" Magnetohydrodynamics 39(3): pp. 245-250, page 245
Chen F. F. "Introduction to Plasma Physics and Controlled Fusion 3rd Edition". Springer International Publishing, Switzerland, 2016, p. 55, pp. 126-131.
Gedalin, M. (1993), "Linear waves in relativistic anisotropic magnetohydrodynamics", Physical Review E, 47 (6): 4354–4357, Bibcode:1993PhRvE..47.4354G, doi:10.1103/PhysRevE.47.4354, PMID 9960513
Hannes Alfvén (1942). "Existence of Electromagnetic–Hydrodynamic Waves". Nature. 150 (3805): 405–406. Bibcode:1942Natur.150..405A. doi:10.1038/150405d0. S2CID 4072220.
Bart de Pontieu (18 December 1997). "Chromospheric Spicules driven by Alfvén waves". Max-Planck-Institut für extraterrestrische Physik. Archived from the original on 16 July 2002. Retrieved 1 April 2012.
Gerhard Haerendel (1992). "Weakly damped Alfven waves as drivers of solar chromospheric spicules". Nature. 360 (6401): 241–243. Bibcode:1992Natur.360..241H. doi:10.1038/360241a0. S2CID 44454309.
Tomczyk, S.; McIntosh, S.W.; Keil, S.L.; Judge, P.G.; Schad, T.; Seeley, D.H.; Edmondson, J. (2007). "Alfven waves in the solar corona". Science. 317 (5842): 1192–1196. Bibcode:2007Sci...317.1192T. doi:10.1126/science.1143304. PMID 17761876. S2CID 45840582.
McIntosh; et al. (2011). "Alfvenic waves with sufficient energy to power the quiet solar corona and fast solar wind". Nature. 475 (7357): 477–480. Bibcode:2011Natur.475..477M. doi:10.1038/nature10235. PMID 21796206. S2CID 4336248.
Karen Fox (27 July 2011). "SDO Spots Extra Energy in the Sun's Corona". NASA. Retrieved 2 April 2012.
Jess, David B.; Mathioudakis, Mihalis; Erdélyi, Robert; Crockett, Philip J.; Keenan, Francis P.; Christian, Damian J. (20 March 2009). "Alfvén Waves in the Lower Solar Atmosphere". Science. 323 (5921): 1582–1585. arXiv:0903.3546. Bibcode:2009Sci...323.1582J. doi:10.1126/science.1168680. hdl:10211.3/172550. ISSN 0036-8075. PMID 19299614. S2CID 14522616.
Srivastava, Abhishek Kumar; Shetye, Juie; Murawski, Krzysztof; Doyle, John Gerard; Stangalini, Marco; Scullion, Eamon; Ray, Tom; Wójcik, Dariusz Patryk; Dwivedi, Bhola N. (3 March 2017). "High-frequency torsional Alfvén waves as an energy source for coronal heating". Scientific Reports. 7 (1): 43147. Bibcode:2017NatSR...743147S. doi:10.1038/srep43147. ISSN 2045-2322. PMC 5335648. PMID 28256538.
Grant, Samuel D. T.; Jess, David B.; Zaqarashvili, Teimuraz V.; Beck, Christian; Socas-Navarro, Hector; Aschwanden, Markus J.; Keys, Peter H.; Christian, Damian J.; Houston, Scott J.; Hewitt, Rebecca L. (2018), "Alfvén Wave Dissipation in the Solar Chromosphere", Nature Physics, 14 (5): 480–483, arXiv:1810.07712, Bibcode:2018NatPh..14..480G, doi:10.1038/s41567-018-0058-3, S2CID 119089600
Lehnert, Bo (15 May 1954). "Magneto-Hydrodynamic Waves in Liquid Sodium". Physical Review. 94 (4): 815–824. Bibcode:1954PhRv...94..815L. doi:10.1103/PhysRev.94.815.
JEPHCOTT, D. F. (13 June 1959). "Alfvén Waves in a Gas Discharge". Nature. 183 (4676): 1652–1654. Bibcode:1959Natur.183.1652J. doi:10.1038/1831652a0. ISSN 0028-0836. S2CID 11487078.
Sonett, C. P.; Smith, E. J.; Judge, D. L.; Coleman, P. J. (15 February 1960). "Current Systems in the Vestigial Geomagnetic Field: Explorer VI". Physical Review Letters. 4 (4): 161–163. Bibcode:1960PhRvL...4..161S. doi:10.1103/PhysRevLett.4.161.
Sugiura, Masahisa (December 1961). "Evidence of low-frequency hydromagnetic waves in the exosphere". Journal of Geophysical Research. 66 (12): 4087–4095. Bibcode:1961JGR....66.4087S. doi:10.1029/jz066i012p04087. ISSN 0148-0227.
Motz, Robin O. (1966). "Alfvén Wave Generation in a Spherical System". Physics of Fluids. 9 (2): 411. Bibcode:1966PhFl....9..411M. doi:10.1063/1.1761687. ISSN 0031-9171.
Hollweg, J. V. (1974). "Hydromagnetic Waves in Interplanetary Space". Publications of the Astronomical Society of the Pacific. 86 (513): 561. Bibcode:1974PASP...86..561H. doi:10.1086/129646. ISSN 1538-3873.
Mendis, D. A.; Ip, W. -H. (March 1977). "The ionospheres and plasma tails of comets". Space Science Reviews. 20 (2): 145–190. Bibcode:1977SSRv...20..145M. doi:10.1007/bf02186863. ISSN 0038-6308. S2CID 119883598.
Roberts, B.; Edwin, P. M.; Benz, A. O. (1984). "Coronal oscillations". Astrophysical Journal. 279 (2): 857–865. Bibcode:1984ApJ...279..857R. doi:10.1086/161956. ISSN 0004-637X.
Aschwanden, Markus J.; Fletcher, Lyndsay; Schrijver, Carolus J.; Alexander, David (1999). "Coronal Loop Oscillations Observed with the Transition Region and Coronal Explorer" (PDF). The Astrophysical Journal. 520 (2): 880. Bibcode:1999ApJ...520..880A. doi:10.1086/307502. ISSN 0004-637X.
Tomczyk, S.; McIntosh, S. W.; Keil, S. L.; Judge, P. G.; Schad, T.; Seeley, D. H.; Edmondson, J. (31 August 2007). "Alfvén Waves in the Solar Corona". Science. 317 (5842): 1192–1196. Bibcode:2007Sci...317.1192T. doi:10.1126/science.1143304. ISSN 0036-8075. PMID 17761876. S2CID 45840582.
Doorsselaere, T. Van; Nakariakov, V. M.; Verwichte, E. (2008). "Detection of Waves in the Solar Corona: Kink or Alfvén?". The Astrophysical Journal Letters. 676 (1): L73. Bibcode:2008ApJ...676L..73V. doi:10.1086/587029. ISSN 1538-4357.
"Science: 318 (5856)". Science. 318 (5856). 7 December 2007. ISSN 0036-8075.
Cirtain, J. W.; Golub, L.; Lundquist, L.; Ballegooijen, A. van; Savcheva, A.; Shimojo, M.; DeLuca, E.; Tsuneta, S.; Sakao, T. (7 December 2007). "Evidence for Alfvén Waves in Solar X-ray Jets". Science. 318 (5856): 1580–1582. Bibcode:2007Sci...318.1580C. doi:10.1126/science.1147050. ISSN 0036-8075. PMID 18063786. S2CID 39318753.
Okamoto, T. J.; Tsuneta, S.; Berger, T. E.; Ichimoto, K.; Katsukawa, Y.; Lites, B. W.; Nagata, S.; Shibata, K.; Shimizu, T. (7 December 2007). "Coronal Transverse Magnetohydrodynamic Waves in a Solar Prominence". Science. 318 (5856): 1577–1580. arXiv:0801.1958. Bibcode:2007Sci...318.1577O. doi:10.1126/science.1145447. ISSN 0036-8075. PMID 18063785. S2CID 121422620.
Pontieu, B. De; McIntosh, S. W.; Carlsson, M.; Hansteen, V. H.; Tarbell, T. D.; Schrijver, C. J.; Title, A. M.; Shine, R. A.; Tsuneta, S. (7 December 2007). "Chromospheric Alfvénic Waves Strong Enough to Power the Solar Wind". Science. 318 (5856): 1574–1577. Bibcode:2007Sci...318.1574D. doi:10.1126/science.1151747. ISSN 0036-8075. PMID 18063784. S2CID 33655095.
Kaghashvili, Edisher Kh.; Quinn, Richard A.; Hollweg, Joseph V. (2009). "Driven Waves as a Diagnostics Tool in the Solar Corona". The Astrophysical Journal. 703 (2): 1318. Bibcode:2009ApJ...703.1318K. doi:10.1088/0004-637x/703/2/1318.
Thierry Alboussière; Philippe Cardin; François Debray; Patrick La Rizza; Jean-Paul Masson; Franck Plunian; Adolfo Ribeiro; Denys Schmitt (2011). "Experimental evidence of Alfvén wave propagation in a Gallium alloy". Phys. Fluids. 23 (9): 096601. arXiv:1106.4727. Bibcode:2011PhFl...23i6601A. doi:10.1063/1.3633090. S2CID 2234120.

11. The electromagnetodynamics of fluids by W F Hughes and F J Young, pp. 159 - 161, p. 308, p. 311, p. 335, p. 446 John Wiley & Sons, New York, LCCC #66-17631

External links

Mysterious Solar Ripples Detected Dave Mosher 2 September 2007 Space.com
EurekAlert! notification of 7 December 2007 Science special issue
EurekAlert! notification: "Scientists find solution to solar puzzle"

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[1] Iwai, K; Shinya, K,; Takashi, K. and Moreau, R. (2003) "Pressure change accompanying Alfvén waves in a liquid metal" Magnetohydrodynamics 39(3): pp. 245-250, page 245

[2] Chen F. F. "Introduction to Plasma Physics and Controlled Fusion 3rd Edition". Springer International Publishing, Switzerland, 2016, p. 55, pp. 126-131.

[3] Gedalin, M. (1993), "Linear waves in relativistic anisotropic magnetohydrodynamics", Physical Review E, 47 (6): 4354–4357, Bibcode:1993PhRvE..47.4354G, doi:10.1103/PhysRevE.47.4354, PMID 9960513

[4] Hannes Alfvén (1942). "Existence of Electromagnetic–Hydrodynamic Waves". Nature. 150 (3805): 405–406. Bibcode:1942Natur.150..405A. doi:10.1038/150405d0. S2CID 4072220.

[5] Bart de Pontieu (18 December 1997). "Chromospheric Spicules driven by Alfvén waves". Max-Planck-Institut für extraterrestrische Physik. Archived from the original on 16 July 2002. Retrieved 1 April 2012.

[6] Gerhard Haerendel (1992). "Weakly damped Alfven waves as drivers of solar chromospheric spicules". Nature. 360 (6401): 241–243. Bibcode:1992Natur.360..241H. doi:10.1038/360241a0. S2CID 44454309.

[7] Tomczyk, S.; McIntosh, S.W.; Keil, S.L.; Judge, P.G.; Schad, T.; Seeley, D.H.; Edmondson, J. (2007). "Alfven waves in the solar corona". Science. 317 (5842): 1192–1196. Bibcode:2007Sci...317.1192T. doi:10.1126/science.1143304. PMID 17761876. S2CID 45840582.

[8] McIntosh; et al. (2011). "Alfvenic waves with sufficient energy to power the quiet solar corona and fast solar wind". Nature. 475 (7357): 477–480. Bibcode:2011Natur.475..477M. doi:10.1038/nature10235. PMID 21796206. S2CID 4336248.

[9] Karen Fox (27 July 2011). "SDO Spots Extra Energy in the Sun's Corona". NASA. Retrieved 2 April 2012.

[:0-10] Jess, David B.; Mathioudakis, Mihalis; Erdélyi, Robert; Crockett, Philip J.; Keenan, Francis P.; Christian, Damian J. (20 March 2009). "Alfvén Waves in the Lower Solar Atmosphere". Science. 323 (5921): 1582–1585. arXiv:0903.3546. Bibcode:2009Sci...323.1582J. doi:10.1126/science.1168680. hdl:10211.3/172550. ISSN 0036-8075. PMID 19299614. S2CID 14522616.

[:1-11] Srivastava, Abhishek Kumar; Shetye, Juie; Murawski, Krzysztof; Doyle, John Gerard; Stangalini, Marco; Scullion, Eamon; Ray, Tom; Wójcik, Dariusz Patryk; Dwivedi, Bhola N. (3 March 2017). "High-frequency torsional Alfvén waves as an energy source for coronal heating". Scientific Reports. 7 (1): 43147. Bibcode:2017NatSR...743147S. doi:10.1038/srep43147. ISSN 2045-2322. PMC 5335648. PMID 28256538.

[:2-12] Grant, Samuel D. T.; Jess, David B.; Zaqarashvili, Teimuraz V.; Beck, Christian; Socas-Navarro, Hector; Aschwanden, Markus J.; Keys, Peter H.; Christian, Damian J.; Houston, Scott J.; Hewitt, Rebecca L. (2018), "Alfvén Wave Dissipation in the Solar Chromosphere", Nature Physics, 14 (5): 480–483, arXiv:1810.07712, Bibcode:2018NatPh..14..480G, doi:10.1038/s41567-018-0058-3, S2CID 119089600

[13] Lehnert, Bo (15 May 1954). "Magneto-Hydrodynamic Waves in Liquid Sodium". Physical Review. 94 (4): 815–824. Bibcode:1954PhRv...94..815L. doi:10.1103/PhysRev.94.815.

[14] JEPHCOTT, D. F. (13 June 1959). "Alfvén Waves in a Gas Discharge". Nature. 183 (4676): 1652–1654. Bibcode:1959Natur.183.1652J. doi:10.1038/1831652a0. ISSN 0028-0836. S2CID 11487078.

[15] Sonett, C. P.; Smith, E. J.; Judge, D. L.; Coleman, P. J. (15 February 1960). "Current Systems in the Vestigial Geomagnetic Field: Explorer VI". Physical Review Letters. 4 (4): 161–163. Bibcode:1960PhRvL...4..161S. doi:10.1103/PhysRevLett.4.161.

[16] Sugiura, Masahisa (December 1961). "Evidence of low-frequency hydromagnetic waves in the exosphere". Journal of Geophysical Research. 66 (12): 4087–4095. Bibcode:1961JGR....66.4087S. doi:10.1029/jz066i012p04087. ISSN 0148-0227.

[17] Motz, Robin O. (1966). "Alfvén Wave Generation in a Spherical System". Physics of Fluids. 9 (2): 411. Bibcode:1966PhFl....9..411M. doi:10.1063/1.1761687. ISSN 0031-9171.

[18] Hollweg, J. V. (1974). "Hydromagnetic Waves in Interplanetary Space". Publications of the Astronomical Society of the Pacific. 86 (513): 561. Bibcode:1974PASP...86..561H. doi:10.1086/129646. ISSN 1538-3873.

[19] Mendis, D. A.; Ip, W. -H. (March 1977). "The ionospheres and plasma tails of comets". Space Science Reviews. 20 (2): 145–190. Bibcode:1977SSRv...20..145M. doi:10.1007/bf02186863. ISSN 0038-6308. S2CID 119883598.

[20] Roberts, B.; Edwin, P. M.; Benz, A. O. (1984). "Coronal oscillations". Astrophysical Journal. 279 (2): 857–865. Bibcode:1984ApJ...279..857R. doi:10.1086/161956. ISSN 0004-637X.

[21] Aschwanden, Markus J.; Fletcher, Lyndsay; Schrijver, Carolus J.; Alexander, David (1999). "Coronal Loop Oscillations Observed with the Transition Region and Coronal Explorer" (PDF). The Astrophysical Journal. 520 (2): 880. Bibcode:1999ApJ...520..880A. doi:10.1086/307502. ISSN 0004-637X.

[22] Tomczyk, S.; McIntosh, S. W.; Keil, S. L.; Judge, P. G.; Schad, T.; Seeley, D. H.; Edmondson, J. (31 August 2007). "Alfvén Waves in the Solar Corona". Science. 317 (5842): 1192–1196. Bibcode:2007Sci...317.1192T. doi:10.1126/science.1143304. ISSN 0036-8075. PMID 17761876. S2CID 45840582.

[23] Doorsselaere, T. Van; Nakariakov, V. M.; Verwichte, E. (2008). "Detection of Waves in the Solar Corona: Kink or Alfvén?". The Astrophysical Journal Letters. 676 (1): L73. Bibcode:2008ApJ...676L..73V. doi:10.1086/587029. ISSN 1538-4357.

[24] "Science: 318 (5856)". Science. 318 (5856). 7 December 2007. ISSN 0036-8075.

[25] Cirtain, J. W.; Golub, L.; Lundquist, L.; Ballegooijen, A. van; Savcheva, A.; Shimojo, M.; DeLuca, E.; Tsuneta, S.; Sakao, T. (7 December 2007). "Evidence for Alfvén Waves in Solar X-ray Jets". Science. 318 (5856): 1580–1582. Bibcode:2007Sci...318.1580C. doi:10.1126/science.1147050. ISSN 0036-8075. PMID 18063786. S2CID 39318753.

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