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A cluster of double layers forming in an Alfvén wave, about a sixth of the distance from the left. Legend: * Red=electrons * green=ions * yellow=electric potential * orange=parallel electric field * pink=charge density * blue=magnetic field
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.
Therefore, the phase velocity of an electromagnetic wave in such a medium is
is the Alfvén velocity. If , then . On the other hand, when , then . 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
where is the ion number density and is the ion mass.
In plasma physics, the Alfvén time is an important timescale for wave phenomena. It is related to the Alfvén velocity by:
where denotes the characteristic scale of the system. For example, could be the minor radius of the torus in a tokamak.
In 1993, Gedalin derived the Alfvén wave velocity using relativistic magnetohydrodynamics to be
where is the total energy density of plasma particles, is the total plasma pressure, and is the magnetic pressure. In the non-relativistic limit , and we immediately recover the expression from the previous section.
Cold plasma floating in the corona above the solar limb. Alfvén waves were observed for the first time, extrapolated from fluctuations of the plasma.
The coronal heating problem
The study of Alfvén waves began from the coronal heating problem, a longstanding question in heliophysics. It was unclear as to why the temperature of the sun's corona is hot (about one million degrees Kelvin) compared to its surface (i.e. the photosphere), which is only a few thousand degrees. 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 quotes:
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, 
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 and Haerendel 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. 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. 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. 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. 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. In 2017, Srivastava et al. 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. 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.
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.
1977: Mendis and Ip suggest the existence of hydromagnetic waves in the coma of Comet Kohoutek.
1984: Roberts et al. predict the presence of standing MHD waves in the solar corona and opens the field of coronal seismology.
1999: Aschwanden et al. 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. However, these observations turned out to be kink waves of coronal plasma structures.
2007: A special issue on the Hinode space observatory was released in the journal Science. Alfvén wave signatures in the coronal atmosphere were observed by Cirtain et al., Okamoto et al., and De Pontieu et al. 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.
2011: Alfvén waves are shown to propagate in a liquid metal alloy made of Gallium.
2017: 3D numerical modelling performed by Srivastava et al. show that the high-frequency (12-42 mHz) Alfven waves detected by the Swedish Solar Telescope can carry substantial energy to heat the Sun's inner corona.
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.
^ abGrant, 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, Bibcode:2018NatPh..14..480G, doi:10.1038/s41567-018-0058-3
Aschwanden, M. J.; Fletcher, L.; Schrijver, C. J.; Alexander, D. (1999), "Coronal Loop Oscillations Observed with the Transition Region and Coronal Explorer", The Astrophysical Journal, 520 (2): 880-894, Bibcode:1999ApJ...520..880A, doi:10.1086/307502
Mancuso, S.; Spangler, S. R. (1999), "Coronal Faraday Rotation Observations: Measurements and Limits on Plasma Inhomogeneities", The Astrophysical Journal, 525 (1): 195-208, Bibcode:1999ApJ...525..195M, doi:10.1086/307896
------ (1988b), "Application of Nonlinear Dynamical Invariants in a Single Electromagnetic Wave to the Study of the Alfvén-Ion-Cyclotron Instability", Physics of Fluids, 31 (6): 1456-1464, Bibcode:1988PhFl...31.1456O, doi:10.1063/1.866736
Srivastava, Abhishek K.; Shetye, Juie; Murawski, Krzysztof; Doyle, John Gerard; Stangalini, Marco; Scullion, Eamon; Ray, Tom; Wójcik, Dariusz Patryk; Dwivedi, Bhola N. (2017), "High-frequency torsional Alfvén waves as an energy source for coronal heating", Scientific Reports, 7: id.43147, Bibcode:2017NatSR...743147S, doi:10.1038/srep43147
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, Bibcode:2018NatPh..14..480G, doi:10.1038/s41567-018-0058-3