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. 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 of a magnetized plasma is given by where is the magnetic field strength, is the speed of light, is the permeability of the vacuum, and the mass-density is totalled over all charged plasma particles, each species, indexed by , with ion-number density and individual ionic mass ; the sum includes both electrons  and (few types of) ions. The phase velocity of an electromagnetic wave in such a medium is For the case of an Alfvén wave where is the Alfvén wave group velocity. If then On the other hand, when then That is, at high field or low density, the group 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 . Alfvén time. 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. Relativistic case. 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. 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: 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. 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 the Hinode era in 2007 for the 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 flux tubes. 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. After the seminal work of Jess "et al". (2009), 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 (local thermodynamic equilibrium) 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.