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 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 is the total mass density of the charged plasma particles. Here, goes over all plasma species, both electrons and ions.
Therefore, the phase velocity of an electromagnetic wave in such a medium is
or
where
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.

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 compared to its surface, 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 stated:
jets.
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 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 above chromospheric bright-points. They claimed first direct detection of the long-period incompressible torsional Alfvén waves in the lower solar atmosphere. After the seminal work of Jess et al., 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.

Historical timeline