Alfvén wave

An Alfvén wave, named after Hannes Alfvén, is a type of magnetohydrodynamic wave.^[1]

1 Definition
2 Alfvén velocity
3 Alfvén time
4 Relativistic case
5 History
6 See also
7 References
8 Further reading
9 External links

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 permittivity $\epsilon\,$ of a magnetized plasma is given by

$\epsilon = 1 %2B \frac{c^2 \mu_0 \rho}{B^2}~,$

where $B\,$ is the magnetic field strength, $c\,$ is the speed of light, $\mu_0\,$ is the permeability of the vacuum, and $\rho = \Sigma n_s m_s\,$ 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 velocity of an electromagnetic wave in such a medium is

$v = c/\sqrt{\epsilon} = \frac{c}{\sqrt{1 %2B \frac{c^2 \mu_0 \rho}{B^2}}}~,$

$v = \frac{v_A}{\sqrt{1 %2B v_A^2/c^2}}~,$

where

$v_A = \frac{B}{\sqrt{\mu_0 \rho}}$

is the Alfvén velocity. If $v_A \ll c$ , then $v \approx v_A$ . On the other hand, when $v_A \approx c$ , then $v \approx c$ . 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 CGS

$\qquad \ \approx (2.18\times10^{11}\,\mbox{cm/s})\,(m_i/m_p)^{-1/2}\,(n_i/{\rm cm}^{-3})^{-1/2}\,(B/{\rm gauss})$

where $n_i\,$ is the ion number density and $m_i\,$ is the ion mass.

Alfvén time

In plasma physics, the Alfvén time $\tau_A$ 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 $a$ is the minor radius of the torus in a tokamak.

Relativistic case

The general Alfvén wave velocity is defined by Gedalin (1993):^[2]

$v = \frac{c}{\sqrt{1 %2B \frac{e %2B P}{2 P_m}}}~,$

where

$e\,$ is the total energy density of plasma particles, $P\,$ is the total plasma pressure, and $P_m = B^2/2\mu_0\,$ is the magnetic field pressure. In the non-relativistic limit $P \ll e \approx \rho c^2$ , and we immediately get the expression from the previous section.

History

How this phenomenon became understood

1942: Alfvén suggests the existence of electromagnetic-hydromagnetic waves in a paper published in Nature.
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 Alex 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
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
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
1960: Sugiura suggests evidence of hydromagnetic waves in the Earth's magnetic field
1961: Normal Alfvén modes and resonances in liquid sodium are studied by Jameson
1966: R.O.Motz generates and observes Alfven waves in mercury
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: Hollweg suggests the existence of hydromagnetic waves in interplanetary space
1974: Ip and Mendis suggests the existence of hydromagnetic waves in the coma of Comet Kohoutek.
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.
2007: Tomczyk, et al., report 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. These waves were interpreted as propagating kink waves by Van Doorsselaere et al. (2008)
2007: Alfvén wave discoveries appear in articles by Jonathan Cirtain and colleagues, Takenori J. Okamoto and colleagues, and Bart De Pontieu and colleagues. De Pontieu’s team also shows that the energy associated with the waves is sufficient to heat the corona and accelerate the solar wind. These results appear in a special collection of 10 articles, by scientists in Japan, Europe and the United States, in the 7 December issue of the journal Science. It was demonstrated that those waves should be interpreted in terms of kink waves of coronal plasma structures by Van Doorsselaere, et al. (2008); Ofman and Wang (2008); and Vasheghani Farahani, et al. (2009).
2011: Experimental evidence of Alfvén wave propagation in a Gallium alloy^[3]

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
^ 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
^ doi:10.1063/1.3633090
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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"