# Alfvén waves

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## Alfvén waves

Propagating oscillations in electrically conducting fluids or gases in which a magnetic field is present. Magnetohydrodynamics deals with the effects of magnetic fields on fluids and gases which are efficient conductors of electricity. Molten metals are generally good conductors of electricity, and they exhibit magnetohydrodynamic phenomena. Gases can be efficient conductors of electricity if they become ionized. Ionization can occur at high temperatures or through the ionizing effects of high-energy (usually ultraviolet) photons. A gas which consists of free electrons and ions is called a plasma. Most gases in space are plasmas, and magnetohydrodynamic phenomena are expected to play a fundamental role in the behavior of matter in the cosmos. *See* Plasma (physics)

Waves are a particularly important aspect of magnetohydrodynamics. They transport energy and momentum from place to place and may, therefore, play essential roles in the heating and acceleration of cosmical and laboratory plasmas. A wave is a propagating oscillation. If waves are present, a given parcel of the fluid undergoes oscillations about an equilibrium position. The parcel oscillates because there are restoring forces which tend to return it to its equilibrium position. In an ordinary gas, the only restoring force comes from the thermal pressure of the gas. This leads to one wave mode: the sound wave. If a magnetic field is present, there are two additional restoring forces: the tension associated with magnetic field lines, and the pressure associated with the energy density of the magnetic field. These two restoring forces lead to two additional wave modes. Thus there are three magnetohydrodynamic wave modes. However, each restoring force does not necessarily have a unique wave mode associated with it. Put another way, each wave mode can involve more than one restoring force. Thus the usual sound wave, which involves only the thermal pressure, does not appear as a mode in magnetohydrodynamics. The three modes have different propagation speeds, and are named fast mode (F), slow mode (S), and intermediate mode (I). The intermediate mode is sometimes called the Alfvén wave, but some scientists refer to all three magnetohydrodynamic modes as Alfvén waves. The intermediate mode is also called the shear wave. Some scientists give the name magnetosonic mode to the fast mode.

#### Basic equations

The magnetohydrodynamic wave modes are analyzed by using the magnetohydrodynamic equations for the motion of a conducting fluid in a magnetic field, combined with Maxwell's equations and Ohm's law. *See* Maxwell's equations

It is possible to combine Ohm's law with Faraday's law of induction. The resultant equation is called the magnetohydrodynamic induction equation, which is the mathematical statement of the “frozen-in” theorem. This theorem states that magnetic field lines can be thought of as being frozen into the fluid, with the proviso that the fluid is always allowed to slip freely along the field lines. It is the coupling between the fluid and the magnetic field which makes magnetohydrodynamic waves possible. The oscillating magnetic field lines cause oscillations of the fluid parcels, while the fluid provides a mass loading on the magnetic field lines. This mass loading has the effect of slowing down the waves, so that they propagate at speeds much less than the speed of light (which is the propagation speed of waves in a vacuum). *See* Electromagnetic radiation, Light

#### Linearization of equations

Unfortunately, the basic equations are too difficult to be of much use because some of them are nonlinear; that is, they contain products of the quantities for which a solution is sought. Nonlinear magnetohydrodynamics is still only in its infancy, and only a few specialized solutions are known. In order to get solvable equations, scientists accept the limitation of dealing with small-amplitude waves and linearize the equations, so that products of the unknowns are removed. Fortunately, much can still be learned from this procedure; the resulting equations have solutions which are harmonic in time and space. *See* Harmonic motion

#### Intermediate mode

The motions in this mode are pure shears. There is no compression of the plasma. The tension in the magnetic field lines is the only restoring force involved in the propagation of the wave. This mode is therefore closely analogous to the propagation of waves on a string.

Because these waves channel energy along magnetic fields, they may be responsible for the observed fact that cosmical plasmas are strongly heated in the presence of magnetic fields.

#### Fast mode

This mode is difficult to analyze. However, many cosmical and laboratory plasmas satisfy the strong-magnetic-field case where the fast mode is more easily understood.

Fast waves are compressive, and the magnetic field strength fluctuates as well. Thus fast waves are governed by the two restoring forces associated with the tension and pressure in the magnetic field.

The fast mode can propagate energy across the magnetic field.

#### Slow mode

Like the fast mode, the slow mode is difficult to study in general, and the discussion will again be confined to strong magnetic fields. The slow mode in a strong field is equivalent to sound waves which are guided along the strong magnetic field lines. The strong magnetic field lines can be thought of as a set of rigid pipes which allow free fluid motion along the pipes, but which restrict motion in the other two directions. The motions on the individual pipes are not coupled together, and thus the slow mode is analogous to the sound waves on a set of independent organ pipes. The slow mode channels energy along the magnetic field. Because the sound speed is small, by assumption, the slow mode transmits energy less effectively than the fast or intermediate modes.

#### Nonlinear effects

Only small-amplitude waves have been considered. Real waves have finite amplitude, and nonlinear effects can sometimes be important. One such effect is the tendency of waves to steepen, ultimately forming magnetohydrodynamic shock waves and magnetohydrodynamic discontinuities. There is an abundance of magnetohydrodynamic discontinuities in the solar wind. *See* Shock wave

It is also possible that waves can degenerate into turbulence. There are indications that this too happens in the solar wind. *See* Turbulent flow

#### Surface waves

Only waves in a spatially uniform background have been considered. While the analysis of magnetohydrodynamic waves in a nonuniform background is complicated, it is possible to consider an extreme limit, in which the background is uniform except at certain surfaces where it changes discontinuously. Surfaces can support magnetohydrodynamic waves, which are in some respects similar to waves on the surface of a lake. These waves may play important roles in heating cosmical and laboratory plasmas. *See* Magnetohydrodynamics

## Alfvén waves

(al-**ven**) Disturbances transmitted through a plasma in the presence of a magnetic field. The direction of propagation is parallel to the mean magnetic field, with the plasma particles vibrating at right angles to this direction. The speed of propagation, the

*Alfvén speed*, depends on the magnetic field strength and the plasma density. The waves interact with the plasma particles, for example by exerting radiation pressure on the plasma. As the waves dissipate, the plasma particles are heated and accelerated. Alfvén waves are a type of magnetohydrodynamic (MHD) wave. They have been directly observed in the solar wind, particularly in the high-speed streams, and in planetary magnetospheres.