In plasma physics, an Alfvén wave, named after Hannes Alfvén, is a type of plasma 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 is a low-frequency (compared to the ion gyrofrequency) travelling oscillation of the ions and magnetic field in a plasma. The ion mass density provides the inertia and the magnetic field line tension provides the restoring force. Alfvén waves propagate in the direction of the magnetic field, and the motion of the ions and the perturbation of the magnetic field are transverse to the direction of propagation. However, Alfvén waves existing at oblique incidences will smoothly change into magnetosonic waves when the propagation is perpendicular to the magnetic field. Alfvén waves are 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 in SI in Gauss 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 The Alfvén wave velocity in relativistic magnetohydrodynamics is 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