A guiding centre Vlasov equation and its application to Alfvén waves

Guiding centre approximations are used to derive the dielectric tensor of a collisionless plasma. This approximate dielectric tensor is used to obtain the dispersion relation of Alfvén waves in a warm plasma. In a ‘low/ β’ equilibrium plasma Alfvén waves are shown to suffer considerable Landau damping if the propagation vector is almost perpendicular to the magnetic field. In a non- equilibrium plasma Alfvén waves can be generated by ‘negative Landau damping’ even if β is low. For sufficiently high β the well-known ‘garden hose’ instability can occur and is then probably dominant. The importance of these two instabilities in the magnetosphere and in the solar wind is discussed.