Low-frequency Alfvén waves and parametric instabilities in fluid and kinetic plasmas

In this tutorial, a brief overview of Alfvén wave solutions in fluid and kinetic systems and parametric instabilities of Alfvén waves is provided. As a class of relaxed states in fluid systems, we revisit the Alfvén wave solution. Then, Alfvén wave solutions of the Vlasov system and accompanying relations such as the one between temperature anisotropy and the wave frequency are shown. Parametric instabilities play a substantial role in the damping of Alfvén waves when collisionless damping of low-frequency Alfvén waves through Landau/cyclotron resonance is weak. This tutorial focuses mostly on linear analysis of parametric instabilities.