Theoretical approaches to low-frequency magnetized turbulence in collisionless and weakly collisional astrophysical plasmas are reviewed. The proper starting point for an analytical description of these plasmas is kinetic theory, not fluid equations. The anisotropy of the turbulence is used to systematically derive a series of reduced analytical models. Above the ion gyroscale, it is shown rigourously that the Alfvén waves decouple from the electron-density and magnetic-field-strength fluctuations and satisfy the reduced MHD equations. The density and field-strength fluctuations (slow waves and the entropy mode in the fluid limit), determined kinetically, are passively mixed by the Alfvén waves. The resulting hybrid fluid-kinetic description of the low-frequency turbulence is valid independently of collisionality. Below the ion gyroscale, the turbulent cascade is partially converted into a cascade of kinetic Alfvén waves, damped at the electron gyroscale. This cascade is described by a pair of fluid-like equations, which are a reduced version of the electron MHD. The development of these theoretical models is motivated by observations of the turbulence in the solar wind and interstellar medium. In the latter case, the turbulence is spatially inhomogeneous and the anisotropic Alfvénic turbulence in the presence of a strong mean field may coexist with isotropic MHD turbulence that has no mean field.
Three-dimensional computation of drift Alfvén turbulence