ABSTRACT
This chapter explains the specificity of allowing for inertial and magneto-viscosity effects, and discusses the transverse heat conductivity and the transverse viscosity. It analyses a suitable sub-set for the problem of instabilities in an inhomogeneous finite–β plasma, and describes some of the most familiar hydrodynamic approaches allowing for the terms essential for this problem. The chapter explores Grad's approach by allowing for some additional moments of the distribution function. Instabilities in an inhomogeneous finite–β plasma can be studied on the basis of two approaches: kinetic, and hydrodynamic. The different ways of truncating the series of equations lead to a closed transport equation set with different degree of accuracy suitable for describing specific phenomena. When the cyclotron frequency is large compared with the collision frequency, the multimoment equation set is significantly simplified. The gyro-relaxation effect is also important for the problem of collisional finite–β plasma instabilities.
