We present a theoretical overview of low-frequency waves and instabilities in collisionless, multi-component plasmas with gyrotropic (\documentclass[12pt]{minimal}
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\begin{document}$$P_\perp \not= P_\parallel $$\end{document}) thermal pressure. We show that the complete dispersion relation can be obtained in the framework of a mixed magnetohydrodynamic (MHD)-kinetic formalism, which uses the MHD mass, momentum, and induction equations, together with the kinetically corrected version of the double-adiabatic equations of state. The complete dispersion relation contains not only the three standard modes (fast, slow, and Alfvén) from double-adiabatic MHD, but also the mirror mode from kinetic theory. We examine the stability properties of these four modes, firstly in the case of a uniform medium, and secondly in the case of a stratified and rotating medium. We also discuss the connections with the quasi-interchange modes (interchange and translation) often referred to in the context of magnetospheric physics.
