Abstract. Temperature or pressure anisotropies are characteristic of space plasmas, standard magnetohydrodynamic (MHD) model for describing large-scale plasma phenomena however usually assumes isotropic pressure. In this paper we examine the characteristics of MHD waves, fire-hose and mirror instabilities in anisotropic homogeneous magnetized plasmas. The model equations are a set of gyrotropic MHD equations closed by the generalized Chew-Goldberger-Low (CGL) laws with two polytropic exponents representing various thermodynamic conditions. Both ions and electrons are allowed to have separate plasma beta, pressure anisotropy and energy equations. The properties of linear MHD waves and instability criteria are examined and numerical examples for the nonlinear evolutions of slow waves, fire-hose and mirror instabilities are shown. One significant result is that slow waves may develop not only mirror instability but also a new type of compressible fire-hose instability. Their corresponding nonlinear structures thus may exhibit anticorrelated density and magnetic field perturbations, a property used for identifying slow and mirror mode structures in the space plasma environment. The conditions for nonlinear saturation of both fire-hose and mirror instabilities are examined.

Recently the nonlinear magnetohydrodynamic ͑MHD͒ model corrected by pressure anisotropy and Hall current arising from the ion inertia is applied to construct slow or mirror mode structures and compare with the observation in the Earth's magnetosphere ͓K. Stasiewicz, Phys. Rev. Lett. 93, 125004 ͑2004͔͒. A serious issue is also raised of whether the Hall MHD model is appropriate for describing the high beta plasma and is capable of reproducing the wave dispersion found in kinetic theory ͓O. Pokhotelov et al., Phys. Rev. Lett. 95, 129501 ͑2005͔͒. In this paper we give an overview of the characteristics of linear slow mode waves and mirror instability within the context of the gyrotropic Hall MHD model closed with various energy equations. The properties examined include the phase speed, the compressibility of plasma density and magnetic field, the magnetic polarization of slow mode waves, as well as the criteria for mirror instability and their dependences on pressure anisotropy, plasma beta, propagation angle, and energy closures. The analyses help to clarify the applicability and limitation with applying the gyrotropic Hall MHD model to the observed slow or mirror mode structures.

[1] The linear theory and nonlinear evolution of parallel or classical fire hose instability previously studied based on hybrid particle simulations are examined within the framework of a gyrotropic Hall magnetohydrodynamic (MHD) model that incorporates the ion inertial effects arising from the Hall current but neglects the electron inertia in the generalized Ohm's law. Both the ion cyclotron and whistler waves become fire hose unstable for b k − b ? > 2 + l i 2 k 2 /2 with right-handed circular polarization, where l i and k are the ion inertial length and wave number, respectively, implying that the ion inertia plays a stabilizing role in parallel fire hose instability. Substantial noncoplanar components of the magnetic field and flow velocity may develop as a result of Hall current. The evolution characteristics of magnetic field fluctuations may depend on the Hall parameter h = l i /l r , where l r is the resistive length. For moderately and highly dispersive cases the instability may be purely growing in the early stage with the development of soliton-like structures with depressed magnetic field on the order of a few ion inertial lengths and become propagating with right-handed circular polarization in the late phase when the waves have inversely cascaded to large-amplitude Alfvén waves of longer wavelengths. Oscillatory and damping behavior resembling those found in the kinetic model may be reproduced for single-mode perturbations of long wavelengths with h ≤ 1. The saturated magnetic field is found to comply with the quasi-linear theory.

Pressure anisotropy may modify the characteristics of magnetohydrodynamic (MHD) waves, in particular, the slow mode wave and the corresponding shocks and discontinuities. In this study the formation of slow shocks (SSs) in anisotropic plasmas is examined by solving the gyrotropic MHD and Hall MHD equations numerically for one‐dimensional Riemann problem. The MHD shocks and discontinuities are generated by imposing a finite normal magnetic field on the Harris type current sheet with a guide magnetic By component. It is shown that anomalous SSs moving faster than the intermediate wave or with positive density‐magnetic field correlation may be generated in gyrotropic MHD and Hall MHD models. Moreover, for some parameter values SSs may exhibit upstream wave trains with right‐handed polarization in contrast with the earlier prediction that SSs shall possess downstream left‐hand polarized wave trains based on the isotropic Hall MHD theory. For the cases of By ≠ 0, SSs with increased density and decreased magnetic field followed by noncoplanar intermediate mode or rotational discontinuity (RD)‐like structures similar to the compound SS‐RD structures observed in space plasma environments may possibly form in symmetric and asymmetric current layers. The Walén relation of these anomalous RDs without the correction of pressure anisotropy may significantly be violated.