Effects of Hall current and electron temperature anisotropy on proton fire-hose instabilities

The standard magnetohydrodynamic (MHD) theory predicts that the Alfvén wave may become fire-hose unstable for β - β > 2. In this study, we examine the proton fire-hose instability (FHI) based on the gyrotropic two-fluid model, which incorporates the ion inertial effects arising from the Hall current and electron temperature anisotropy but neglects the electron inertia in the generalized Ohm's law. The linear dispersion relation is derived and analyzed which in the long wavelength approximation, λ i k → 0 or α e = μ 0 (p e - p e) / B 2 = 1, recovers the ideal MHD model with separate temperature for ions and electrons. Here, λ i is the ion inertial length and k is the wave number. For parallel propagation, both ion cyclotron and whistler waves become propagating and growing for β - β > 2 + λ i 2 k 2 (α e - 1) 2 / 2. For oblique propagation, the necessary condition for FHI remains to be β - β > 2 and there exist one or two unstable fire-hose modes, which can be propagating and growing or purely growing. For large λ i k values, there exists no nearly parallel FHI leaving only oblique FHI and the effect of α e > 1 may greatly enhance the growth rate of parallel and oblique FHI. The magnetic field polarization of FHI may be reversed due to the sign change associated with (α e - 1) and the purely growing FHI may possess linear polarization while the propagating and growing FHI may possess right-handed or left-handed polarization.