General formulation for magnetohydrodynamic wave propagation, fire-hose, and mirror instabilities in Harris-type current sheets

Harris-type current sheets with the magnetic field model of B ⇀ = B x (z) x + B y (z) y have many important applications to space, astrophysical, and laboratory plasmas for which the temperature or pressure usually exhibits the gyrotropic form of p ↔ = p ∥ bb + p (I ↔ - bb). Here, p ∥ and p are, respectively, to be the pressure component along and perpendicular to the local magnetic field, b= B → / B. This study presents the general formulation for magnetohydrodynamic (MHD) wave propagation, fire-hose, and mirror instabilities in general Harris-type current sheets. The wave equations are expressed in terms of the four MHD characteristic speeds of fast, intermediate, slow, and cusp waves, and in the local (k ∥, k, z) coordinates. Here, k ∥ and k are, respectively, to be the wave vector along and perpendicular to the local magnetic field. The parameter regimes for the existence of discrete and resonant modes are identified, which may become unstable at the local fire-hose and mirror instability thresholds. Numerical solutions for discrete eigenmodes are shown for stable and unstable cases. The results have important implications for the anomalous heating and stability of thin current sheets.