Kinetic Simulations of Proton Mirror Instability: Phase Relations and Thermodynamics

Mirror-mode waves driven by the large temperature anisotropy of T _⊥/T _∣∣ > 1 have been widely observed in the solar wind, planetary magnetosheaths, heliosheath, etc. Recent studies have shown that the phase relations and thermodynamics of the mirror waves observed in the terrestrial magnetosheath may well be interpreted by the linear mixed kinetic-MHD theory of proton mirror instability. In particular, the energy laws possess the form of double-polytropic closures with the thermodynamic exponents being functions of β _⊥,∣∣ = p _⊥,∣∣/(B ²/2μ ₀). In this study, we examine the time evolution of proton mirror instability based on the hybrid particle simulations for twenty sets of β _⊥,∣∣ values. Quantitative comparisons between the kinetic simulations, linear Vlasov theory, and observations are made in terms of the growth rates, phase relations, thermodynamic conditions, etc., which show high agreements. In particular, the dependences of various compressibility and thermodynamic exponents on β _⊥,∣∣ are confirmed by the kinetic simulations, which show that the polytropic exponents are in the ranges of γ _⊥ = 0.64 ± 0.21, and γ _∣∣ = 1.07 ± 0.12 consistent with the theoretical predictions and mirror observations of γ _⊥ < 1 and γ _∣∣ ≳ 1. It is shown that the observed features, including the various perturbations and wavelengths, may indeed be reproduced by the nonlinear simulations. The saturated temperature anisotropy β _⊥/β _∣∣ and plasma β _∣∣ show an anticorrelation, which may well be fitted by the modified mirror instability threshold of γ ∣ ∣ β ∣ ∣ = β ⊥ 2 / 2 + γ ⊥ β ⊥ with γ _⊥ ≈ 0.8, γ _∣∣ ≈ 1.3, and the saturated magnetic field of δ B/B ≈ 0.26 ∼ 0.97 increases with increasing values of β ≈ 1.6 ∼ 8.3.