Do There Exist Energy Closures to the Observed Mirror Waves?

Low-frequency mirror waves are frequently observed in the planetary magnetosheaths for the plasma conditions of T⊥>T|| and β≥1. This study first shows that the thermodynamic conditions of the observed mirror events may well be described by a set of double-polytropic (DP) laws first proposed by Hau and Sonnerup (1993, https://doi.org/10.1029/93gl01706), d(p⊥/ρBγ⊥-1)/dt=0 and d(p||Bγ||-1/ργ||)/dt=0, with the polytropic values being in the ranges of γ⊥=0.67∼0.94 and γ||=0.99∼1.17. The physical basis of the empirical thermodynamic relations is then examined based on the mixed Vlasov-MHD formulation for low-frequency linear mirror instability (Ferrière & Andre, 2002, https://doi.org/10.1029/2002JA009273) which shows that the theoretical energy equations possess the DP forms with γ⊥ and γ|| being functions of β⊥,||=p⊥,||/(B2/2μ0); in particular, for mirror unstable cases γ⊥<1 and γ||>1. The theoretical predictions are found to be in high agreements with the empirical polytropic values with the average errors of ∼6.5%.