Proton resonant firehose instability: Temperature anisotropy and fluctuating field constraints

Abstract

The electromagnetic proton firehose instability may grow in a plasma if the proton velocity distribution is approximately bi‐Maxwellian and T ‖ p > T ⊥ p , where the directional subscripts denote directions relative to the background magnetic field. Linear Vlasov dispersion theory in a homogeneous electron‐proton plasma implies an instability threshold condition at constant maximum growth rate 1 − T ⊥ p / T ‖ p = S p /β ‖ p α p over 1 < β ‖ p ≤ 10 where and B o is the background magnetic field. Here S p and α p are fitting parameters and α p ≃ 0.7. One‐ and two‐dimensional initial value hybrid simulations of this growing mode are carried out under proton cyclotron resonant conditions in a homogeneous plasma on the initial domain 2 ≲ β ‖ p ≤ 100. The two‐dimensional simulations show that enhanced fluctuations from this instability impose a bound on the proton temperature anisotropy of the form of the above equation with the fluid theory result α p ≃ 1.0. On this domain both one‐ and two‐dimensional simulations yield a new form for the upper bound on the fluctuating field energy density from the proton resonant firehose instability where S B and α B are empirical parameters which are functions of the initial growth rate. This logarithmic behavior is qualitatively different from a fluid theory prediction and, like the anisotropy bound, should be subject to observational verification in any sufficiently homogeneous plasma in which the proton velocity distribution is approximately bi‐Maxwellian.