Proton resonant firehose instability: Temperature anisotropy and fluctuating field constraints

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 = Sp/β‖pαp over 1 < β‖p ≤ 10 where and Bo is the background magnetic field. Here Sp and αp are fitting parameters and αp ≃ 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 αp ≃ 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 SB and αB 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.