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-parallel to p > T-perpendicular to 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 l-T-perpendicular to p/T-parallel to p = S-p/beta(parallel to p)(alpha p) over 1 < beta(parallel to p) less than or equal to 10 where beta(parallel to p) = 8 pi n(p)T(parallel to p)/B-0(2) and B-0 is the background magnetic field. Here S-p and alpha(p) are fitting parameters and alpha(p) similar or equal to 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 less than or similar to beta(parallel to p) less than or equal to 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 alpha(p) similar or equal to 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 /delta B/(2) / B-0(2) = S-B + alpha(B)ln (beta(parallel to p)) where S-B and alpha(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.