Proton temperature anisotropy upper bound

The electromagnetic proton cyclotron instability and the mirror instability are driven by the proton temperature anisotropy T-perpendicular to p/T-parallel to p > 1, where perpendicular to and parallel to denote directions relative to the background magnetic field. Linear theory and one-dimensional hybrid simulations imply that the former mode grows more rapidly over 0.05 less than or equal to beta(parallel to p) less than or equal to 5 and that wave-particle scattering by its enhanced fluctuations imposes an upper bound on the temperature anisotropy of the form T-perpendicular to p/T-parallel to p - 1 = S-p/beta(parallel to p)alpha(p) where beta(parallel to p) = 8 pi n(p)T(parallel to p)/B-o(2) and B-o is the background magnetic field. Here S-p and alpha(p) are fitting parameters, and 0.4 less than or similar to alpha(p) less than or similar to 0.5. This paper describes results from more general two-dimensional hybrid simulations, which permit both instabilities to grow simultaneously. These simulations confirm the one-dimensional results on the initial domain 0.05 less than or equal to beta(parallel to p) similar or equal to 5; enhanced fluctuations display the properties of the proton cyclotron instability and alpha(p) similar or equal to 0.4. On this domain the two-dimensional simulations also yield an upper bound for the fluctuating field energy density of the form \delta B\(2)/B-o(2) = Sigma k\delta B-k\(2)/B-o(2) = S-B beta(parallel to p)(alpha B) with fitting parameter 0.5 less than or similar to alpha(B) less than or similar to 1. The simulations on the initial domain 10 less than or equal to beta(parallel to p) less than or equal to 100 show spectral characteristics of both instabilities and exhibit a more stringent bound on the proton anisotropy, in agreement with observations in the terrestrial magnetosheath.