  We use particle-in-cell (PIC) simulations to study the nonlinear evolution of
ion velocity space instabilities in an idealized problem in which a background
velocity shear continuously amplifies the magnetic field. We simulate the
astrophysically relevant regime where the shear timescale is long compared to
the ion cyclotron period, and the plasma beta is ~ 1-100. The background field
amplification in our calculation is meant to mimic processes such as turbulent
fluctuations or MHD-scale instabilities. The field amplification continuously
drives a pressure anisotropy with the perpendicular pressure larger than the
parallel pressure, and the plasma becomes unstable to the mirror and ion
cyclotron instabilities. In all cases, the nonlinear state is dominated by the
mirror instability, not the ion cyclotron instability, and the plasma pressure
anisotropy saturates near the threshold for the linear mirror instability. The
magnetic field fluctuations initially undergo exponential growth but saturate
in a secular phase in which the fluctuations grow on the same timescale as the
background magnetic field (with \delta B ~ 0.3 <B> in the secular phase). At
early times, the ion magnetic moment is well-conserved but once the fluctuation
amplitudes exceed \delta B ~ 0.1 <B>, the magnetic moment is no longer
conserved but instead changes on a timescale comparable to that of the mean
magnetic field. We discuss the implications of our results for
low-collisionality astrophysical plasmas, including the near-Earth solar wind
and low-luminosity accretion disks around black holes.
