Hybrid-kinetic numerical simulations of firehose and mirror instabilities in a collisionless plasma are performed in which pressure anisotropy is driven as the magnetic field is changed by a persistent linear shear $S$. For a decreasing field, it is
found that mostly oblique firehose fluctuations grow at ion Larmor scales and saturate with energies $sim$$S^{1/2}$; the pressure anisotropy is pinned at the stability threshold by particle scattering off microscale fluctuations. In contrast, nonlinear mirror fluctuations are large compared to the ion Larmor scale and grow secularly in time; marginality is maintained by an increasing population of resonant particles trapped in magnetic mirrors. After one shear time, saturated order-unity magnetic mirrors are formed and particles scatter off their sharp edges. Both instabilities drive sub-ion-Larmor--scale fluctuations, which appear to be kinetic-Alfv{e}n-wave turbulence. Our results impact theories of momentum and heat transport in astrophysical and space plasmas, in which the stretching of a magnetic field by shear is a generic process.
In weakly collisional extragalactic plasmas such as the intracluster medium, viscous stress and the rate of change of the magnetic-field strength are proportional to the local pressure anisotropy, so subject to constraints imposed by the pressure-ani
sotropy-driven mirror and firehose instabilities and controlled by the local instantaneous plasma beta. The dynamics of such plasmas is dramatically different from a conventional MHD fluid. The plasma is expected to stay locally in a marginal state with respect to the instabilities, but how it does this is an open question. Two models of magnetic-field evolution are investigated. In the first, marginality is achieved via suppression of the rate of change of the field. In the second, the instabilities give rise to anomalous collisionality, reducing pressure anisotropy to marginal - at the same time decreasing viscosity and so increasing the turbulent rate of strain. Implications of these models are studied in a simplified 0D setting. In the first model, the field grows explosively but on a time scale that scales with initial beta, while in the second, dynamical field strength can be reached in one large-scale turbulence turn-over time regardless of the initial seed. Both models produce very intermittent fields. Both also suffer from strong constraints on their applicability: for typical cluster-core conditions, scale separation between the fluid motions and the microscale fluctuations breaks down at beta~10^5-10^4. At larger beta (weaker fields), a fully collisionless plasma dynamo theory is needed in order to justify the growth of the field from a tiny primordial seed. However, the models discussed here are appropriate for studying the structure of the currently observed field as well as large-scale dynamics and thermodynamics of the magnetized ICM or similarly dilute astrophysical plasmas.
