• Firehose instability
• pressure anysotropy
• collisionless magnetized plasma

It is well known that in many cases the dynamics of a magnetized plasma involve a very large number of scale lengths and frequencies which, even if separated by several orders of magnitudes, are intrinsically connected one each other. Such a system cannot be studied by solving numerically the whole dynamics, but it is necessary to make appropriate approximations in order to simplify the analysis and work out a model able to capture the main features that characterize the evolution
of the system. Therefore, a suitable fluid modeling, capable of including the main kinetic effects, is of primary importance for the description of space and laboratory plasmas. Moreover, space and laboratory plasma are characterized by the presence of a background magnetic field which introduces conditions for a more complex dynamics, globally modifying the symmetry of the system. From a theoretical point of view, the set of fluid equations cannot be closed by means of the usual adiabatic relation on the pressure term, because of the different behaviour along or perpendicular to the magnetic field. Therefore, the pressure tensor cannot be longer assumed as isotropic.
An appropriate alternative is to adopt a model based on the double adiabatic invariants of particle motion. This theory, proposed by Chew, Goldberger and Low (CGL), gives two conservative equations for the parallel and the perpendicular pressures, allowing the system to follow a different evolution for each component. Depending if the level of pressure anisotropy overcomes some given threshold, the
magnetized plasma becomes unstable under two important anisotropy instabilities: the firehose instability (FHI) and the mirror instability (MI). These instabilities tend to compensate the pressure unbalance changing the equilibrium configuration of the system.
The development of a pressure anisotropy is a relevant process which may be involved in many problems of space and laboratory plasmas. The phenomenon analysed in this work is the non-linear evolution of vortices generated by the Kelvin-Helmholtz instability (KHI), which occurs in the solar wind-magnetosphere interaction. Multi-spacecraft measurements have provided unambiguous evidence for rolled-up vortices as well as observed boundary layer features resulting from the suggested KH instability. This important mechanism is supposed to be the best candidate for the formation of a mixing layer along the low latitude flanks of the magnetosphere during the northwards conditions (when the solar-wind field and geomagnetic field are parallel), providing an efficient way for the solar wind plasma to enter the magnetospheric region. Indeed, the magnetic field line reconnection, which is the dominant process in the southwards conditions, cannot explain experimental data in the opposite northwards configuration, when the mixing process is larger than expected. The vortex motion is in turn the source of secondary magneto-fluid instabilities, e.g. magnetic reconnection, KH, Rayleigh-Taylor. The characteristic time scales of these secondary instabilities have a crucial role in the competition with the vortex pairing hydrodynamic process, driving a disruptive process inside and outside the vortices. During this non-linear dynamics, the firehose instability may have an important role in the evolution of the magnetic field configuration, considerably increasing the small in-plane component and bringing the system in a more complex turbolent state.
The present work has the purpose to study the development of the pressure anisotropy during the formation of vortex structures driven by an initial shear velocity configuration and their non-linear evolution. The simplified 2D dynamics is defined by the velocity field and the perpendicular inhomogeneous direction. The shear velocity length and the magnetofast Mach number characterize the vortex motion. Several initial configurations, concerning the subsonic and supersonic regimes and the presence of a density gradient, have been investigated during the linear KHI development and during the evolution of its non-linear stage. More attention has been given to the compressibility property connected to the pressure anisotropy evolution. In particular, it has been observed that in the supersonic regime the pressure anisotropy is large enough to be able to drive the FH instability. Moreover, its linear growth can occur on a much smaller time-scale compared to the vortex dynamics, competiting with the pairing and the secondary instabilities processes. Finally, in this work we have also started to consider the full 3D system. In particular, we have focused our attention on the development of the FH instability. The preliminary work is limited to the analysis of the linear growth rate of the FH instability assuming as initial state several anisotropy conditions.