• collisionless magnetized plasma
• Kelvin-Helmholtz instability
• Finite Larmor Radius

In many situations, the dynamics observed in collisionless magnetized plasmas are characterized by an impressive number of scale lengths and
frequencies, from fluid to kinetic.
On this aspect, the terrestrial environment, from the Magnetosphere to the Solar Wind, is a
natural laboratory for the study of multi-scale, collisionless plasma dynamics.

A relevant example of multiple-scale behavior of magnetized plasmas is provided by the non-linear
evolution of the vortices generated by the Kelvin-Helmholtz (K-H) instability and the consequent
formation of a mixing layer along the flank Magnetosphere at low latitude. In this region, the K-H
instability, driven by the velocity shear between the Magnetosheath and Magnetospheric plasma, generates
fully rolled-up MHD vortices. These vortices are in turn the source of secondary magneto-fluid instabilities,
e.g. magnetic reconnection, Rayleigh-Taylor, K-H etc. These secondary instabilities compete with the classical
vortex pairing hydro-process, eventually leading to small-scale dynamics. In a simplified 2D geometry,
the formation and the early non linear evolution of a vortex of typical width of
the order of a few ion skin depth (or Larmor radius) is essentially a MHD process depending only on the
initial velocity and magnetic field. The dynamics develops in the plane defined by the velocity field and its
inhomogeneity direction and is characterized by the value of the velocity shear length and by the sound and
Alfv\'en Mach numbers.
If the value of the in-plane magnetic field parallel to the initial flow is "low" enough,
namely the in-plane Alfv\'en Mach number is greater than five, the motion of the vortices advects the magnetic
field and thus stretches and roll-up the magnetic field lines.
As a result, the large-scale MHD evolution
spontaneously generates small scales.
Therefore, although large-scale vortices are essentially MHD structures, their motion is capable
to build up sub - ion skin depth dynamics.
Moreover, for $\beta\sim1$ plasmas , the kinetic (Larmor radius) and inertial (skin
depth) lengths are approximatively the same and thus kinetic effects can't be neglected anymore.

The above discussion, even if very schematic and simplified, shows the necessity of adopting a kinetic model capable
of capturing at least the main physical processes at play.
Two main strategies can be carried out. The first one is to try to model, still in a fluid framework, the main kinetic
effects, e.g. Finite Larmor Radius effects and/or linear Landau damping.
The second strategy is to make use of fully kinetic simulations (e.g. Lagrangian (PIC) codes or Eulerian
Vlasov codes). However, in this second approach major difficulties arise, e.g. the need of an impressive number of mesh points
to resolve the multi-scale physics and the difficulty to obtain an initial condition for the distribution function
corresponding to a Vlasov equilibrium for modeling the transition between the solar wind and the magnetosphere plasma.\\

The present work is part of the first approach, e.g the fluid modeling of the main kinetic effects due to Finite Larmor Radius (FLR).
It can be roughly divided in two parts: the first one concerns fluid models and the theoretical modeling of kinetic effects, while the
second one is about the numerical implementation of the model adopted and simulations results.

After a discussion on the several fluid models one could adopt, we present how the FLR contributions are derived.
In doing this, we focus our attention on the properties of these terms under magnetic field inversion. Also, we give a more general
formulation of these terms for whatever magnetic field orientation is taken. Both these two features are not found in the existing
literature and are particularly relevant for systems in which the magnetic field directions changes during the evolution.

Then we present governing equations implemented in the numerical codes, from the original Two-Fluids (TF) version to
the FLR-Landau-fluid (FLRLF) version, developed during this thesis.

The first original contribution presented in this work is the extension to an anisotropic (i.e. gyrotropic) electronic pressure
tensor starting from an isotropic (i.e. scalar) pressure adopted in the previous model. We also derive from the pressure
tensor equation the FLR corrections then implemented in the numerical code for both ionic and electronic pressure tensor.
Ion's FLR corrections are included in the equation of motion,
while corrections for both species are included, together with gyrotropic pressures for both species, in generalized Ohm's law.
Retaining ionic FLR terms in momentum equation is relevant in the initial configuration problem and for the K-H instability
development.
Moreover, adopting gyrotropic pressures with explicit evolution equations, could be relevant for anisotropy-driven secondary
instabilities (e.g. fire-hose instability) and, together with retaining the full pressure tensors in the generalized Ohm's law, could
be relevant in magnetic reconnection and dynamo problem.

After a discussion of the plasma stability for configurations including FLR terms, we also give a brief description of
the basic numerical procedures and of boundary conditions implemented in the code.

Finally, a detailed analysis of the numerical simulations results is done. The analysis could be divided in two different parts:
the first one concerns the initialization problem when FLR terms are taken into accounts, while the second one is focused
on the K-H instability development.

About the initialization problem, we demonstrate analytically and numerically that a MHD-type configuration is no more an
equilibrium initialization when FLR corrections are retained in the equation of motion.
This is particularly important since the rapid readjustment of the large-scale system to a new equilibrium configuration modifies a
relevant parameter for the K-H instability such as the velocity shear width.
A simple fluid approach to the equilibrium configuration problem in presence of FLR terms, together with a convergent
approximation method to solve it, are developed and a set of equilibrium profiles is given. We then show that adopting these profiles
the initialization problem is solved, recovering the control over the initialization parameters.\\
Moreover, by means of these simulations (and also analytically), different implicit polytropic closure relations for the gyrotropic
pressures are proved to exist in the plane perpendicular to the magnetic field and along it.

K-H simulations are focused on the development of the instability, both in its linear and nonlinear phase.
We first point out the role of compressibility effects arising when an anisotropic description of the
pressures is adopted, even if FLR corrections are neglected.
Several sound and Alfv\'en Mach number regimes are investigated, showing the dependence of such effects by these parameters
and thus on the closure problem. Both an uniform and a density jump configurations are adopted as initial configurations.\\
After that, FLR terms and their effects are taken into accounts. We show how such terms affect the development of the instability
in the linear and nonlinear phase, introducing in the system a natural asymmetry which depends on the sign of the scalar product of
the magnetic field with the fluid vorticity, $\mathbf{\Omega}\cdot\mathbf{B}$.
In particular, we show that in the nonlinear phase also a hydrodynamic-like effects would compete with FLR effects .
This influences the vortex pairing process and could be interpreted in terms of a quantity called
generalized vorticity.

At the end, conclusions about all these effects and their relevance to the modeling of collisionless plasma dynamics are
drawn. We also discuss applications to the solar wind-magnetosphere interaction problem. A qualitative discussion of preliminary,
but impressive results concerning secondary anisotropy-driven instability (fire-hose instability) and the dynamo problem are presented.
These will be part of future works based on the application of this new model in 2D and in 3D geometry.