• magnetosphere flank instability
• reconnection kelvi

At the beginning of the space exploration, in the late Fifties, it was just becoming clear that the surroundings of our planet are not simply some "void" in which asteroids and comets fluctuate with no resistance according to Einstein's gravitational laws, but rather a complex dynamical environment rich in a variety of non-trivial electromagnetic phenomena. From that moment onwards, observations have been recorded and theorical tools have been developed to merge lots of separate pieces of evidence into the consistent frame of some widely accepted, large scale picture. In such model we distinguish two main components: a "solar wind" of protons and electrons streaming outwards from the sun's corona and expanding with our star's magnetic field, alongside with an Earth-solidal region called "magnetosphere" in which it is our planet's magnetism that characterizes particle dynamics, to a great extent keeping out coronal material and thus defining a clear "bubble" of different composition and physical properties. The dynamical evolution of the sun changes the local orientation of the stellar magnetic field as well as the solar wind flux intensity, resulting in a constant re-shaping of the magnetosphere which is usually more evident in the so-called "tail" region (pointing away from the sun) but to some degree involves the whole of this system.
While this general electromagnetic overview of our space neighbourhood is nowadays quite clear, finer details often keep eluding our full understanding even after the advent of purpose-built spacecraft and computer-simulated models, maintaining open questions and unresolved points in the overall picture. The non-triviality of many of these questions follows directly from the fact that this system displays complex three-dimensional ever-changing local configurations, evolving in an inherently non-linear fashion over a wide span of intertwined scale lengths. On the observational side, these complications make difficult to interpret in-situ collected data because no method can fully reconstruct the tridimensional non-stationary environment from streams of values recorded by satellite probes along their trajectories (that are nearly monodimensional cuts). On the theorical side, given that the system's global dimensions are enormous when compared to the smallest characteristic scale lengths, it is necessary to choose either to reproduce the whole of magnetosphere losing all processes starting as fine dynamics (even if some of them could end up in nonnegligible global effects due to the intrinsic strong nonlinearity in the system), or to replicate the whole of physics trough an impressive number of frequencies and scales, but forcedly dealing with limited portions of the environment only (and thus complicating choices as fundamental as: how much of the system to include, which boundary conditions need to be set and what kind of initial configuration is the best suited for each particular study). One of such standing issues in magnetospheric physics is the dynamical modeling of the so-called "magnetic boundary" between solar wind and magnetosphere, where the two environments interact both at material and magnetic level.
Previous investigations have clarified that the flowing of solar particles primarly perturbes the magnetic boundary on its flanks, acting in much the same way as wind blowing over the ocean excites surface waves: the charged particles stream induces the draping of the outermost magnetospheric lines in a scenario of increasing complexity as we move anti-sunward along the boundary. Such process can be understood as an example of Kelvin-Helmholtz instability, which is the phenomenon excited each time two neighboring fluids present sufficiently different mean velocities, if we assume a fluid modeling for the streaming solar wind and the Earth-solidal magnetosphere. While developing into the characteristic configuration of vortices, this phenomenon can trigger a wide range of secondary processes that in turn may noticeably influence the overall evolution. Thanks to the system's inherent strong non-linearity, the feedback of the secondary dynamics on the primary vortex structures can be effective to the point of destroying the latter, preventing an ordered development and resulting in wide turbulent mixing regions that can account for some portion of the observed cross-boundary plasma transport. The effectiveness of this material exchange is found to depend strongly on the setting of characteristic parameters in the system, as these strongly influence which kind of secondary phenomena can develop alongside with the primary perturbation: secondary Kelvin-Helmholtz processes, Rayleigh-Taylor phenomena (fluid instabilities given by density difference in a gravity-like external force field), vortex pairings (that is, close couples of vortices merge giving rise to bigger structures), formation of magnetized shocks (regions of strong gradients in which fluid quantities experience a finite discontinuity) and magnetic reconnections (the local rearranging of the magnetic line topology) are among the observed possibilities.
While such rich and variegate dynamics have stimulated a wide class of studies focusing on the two-dimensional investigation of primary Kelvin-Helmholtz vortex structures and their nonlinear development, such analysis needs nonetheless to be merged into some coherent global picture of the overall flank boundary. Three dimensional studies in which the complex latitudinal dynamics of the flank has been addressed have shown that vortex evolution can deeply influence the system also far from the equatorial region, where we have the strongest Kelvin-Helmholtz growth, due to magnetic tension effectively transporting momentum and energy along field lines. We stress that the nonlinear phenomena excited because of this mechanism on the shoulder of the primary vortices can play a key role in the context of the global evolution of the magnetospheric flanks. For example, mid-latitude reconnection events can induce a cross-boundary particle transport which has been extimated to be of the same order of magnitude of the material fluxes observed).
In this work we shall investigate the latitude-wide flank picture of the dynamics induced by Kelvin-Helmholtz vortices and their nonlinear evolution, the magnetic field displaying a sheared configuration between terrestrial and solar wind regions. To introduce the modeling of this system and present the phenomena we want to study, in chapter 2 we will review some aspects of the magnetohydrodynamic theory by which we model space plasmas (Sec. 2.1), then we shall present analytical simplified studies of the Kelvin-Helmholtz instability (Sec. 2.2) and magnetic reconnection processes (Sec. 2.3). In the second part of the thesis, chapter 3, a thorough account of results from previous studies will be outlined (Sec. 3.1). At last, we shall present our analytical and numerical study (Sec. 3.2). Trough it, plasma is modelized as two equal density and oppositely charged fluids, implemented in a simulation box representative of a portion of the magnetospheric flank thanks to appropriate boundary conditions and an initial quasi-equilibrium state. The use of an appropriate tracer highlights the draping of the boundary regions under the instability, while a proxy diagnostic points out the positioning of reconnection processes. We have performed several simulations using different magnetic field orientations, noting that the Kelvin-Helmholtz instability appears for a large variety of interplanetary magnetic field orientations, the shear between the Earth's and the sun's magnetic lines not preventing its growth but rather slowing it down. Moreover, we observe that the instability wavevectors are tilted with respect to the plane of maximal velocity shear (that in our case is the equatorial one), that the latitude band affected by Kelvin-Helmholtz processes shifts from the equatorial region into one hemisphere while secondary fluid instabilities and reconnection develop preferably in the opposite hemisphere. These inherently three-dimensional results, which a mainly qualitative analysis explains in terms of the magnetic field lines bending under different fluid advection, agree with in-situ observations relative to ideal and nonideal processes positioning. This overall scheme of ideal and nonideal processes is generally mantained even trough the late phase of nonlinear evolution, the only relevant difference being the development of a great number of doubly reconnected lines (which suggest a possibly relevant, steady material exchange process between the two environments).
Thanks to this general wide latitude inquiry on the magnetospheric flanks we have been able to build up a strong and comprehensive model of their dynamics, an overall wide picture coherent with results of several previous peculiar studies focusing on this system or parts of it. This work can be considered as the reference first step in order to reproduce the flank dynamics using "real" parameters obtained from satellite observations.