Role of the Pressure Tensor on the Evolution of the Non-Linear Kelvin-Helmoltz instability

In the late Fifties the advent of space exploration revealed that most of the space in the Solar System is filled by a tenuous globally neutral plasma composed of charged particels, mostly electrons and protons interacting via electromagnetic forces. Such particles are ejected from the Solar corona and stream in the so-called solar wind, which expands throughout all our planetary system. Because of the low collisionality of a plasma and the presence of magnetic fields, each particle performs nearly a circular orbits around field lines while it flows along the magnetic field away from the Sun. The solar wind impacts the Earth and in particular its magnetic field, but the frozen in law forbids mixing between magnetospheric and solar wind plasmas. As a consequence, the Earth’s magnetic field acts as shield against the solar wind. Nevertheless, observations from satellites show evidence of solar wind plasma inside the Earth’s magnetosphere. To explain this inconsistency the phenomenon of magnetic reconnection is invoked. Indeed, reconnection is the only process able to change the magnetic field topology when, roughly speaking at least in 2D, the magnetic field inverts its direction. This is the case, in certain regions, when the Interplanetary Magnetic Field (IMF) points southward. On the other hand, when the IMF points northward, reconnection cannot occurs, but nevertheless there are injections of solar plasma recorded near the equatorial plane. A possible solution is the development of a ”chain” of Kelvin-Helmoltz vortices along the flanks of the magnetosphere developing near the equatorial plane. Indeed, the flow of the solar wind around the magnetosphere gives raise to a strong velocity gradient corresponding to a Kelvin-Helmoltz-unstable equilibrium. Once the vortices are rolled-up, it is possible to create dynamically the conditions for the development of magnetic reconnection. In this Thesis we analyze the non-linear evolution of a Kelvin-helmoltz chain of vortices by means of 2D numerical simulations in order to investigate the development of secondary instabilities which may raise on the shoulder of the primary Kelvin-Helmoltz instability vortices and evolve until they disrupt the vortices itselves. Our main objective here is to study the role of the pressure tensor on the system evolution by using different plasma models. In the literature a scalar pressure is usually assumed even if a magnetic field is present, which change the response of the plasma along and perpendicular to the field; thus a tensor form is more appropriate. We aim at comparing the classic evolution of the Kelvin-Helmoltz instability obtained with a scalar pressure with a more general model which takes into account the presence of the magnetic field and the effect of particle gyration in a fluid modeling. We have organized the Thesis as follow. In Chapter 1 an overall introduction of the subject is given. In Chapter 2 we give the basic principles of plasma physics and present the differences among the models we want to investigate. In Chapter 3 we review the sta- bility conditions for different sheared velocity profiles in order to have a theoretical model as a reference for the primary as well as secondary instabilities. In Chapter 4 we present the simulations, discuss the difference between them and investigate the development of secondary instabilties. In Chapter 5 we present the conclusions of our analysis