• instability
• Kelvin-Helmoltz
• MHD
• numerical simulation
• plasma physics
• pressure

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.