Section: Scientific Foundations

Charged Particles

Modeling of Plasma Confinement

Plasmas, the fourth state of the matter, play an important role in many branches of physics, such as thermonuclear fusion research, astrophysics and space physics. A plasma is a (partially) ionized gas where charged particles interact via electromagnetic fields. Since the announcement of the creation of the experimental fusion reactor ITER, and with the progress on the ICF(Inertial Confinement Fusion) program, plasmas and their modelling got a renewed interest.

The nuclear fusion mechanisms result from the strong confinement of charged particles, either by inertial confinement (nuclear fusion reactions are initiated by heating and compressing a target - a pellet that most often contains deuterium and tritium - by the use of intense laser or ion beams) or by the - more promising - magnetic fusion confinement. The tokamaks are experimental devices which produce a toroidal magnetic field for confining a plasma.

The description of these phenomena is extremely complex and leads to delicate problems in mathematical analysis and numerical simulation. Actually, plasmas may be described with various levels of detail. The simplest possibility is to treat the plasma as a single fluid governed by the Navier Stokes Equations. A more general description is the two-fluid picture, where the ions and electrons are considered to be distinct. If electric or magnetic fields are present, then the Maxwell equations will be needed to describe them. The coupling of the description of a conductive fluid to electromagnetic fields is known generally as magnetohydrodynamics, or simply MHD.

For some cases the fluid description is not sufficient. Then, the kinetic models may become useful. Kinetic models include information on distortions of the velocity distribution functions with respect to a Maxwell-Boltzmann distribution. This may be important when currents flow, when waves are involved, or when gradients are very steep.

The main mathematical difficulties are therefore linked to the conjunction of the following elements

• These two types of models are strongly nonlinear.

• The unknowns depend on the time and space variables and, in the case of kinetic models, also on the velocity variables. Therefore, we can be led to work with variables of 1 + 3 + 3 dimensions.

• There exist many very different scales (time scale, characteristic length, etc).

The numerical resolution of a complete system of equations, with meshes adapted to the lower scales, leads to prohibitive computational costs both in terms of memory and time. The derivation of new reduced models, corresponding to relevant asymptotic regimes (high magnetic field for example), is therefore a crucial issue. Moreover, very serious efforts must be done on the numerical methods that are used in order to reproduce the typical phenomena. This work depends on the one hand on seriously thinking over the models, the physical parameters, their typical respective scales, and on the other hand over some arguments of asymptotic analysis, which can particularly call on deterministic or random homogeneization.

Spacecraft Environment

Satellites in geostationary and low Earth orbits naturally evolve in a plasma. This ionized environment induces some perturbations which may lead to many kind of faults and to the partial or complete loss of a mission. The satellites are covered by dielectric coatings in order to protect them against thermal radiations. Electrons and ions species of the space plasma interact with the external surfaces of the satellite and modify their electrostatic charges. This effect produces potential differences between the satellite surfaces and its electric mass. When the electric field exceeds a certain level, an electrostatic discharge appears. This electric current pulse is able to disrupt the equipments, to damage the external surfaces and even to destroy some electronic components. The plasma may also be created by an other source : the electric thrusters. This new propulsion device uses the electric energy supplied by solar arrays to speed up charged species. It is more and more used in satellite industry and has preference over the classical chemical propulsion. Indeed, the latter needs a very large amount of propellant inducing an expensive rocket launch. On the one hand, the electric thrusters allow to significantly reduce the satellite weight. On the other hand, it is necessary to understand their potential impacts on the other systems of the satellite.

This line of research, which continues former works of the team CAIMAN at Sophia Antipolis, was the object of a strong collaboration with the Department Research and Technology of the company Thales Alenia Space. N. Vauchelet proposed several evolutions for the SPARCS code, including parallel procedures. A comparizon of different numerical schemes (Finite Volume, Semi-Lagrangian, Back Trajectory) to treat the Vlasov equations of spacecraft charging has been discussed in details. Moreover, the PhD of S. Borghol shows that at LEO or PEO altitudes(Low Earth Orbit and Polar Earth Orbit respectively) (instead of the standard GEO(Geostationary orbit) framework) the Vlasov equations can be replaced by its hydrodynamic limit. Then the question was: how to define boundary conditions for the macroscopic quantities in order to reproduce phenomena due to the kinetic ones? A partial answer to this question was given for the BGK equation and its hydrodynamic limit: the Euler system.

Effective Energy Dissipation Models for Charged Particles

In models of charge transport, say transport of electrons, a phenomenological friction force is generally introduced, which is proportional to the velocity $v$. Our idea is to go back to a more microscopic framework, with a description of the energy exchanges between the electrons and the surrounding medium. In turn, the dissipation of energy by the medium will lead to an effective friction force. The first contributions only model the transport of a unique particle, and we aim at considering now a plasma, through a statistical description. This yields a Vlasov-Poisson-like model. (More precisely, the kinetic equation is coupled to a finite, or infinite, set of oscillators.) This program requires efforts in modelling and analysis, but the questions are also really challenging for numerics, due, on the one hand, to the large number of degrees of freedom involved in the equation, and on the other hand, to the presence of stiff terms. In this way, we expect to be able to shed light on the range of validity of the Ohm law. Similar considerations also apply for heat transport and the derivation of the Fourier law.