The aim of this summer school is to give a global view of tools and needs, in modeling and numerical methods, for the different aspects of ionized flows : kinetic models, fluid models, waves-plasmas interaction and plasma instabilities. Lecture material is made available here for the summer school participants.

### Invited lecturers

### Abstracts

### Tentative schedule

- James D. Callen (University of Wisconsin): Fluid and transport modeling of plasmas
- Stéphane Colombi (IAP): Numerical simulations of large-scale structures in the Universe
- Xavier Garbet (CEA - IRFM): Modelling of Magnetic Fusion Plasmas: from fluid to kinetic description
- Laurent Masse (CEA - DIF): Modeling for Inertial Conﬁnement Fusion
- Jean-Marcel Rax (Ecole Polytechnique): Confinement and Control in Plasmas
- Eric Sonnendrücker (IPP - TUM): Kinetic modeling and approximation
- Vladimir Tikhonchuk (CELIA - Bordeaux 1): Laser plasma interaction in high energy density physics

- James D. Callen: Fluid and transport modeling of plasmas Videos of the lectures
- Xavier Garbet: Modelling of Magnetic Fusion Plasmas: from fluid to kinetic description Video of the lecture
- Jean-Marcel Rax: Confinement and Control in Plasmas
- Eric Sonnendrücker: Kinetic modeling and approximation
- Vladimir Tikhonchuk: Laser plasma interaction in high energy density physics

[Excerpt from the preface of the lecture notes] These lecture notes discuss some basic plasma equations relevant for inertial fusion. However, the focus is on developing equations and models for magnetically-confined fusion plasmas on a hierarchy of time scales that extend to the long time scales for transport of plasma across the confining magnetic field -- e.g., in ITER (http://www.iter.org) the desired plasma confinement time is a few seconds. The present fusion grand challenge is to develop a “predictive capability” for D-T burning plasmas in ITER. Pedagogically, these lecture notes begin from a microscopic particle-based description. The ensemble-averaged plasma kinetic equation is then developed from particle motions and interactions. Next, fluid moment descriptions are developed on successively longer time scales. The emphasis is on developing models "the plasma can understand." That is, the focus is on models whose key properties and predictions have been experimentally validated to some degree. In concert with the modern trend in the physical sciences, SI or mks units are used throughout these lecture notes.

This lecture will present a short overview on kinetic MHD. The advantages and drawbacks of kinetic versus fluid modelling will be summarized. Various techniques to implement kinetic effects in the fluid description will be introduced with increasing complexity: bi-fluid effects, gyroaverage fields, Landau closures. Hybrid formulations, which combine fluid and kinetic approaches will be presented. It will be shown that these formulations raise several difficulties, including inconsistent ordering and choice of representation. The non linear dynamics of an internal kink mode in a tokamak will be used as a test bed for the various formulations. It will be shown that bi-fluid effects can explain to some extent fast plasma relaxations (reconnection), but cannot address kinetic instabilities due to energetic particles. Some results of hybrid codes will be shown. Recent developments and perspectives will be given in conclusion.

Le contrôle d’un plasma, industriel ou thermonucléaire, nécessite l’utilisation de champs électromagnétiques pour : (i) confiner les particules qui présentent une tendance à occuper tous le volume d’espace des phases disponible (2nd principe de la thermodynamique) et (ii) chauffer ces particules, car le contenu énergétique se dissipe sur tous les degrés de liberté de l’environnement (2nd de la thermodynamique). Le confinement est obtenu à travers l’obtention d’un régime (i) adiabatique d’interaction particules/champs et le chauffage à travers les régimes (ii) résonnant et chaotique d’interaction champs/particules. C’est cette vision dynamique : adiabaticité, résonance et chaos, qui constituera le fil directeur de cette présentation de la physique du confinement et du contrôle des plasmas. Physique des Plasmas, JM Rax, Editions DUNOD, Collection Sciences SUP, 426 pages, 3e tirage Septembre 2013. Physique des Tokamaks, JM Rax, Editions de l’Ecole Polytechnique, Collection Physique, 423 pages, 1e tirage Février 2011.

The lecture will be devoted to kinetic plasma models and their numerical approximation. We will start by presenting the hierarchy of models, from microscopic to fluid, used for modelling plasmas and then focus on kinetic models. Some specificities linked to magnetic fusion plasmas, which are confined in a large magnetic field will be emphasised and the gyrokinetic approximation will be recalled. We will also introduced some simple collision operators like the Krook or BGK operator and Fokker-Planck operator. Then we will present two of the most used numerical methods, namely the semi-Lagrangian method and the Particle In Cell method. The semi-Lagrangian method relies on three three numerical tools: operator splitting, backward computation of characteristics and interpolation. We will recall the different methods that are classically used for each step. The Particle In Cell method is a Monte Carlo approximation of the kinetic equation coupled to a grid based approximation of the electromagnetic field. We will recall the principle of Monte Carlo simulation and the main methods to draw particles from a given probability density function. Then we will present the particle advance and the coupling with the field solver. We will also focus and two important variance reduction techniques: importance sampling, which leads to the weighted PIC method, and control variates, which leads to the so-called delta f method. Finally the Monte Carlo approximation of the Fokker-Planck equation leading to the solution of the Langevin stochastic differential equation.

This course provides the basic knowledge in the physics of laser plasma interaction and applications in the high energy density physics, which is characterized by extremely high pressures and temperatures of the matter. An intense laser radiation can create on a target the pressures exceeding 100 Mbars and the temperatures exceeding 107 K. These conditions are relevant for excitation of the nuclear fusion reactions, modeling the astrophysical conditions in laboratory or material processing. Multi-scale numerical modeling is indispensable part of the high energy density physics. Three main issues will be addressed in this course: the laser beam propagation and absorption in hot and expanding plasmas; the energy transport from the absorption region to denser plasma and formation of the ablation flow; the shock formation and target acceleration under the effect of laser radiation. We limit ourselves to the domain of non-relativistic laser intensities which are of major interest for fusion and astrophysical applications. A particular attention is given to the physics of linear and non-linear laser energy absorption and the energy transport with thermal-like and beam-like electron energy distributions. Several applications in the inertial confinement fusion and laboratory astrophysics will be also presented.

Lundi 21 | Mardi 22 | Mercredi 23 | Jeudi 24 | Vendredi 25 | |
---|---|---|---|---|---|

09:00-10:30 | V. Tikhonchuk (laser plasma interaction) |
V. Tikhonchuk (laser plasma interaction) |
L. Masse (ICF modeling) |
V. Tikhonchuk (laser plasma interaction) |
E. Sonnendrücker (kinetic methods) |

10:30-10:50 | Coffee break |
Coffee break |
Coffee break |
Coffee break |
Coffee break |

10:50-12:20 | J.-M. Rax (confinement and control) |
E. Sonnendrücker (kinetic methods) |
X. Garbet (MCF modeling) |
E. Sonnendrücker (kinetic methods) |
E. Sonnendrücker (kinetic methods) |

12:30-14:00 | Lunch |
Lunch |
Lunch |
Lunch |
Lunch |

14:00-16:00 | J.D. Callen (fluid and transport modeling) |
J.D. Callen (fluid and transport modeling) |
S. Colombi (large-scale structures) |
J.D. Callen (fluid and transport modeling) |
J.D. Callen (fluid and transport modeling) |

16:00-16:30 | Coffee break |
Coffee break |
Coffee break |
Coffee break |
Coffee break |

16:30-18:00 | Research projects presentation |
J.-M. Rax (confinement and control) |
J.-M. Rax (confinement and control) |
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18:00-19:30 | Free time / soccer |
Free time / soccer |
Free time / soccer |
Free time / soccer |
Free time / soccer |

19:30-20:30 | Dinner |
Dinner |
Dinner |
Dinner |
Dinner |