Instability of Nonmonotone Magnetic Equilibria of the Relativistic Vlasov-Maxwell System

We consider the question of linear instability of an equilibrium of the Relativistic Vlasov-Maxwell (RVM) System that has a strong magnetic field. Standard instability results deal with systems where there are fewer particles with higher energies. In this paper we extend those results to the class of equilibria for which the number of particles does not depend monotonically on the energy. Without the standard sign assumptions, the analysis becomes significantly more involved.Comment: 46 page

On the Einstein-Vlasov system: Stationary Solutions and Small Data Solutions with Charged and Massless Particles

The Vlasov matter model describes an ensemble of collisionless particles moving through space-time. These particles interact via the gravitational field which they create collectively. In the framework of General Relativity this gravitational field is described by space-time curvature. Mathematically the situation is captured by the Einstein-Vlasov system. If the particles are charged an electro-magnetic field is created as well and the Maxwell equations are coupled to the system in addition. In astrophysics Vlasov matter is widely used to describe galaxies, globular clusters or galaxy clusters. Also in cosmology or plasma physics the Vlasov matter model plays an important role.In this thesis a collection of results on the Einstein-Vlasov system is presented. The Papers I to IV are concerned with stationary solutions and the Papers V and VI contain stability results for Minkowski space-time (the trivial solution of Einstein\u27s field equations describing an empty, flat space-time), i.e.~global existence results for the time evolution problem with small initial data.In Paper I, spherically symmetric, static solutions of the Einstein-Vlasov system with massless particles are constructed. These solutions constitute very thin and highly dense shells of matter with a vacuum region at the center. One can think of these shells as highly energetic, bent light which keeps itself together through the strong gravitational field created by itself. In Paper II, charged particles are considered and the existence of spherically symmetric, static solutions of the Einstein-Vlasov-Maxwell system is proven. It is possible to obtain the large variety of different spherically symmetric, static solutions that are known in the uncharged case, as for example balls, shells and multi-shells. Paper III is concerned with isotropic solutions, i.e.~solutions where the momenta are equally distributed among the particles. In this case Vlasov matter resembles a perfect fluid in many respects. It is shown that a uniqueness result for perfect fluids can be applied to Vlasov matter. This implies that every isotropic, static solution is uniquely determined by the surface potential and in particular spherically symmetric, if its overall pressure is not too high. In Paper IV solutions are constructed where the momenta are not equally distributed among the particles. These solutions have preferred axes of rotation or even an overall angular momentum. This way axially symmetric (but not spherically symmetric), stationary solutions of the Einstein-Vlasov-Maxwell system are obtained.In Paper V, exploiting the convenient conformal invariance properties of massless Vlasov matter, this matter model is integrated into the framework of the conformal Einstein field equations. In this framework, via a conformal rescaling, the physical space-time, which might be a perturbation of Minkowski space-time or de Sitter space-time, is identified with a compact portion of the Einstein-cylinder (or perturbations thereof). This way global Cauchy problems are turned into local Cauchy problems for which methods to obtain local existence are available. A semi-global stability result for Minkowski space-time and a global stability result for de Sitter space-time is obtained this way. In Paper VI the stability of Minkowski space-time for perturbations with massless Vlasov matter is proved with a completely different method, the vector field method for relativistic transport equations. Thereby an asymptotic stability result with very weak assumptions on the initial data is obtained, in particular no compact support assumptions of any kind are necessary for the initial data

Using Synthetic Spacecraft Data to Interpret Compressible Fluctuations in Solar Wind Turbulence

Kinetic plasma theory is used to generate synthetic spacecraft data to analyze and interpret the compressible fluctuations in the inertial range of solar wind turbulence. The kinetic counterparts of the three familiar linear MHD wave modes---the fast, Alfven, and slow waves---are identified and the properties of the density-parallel magnetic field correlation for these kinetic wave modes is presented. The construction of synthetic spacecraft data, based on the quasi-linear premise---that some characteristics of magnetized plasma turbulence can be usefully modeled as a collection of randomly phased, linear wave modes---is described in detail. Theoretical predictions of the density-parallel magnetic field correlation based on MHD and Vlasov-Maxwell linear eigenfunctions are presented and compared to the observational determination of this correlation based on 10 years of Wind spacecraft data. It is demonstrated that MHD theory is inadequate to describe the compressible turbulent fluctuations and that the observed density-parallel magnetic field correlation is consistent with a statistically negligible kinetic fast wave energy contribution for the large sample used in this study. A model of the solar wind inertial range fluctuations is proposed comprised of a mixture of a critically balanced distribution of incompressible Alfvenic fluctuations and a critically balanced or more anisotropic than critical balance distribution of compressible slow wave fluctuations. These results imply that there is little or no transfer of large scale turbulent energy through the inertial range down to whistler waves at small scales.Comment: Accepted to Astrophysical Journal. 28 pages, 7 figure

Axion-induced oscillations of cooperative electric field in a cosmic magneto-active plasma

We consider one cosmological application of an axionic extension of the Maxwell-Vlasov theory, which describes axionically induced oscillatory regime in the state of global magnetic field evolving in the anisotropic expanding (early) universe. We show that the cooperative electric field in the relativistic plasma, being coupled to the pseudoscalar (axion) and global magnetic fields, plays the role of a regulator in this three-level system; in particular, the cooperative (Vlasov) electric field converts the regime of anomalous growth of the pseudoscalar field, caused by the axion-photon coupling at the inflationary epoch of the universe expansion, into an oscillatory regime with finite density of relic axions. We analyze solutions to the dispersion equations for the axionically induced cooperative oscillations of the electric field in the relativistic plasma.Comment: 7 pages, misprints correcte

Verification of Gyrokinetic codes: theoretical background and applications

In fusion plasmas the strong magnetic field allows the fast gyro-motion to be systematically removed from the description of the dynamics, resulting in a considerable model simplification and gain of computational time. Nowadays, the gyrokinetic (GK) codes play a major role in the understanding of the development and the saturation of turbulence and in the prediction of the subsequent transport. Naturally, these codes require thorough verification and validation. Here we present a new and generic theoretical framework and specific numerical applications to test the faithfulness of the implemented models to theory and to verify the domain of applicability of existing GK codes. For a sound verification process, the underlying theoretical GK model and the numerical scheme must be considered at the same time, which has rarely been done and therefore makes this approach pioneering. At the analytical level, the main novelty consists in using advanced mathematical tools such as variational formulation of dynamics for systematization of basic GK code's equations to access the limits of their applicability. The verification of numerical scheme is proposed via the benchmark effort. In this work, specific examples of code verification are presented for two GK codes: the multi-species electromagnetic ORB5 (PIC) and the radially global version of GENE (Eulerian). The proposed methodology can be applied to any existing GK code. We establish a hierarchy of reduced GK Vlasov-Maxwell equations implemented in the ORB5 and GENE codes using the Lagrangian variational formulation. At the computational level, detailed verifications of global electromagnetic test cases developed from the CYCLONE Base Case are considered, including a parametric $\beta$-scan covering the transition from ITG to KBM and the spectral properties at the nominal $\beta$ value.Comment: 16 pages, 2 Figures, APS DPP 2016 invited pape

Detailed Analysis of Filamentary Structure in the Weibel Instability

We present results of a 2D3V kinetic Vlasov simulation of the Weibel instability. The kinetic Vlasov simulation allows us to investigate the velocity distribution of dilute plasmas, in which the effect of collisions between particles is negligible, and has the advantage that the accuracy of the calculated velocity distribution does not depend on the density of plasmas at each point in the physical space. We succeed in reproducing some features of the Weibel instability shown by other simulations, for example, the exponentially growing phase, the saturation of the magnetic field strength, the formation of filamentary structure, and the coalescence of the filaments. Especially, we concentrate on the behavior of the filaments after the saturation of the magnetic field strength and find that there is a kind of quasi-equilibrium states before the coalescence occurs. Furthermore, it is found that an analytical solution for stationary states of the 2D3V Vlasov-Maxwell system can reproduce some dominant features of the quasi-equilibrium, e.g, the configuration of the magnetic field and the velocity distribution at each point. The analytical expression could give a plausible model for the transition layer of a collisionless shock where a strong magnetic field generated by the Weibel instability provides an effective dissipation process instead of collisions between particles.Comment: 9 pages, 11figures, to appear in Ap

High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications

A high order, deterministic direct numerical method is proposed for the nonrelativistic $2D_{\bf x} \times 3D_{\bf v}$ Vlasov-Maxwell system, coupled with Fokker-Planck-Landau type operators. Such a system is devoted to the modelling of electronic transport and energy deposition in the general frame of Inertial Confinement Fusion applications. It describes the kinetics of plasma physics in the nonlocal thermodynamic equilibrium regime. Strong numerical constraints lead us to develop specific methods and approaches for validation, that might be used in other fields where couplings between equations, multiscale physics, and high dimensionality are involved. Parallelisation (MPI communication standard) and fast algorithms such as the multigrid method are employed, that make this direct approach be computationally affordable for simulations of hundreds of picoseconds, when dealing with configurations that present five dimensions in phase space