Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation

One of the main applications in plasma physics concerns the energy production through thermo-nuclear fusion. The controlled fusion requires the confinement of the plasma into a bounded domain and for this we appeal to the magnetic confinement. Several models exist for describing the evolution of strongly magnetized plasmas. The subject matter of this paper is to provide a rigorous derivation of the guiding-center approximation in the general three dimensional setting under the action of large stationary inhomogeneous magnetic fields. The first order corrections are computed as well : electric cross field drift, magnetic gradient drift, magnetic curvature drift, etc. The mathematical analysis relies on average techniques and ergodicity

Variational approach to low-frequency kinetic-MHD in the current coupling scheme

Hybrid kinetic-MHD models describe the interaction of an MHD bulk fluid with an ensemble of hot particles, which is described by a kinetic equation. When the Vlasov description is adopted for the energetic particles, different Vlasov-MHD models have been shown to lack an exact energy balance, which was recently recovered by the introduction of non-inertial force terms in the kinetic equation. These force terms arise from fundamental approaches based on Hamiltonian and variational methods. In this work we apply Hamilton's variational principle to formulate new current-coupling kinetic-MHD models in the low-frequency approximation (i.e. large Larmor frequency limit). More particularly, we formulate current-coupling hybrid schemes, in which energetic particle dynamics are expressed in either guiding-center or gyrocenter coordinates.Comment: v3.0. 30 page

ORB5: a global electromagnetic gyrokinetic code using the PIC approach in toroidal geometry

This paper presents the current state of the global gyrokinetic code ORB5 as an update of the previous reference [Jolliet et al., Comp. Phys. Commun. 177 409 (2007)]. The ORB5 code solves the electromagnetic Vlasov-Maxwell system of equations using a PIC scheme and also includes collisions and strong flows. The code assumes multiple gyrokinetic ion species at all wavelengths for the polarization density and drift-kinetic electrons. Variants of the physical model can be selected for electrons such as assuming an adiabatic response or a ``hybrid'' model in which passing electrons are assumed adiabatic and trapped electrons are drift-kinetic. A Fourier filter as well as various control variates and noise reduction techniques enable simulations with good signal-to-noise ratios at a limited numerical cost. They are completed with different momentum and zonal flow-conserving heat sources allowing for temperature-gradient and flux-driven simulations. The code, which runs on both CPUs and GPUs, is well benchmarked against other similar codes and analytical predictions, and shows good scalability up to thousands of nodes

High-order Discretization of a Gyrokinetic Vlasov Model in Edge Plasma Geometry

We present a high-order spatial discretization of a continuum gyrokinetic Vlasov model in axisymmetric tokamak edge plasma geometries. Such models describe the phase space advection of plasma species distribution functions in the absence of collisions. The gyrokinetic model is posed in a four-dimensional phase space, upon which a grid is imposed when discretized. To mitigate the computational cost associated with high-dimensional grids, we employ a high-order discretization to reduce the grid size needed to achieve a given level of accuracy relative to lower-order methods. Strong anisotropy induced by the magnetic field motivates the use of mapped coordinate grids aligned with magnetic flux surfaces. The natural partitioning of the edge geometry by the separatrix between the closed and open field line regions leads to the consideration of multiple mapped blocks, in what is known as a mapped multiblock (MMB) approach. We describe the specialization of a more general formalism that we have developed for the construction of high-order, finite-volume discretizations on MMB grids, yielding the accurate evaluation of the gyrokinetic Vlasov operator, the metric factors resulting from the MMB coordinate mappings, and the interaction of blocks at adjacent boundaries. Our conservative formulation of the gyrokinetic Vlasov model incorporates the fact that the phase space velocity has zero divergence, which must be preserved discretely to avoid truncation error accumulation. We describe an approach for the discrete evaluation of the gyrokinetic phase space velocity that preserves the divergence-free property to machine precision

Gyrokinetics from variational averaging: existence and error bounds

The gyrokinetic paradigm in the long wavelength regime is reviewed from the perspective of variational averaging (VA). The VA-method represents a third pillar for averaging kinetic equations with highly-oscillatory characteristics, besides classical averaging or Chapman-Enskog expansions. VA operates on the level of the Lagrangian function and preserves the Hamiltonian structure of the characteristics at all orders. We discuss the methodology of VA in detail by means of charged-particle motion in a strong magnetic field. The application of VA to a broader class of highly-oscillatory problems can be envisioned. For the charged particle, we prove the existence of a coordinate map in phase space that leads to a gyrokinetic Lagrangian at any order of the expansion, for general external fields. We compute this map up to third order, independent of the electromagnetic gauge. Moreover, an error bound for the solution of the derived gyrokinetic equation with respect to the solution of the Vlasov equation is provided, allowing to estimate the quality of the VA-approximation in this particular case.Comment: Keywords: averaging methods, Vlasov equation, Lagrangian mechanics, motion of charged particles, magnetized plasma

Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates

In this paper, we present some new results about the approximation of the Vlasov-Poisson system with a strong external magnetic field by the 2D finite Larmor radius model. The proofs within the present work are built by using two-scale convergence tools, and can be viewed as a new slant on previous works of Fr\'enod and Sonnendr\"ucker and Bostan on the 2D finite Larmor Radius model. In a first part, we recall the physical and mathematical contexts. We also recall two main results from previous papers of Fr\'enod and Sonnendr\"ucker and Bostan. Then, we introduce a set of variables which are so-called canonical gyrokinetic coordinates, and we write the Vlasov equation in these new variables. Then, we establish some two-scale convergence and weak-* convergence results

A comparison of Vlasov with drift kinetic and gyrokinetic theories

A kinetic consideration of an axisymmetric equilibrium with vanishing electric field near the magnetic axis shows that del f should not vanish on axis within the framework of Vlasov theory while it can either vanish or not in the framework of both a drift kinetic and a gyrokinetic theories (f is either the pertinent particle or the guiding center distribution function). This different behavior, relating to the reduction of phase space which leads to the loss of a Vlasov constant of motion, may result in the construction of different currents in the reduced phase space than the Vlasov ones. This conclusion is indicative of some limitation on the implications of reduced kinetic theories in particular as concerns the physics of energetic particles in the central region of magnetically confined plasmas.Comment: 9 page