Finite Larmor Radius Approximation Research Articles

Multiscale Numerical Schemes for the Collisional Vlasov Equation in the Finite Larmor Radius Approximation Regime

Multiscale Numerical Schemes for the Collisional Vlasov Equation in the Finite Larmor Radius Approximation Regime

Numerical methods for the two-dimensional Vlasov–Poisson equation in the finite Larmor radius approximation regime

In this paper, we consider numerical methods for solving the two-dimensional Vlasov–Poisson equation in the finite Larmor radius approximation regime. The model describes the behaviour of charged particles under a strong external magnetic field and the finite Larmor radius scaling. We discretise the equation under Particle-in-Cell method, where the characteristics equations are highly oscillatory system in the limit regime. We apply popular numerical integrators including splitting methods, multi-revolution composition methods, two-scale formulation method and limit solver to integrate the characteristics. We then highlight the strengths and drawbacks of each method. Finally, numerical experiments are presented, and comparisons on the accuracy, efficiency and long-time behaviour of the methods are made, pointing to the method with the best performance for this problem.

A new quasilinear formulation for ICRF plasmas in a toroidal geometry

We present a new formulation for quasilinear velocity space diffusion for ICRF plasmas that considers two different aspects: (1) finite Larmor radius approximation and (2) includes the effect of toroidal geometry and constructs a positive definite form. In the first aspect, the Kennel-Engelmann (K-E) quasilinear diffusion coefficients are successfully approximated in a small Larmor radius limit and implemented for the numerical codes (TORIC-CQL3D). In the second aspect, the quasilinear diffusion is reformulated in a toroidal geometry in order to include the parallel dynamics in the inhomogeneous plasmas and magnetic fields. We use these two quasilinear formulations to simulate ITER plasmas with ICRF injection for minority fundamental heating and Tritium second harmonic cyclotron heating.

Finite Larmor radius approximation for collisional magnetic confinement. Part II: the Fokker-Planck-Landau equation

This paper is devoted to the finite Larmor radius approximation of the Fokker-Planck-Landau equation, which plays a major role in plasma physics. We obtain a completely explicit form for the gyroaverage of the Fokker-Planck-Landau kernel, accounting for diffusion and convolution with respect to both velocity and (perpendicular) position coordinates. We show that the new collision operator enjoys the usual physical properties; the averaged kernel balances the mass, momentum, and kinetic energy and dissipates the entropy, globally in velocity and perpendicular position coordinates.

Finite Larmor radius approximation for collisional magnetic confinement. Part I: The linear Boltzmann equation

The subject matter of this paper concerns the derivation of the finite Lar-mor radius approximation, when collisions are taken into account. Several studies are performed, corresponding to different collision kernels : the relaxation and the Fokker-Planck operators. Gyroaveraging the relaxation operator leads to a position-velocity integral operator, whereas gyroaveraging the linear Fokker-Planck operator leads to diffusion in velocity but also with respect to the perpendicular position coordinates.

On the three-dimensional finite Larmor radius approximation: The case of electrons in a fixed background of ions

This paper is concerned with the analysis of a mathematical model arising in plasma physics, more specifically in fusion research. It directly follows, Han-Kwan (2010) [18], where the three-dimensional analysis of a Vlasov–Poisson equation with finite Larmor radius scaling was led, corresponding to the case of ions with massless electrons whose density follows a linearized Maxwell–Boltzmann law. We now consider the case of electrons in a background of fixed ions, which was only sketched in Han-Kwan (2010) [18]. Unfortunately, there is evidence that the formal limit is false in general. Nevertheless, we formally derive from the Vlasov–Poisson equation a fluid system for particular monokinetic data. We prove the local in time existence of analytic solutions and rigorously study the limit (when the inverse of the intensity of the magnetic field and the Debye length vanish) to a new anisotropic fluid system. This is achieved thanks to Cauchy–Kovalevskaya type techniques, as introduced by Caflisch (1990) [7] and Grenier (1996) [14]. We finally show that this approach fails in Sobolev regularity, due to multi-fluid instabilities.

Approximation de rayon de Larmor fini pour les plasmas magnétisés collisionnels

This Note concerns the derivation of the finite Larmor radius approximation, when collisions are taken into account. We concentrate on the Boltzmann relaxation operator whose study reduces to the gyroaverage computation of velocity convolutions. We emphasize that the resulting gyroaverage collision kernel is no local in space anymore and that the standard physical properties (i.e., mass balance, entropy inequality) hold true only globally in space and velocity. This is a first step in this direction and it will allow us to handle more realistic collisional mechanisms, like the Fokker–Planck or Fokker–Planck–Landau kernels (Bostan and Caldini-Queiros (submitted for publication) [3]).

Well-posedness of a diffusive gyro-kinetic model

We study a finite Larmor radius model used to describe the ions distribution function in the core of a tokamak plasma, that consists in a gyro-kinetic transport equation coupled with an electro-neutrality equation. Since the last equation does not provide enough regularity on the electric potential, we introduce a simple linear collision operator adapted to the finite Larmor radius approximation. We next study the two-dimensional dynamics in the direction perpendicular to the magnetic field. Thanks to the smoothing effects of the collision and the gyro-average operators, we prove the global existence of solutions, as well as short time uniqueness and stability.

The three-dimensional finite Larmor radius approximation

Following Frenod and Sonnendrucker (SIAM J. Math. Anal. 32(6) (2001) 1227-1247), we consider the finite Larmor radius regime for a plasma submitted to a large magnetic field and take into account both the quasineutrality and the local thermodynamic equilibrium of the electrons. We then rigorously establish the asymptotic gyrokinetic limit of the rescaled and modified Vlasov-Poisson system in a three-dimensional setting with the help of an averaging lemma.

Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution

A finite Larmor radius approximation is derived from the classical Vlasov equation, in the limit of large (and uniform) external magnetic field. We also provide an heuristic derivation of the electroneutrality equation in the finite Larmor radius setting. Existence and uniqueness of a solution is proven in the stationary frame for solutions depending only on the direction parallel to the magnetic field and factorizing in the velocity variables.

The Finite Larmor Radius Approximation

The presence of a large external magnetic field in a plasma introduces an additional time-scale which is very constraining for the numerical simulation. Hence it is very useful to introduce averaged models which remove this time-scale. However, depending on other parameters of the plasma, different starting models for the asymptotic analysis may be considered. We introduce here a generic framework for our analysis which fits many of the possible regimes and apply it in particular to justify the finite Larmor radius approximation both in the linear case and in the nonlinear case in the plane transverse to the magnetic field.

Numerical simulation of ion cyclotron waves in tokamak plasmas

The code TORIC solves the finite Larmor radius wave equations in the ion cyclotron range of frequencies in arbitrary axisymmetric toroidal geometry. The model used, based on the finite Larmor radius approximation, describes the compressional and torsional Alfvén waves and ion Bernstein waves excited by linear mode conversion. Absorption by the ions occurs at the fundamental and first harmonic of the cyclotron frequency, and by the electrons through Landau and transit time damping. The numerical solution is based on the spectral representation of the wave fields in the poloidal angle , and cubic finite elements in the radial variable . The spectral approach in the poloidal angle allows us to evaluate in a numerically efficient way the integrals over the particle orbits along magnetic field lines which arise when the high frequency (HF) plasma current is obtained by solving the linearized Vlasov equation; the leading effects of toroidicity on cyclotron absorption and spatial dispersion are also taken into account. The code offers a number of options, which allow us to compare the complete finite Larmor radius model with simpler approximations, such as the widely used order reduction algorithm, which replaces the excitation of ion Bernstein waves with an equivalent power sink.

Evaluation of the surface admittance matrix of a plasma in the finite Larmor radius approximation

The boundary conditions appropriate to the finite Larmor radius (FLR) wave equations that describe wave propagation in the ion cyclotron and lower hybrid frequency ranges in plasmas have been derived. These equations possess three independent solutions, one more than the Maxwell equations in vacuum; to establish the additional condition at a plasma-boundary interface needed to define unambiguously the causal solution in the plasma, it is shown that HF diamagnetic currents (proportional to the gradients of the equilibrium plasma parameters) remain finite in the limit in which the thickness of the plasma-vacuum transition is assumed to be negligible compared with the perpendicular wavelength of the incident waves. The discontinuity of the compressional component of the wave field due to these HF surface currents can be expressed in terms of the FLR contributions to the coefficients of the wave equations, so that the FLR boundary conditions can be formulated in a fully explicit manner. It is also shown that these conditions guarantee the continuity of the power flux through the plasma surface. These boundary conditions have been used to obtain analytic expressions for the surface impedance matrix of a plasma in the FLR approximation, assuming that the WKB solutions to the wave equations can be used near the plasma edge. These analytic results can be used to check numerical solutions of the wave equations against a known case, and to speed up the solution of antenna coupling problems in large plasmas

Modelling loop antennas for HF plasma heating in the ion cyclotron frequency range

The technique of impedance and admittance matrices allows a simple and elegant solution of the problem of radiation from antennas of arbitrary shape and orientation in plane stratified geometry. The medium towards which the antenna radiates is completely characterized by the values of these matrices at its surface, which for a hot plasma in the IC frequency range the author obtains numerically in the finite Larmor radius approximation. In vacuum, the wave admittance and impedance matrices obey simple transformation rules, which are used to solve the matching problem for arbitrary source currents. Combined with the variational approach originally developed for the case of fast wave launching by Teilhaber and Jacquinot (1984), this method allows the determination of the self-consistent current distribution in the antenna, its radiation resistance and its reactive impedance. The flexibility of the approach is illustrated by several examples, both without and with plasma loading.