Alain J. Brizard

Nonlinear gyrokinetic theory for finite-beta plasmas

A self‐consistent and energy‐conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell’s equations for finite‐beta plasmas, is derived. The method utilized in the present investigation is based on the Hamiltonian formalism and Lie transformation. The resulting formulation is valid for arbitrary values of k⊥ρi and, therefore, is most suitable for studying linear and nonlinear evolution of microinstabilities in tokamak plasmas as well as other areas of plasma physics where the finite Larmor radius effects are important. Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques.

Nonlinear gyrokinetic Maxwell-Vlasov equations using magnetic co-ordinates

A gyrokinetic formalism using magnetic coordinates is used to derive self-consistent, nonlinear Maxwell-Vlasov equations that are suitable for particle simulation studies of finite-..beta.. tokamak microturbulence and its associated anomalous transport. The use of magnetic coordinates is an important feature of this work as it introduces the toroidal geometry naturally into our gyrokinetic formalism. The gyrokinetic formalism itself is based on the use of the Action-variational Lie perturbation method of Cary and Littlejohn, and preserves the Hamiltonian structure of the original Maxwell-Vlasov system. Previous nonlinear gyrokinetic sets of equations suitable for particle simulation analysis have considered either electrostatic and shear-Alfven perturbations in slab geometry, or electrostatic perturbations in toroidal geometry. In this present work, fully electromagnetic perturbations in toroidal geometry are considered. 26 refs.

Nonlinear gyrofluid description of turbulent magnetized plasmas

Nonlinear gyrofluid equations are obtained from the gyrocenter‐fluid moments of the nonlinear gyrokinetic Vlasov equation, which describes an equilibrium magnetized nonuniform plasma perturbed by electromagnetic field fluctuations (δφ,δA∥,δB∥), whose space‐time scales satisfy the gyrokinetic ordering: ω≪Ωi, ‖k∥‖/k⊥≪1, and e⊥≡(k⊥ρi)2≂O(1). These low‐frequency (reduced) fluid equations contain terms of arbitrary order in e⊥ and take into account the nonuniformity in the equilibrium density and temperature of the ion and electron species, as well as the nonuniformity in the equilibrium magnetic field. From the gyrofluid equations, one can systematically derive nonlinear reduced fluid equations with finite‐Larmor‐radius (FLR) corrections, which contain linear and nonlinear terms of O(e⊥), by expressing the gyrocenter‐fluid moments appearing in the gyrofluid equations in terms of the particle‐fluid moments, and then keeping terms up to O(e⊥) in the e⊥ expansion of the gyrofluid equations. By using gyrocenter‐f...

Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks

The nonlinear gyrokinetic Vlasov equation is derived for an arbitrary magnetized plasma in a local reference frame moving with the nonuniform equilibrium fluid velocity u(r). The derivation of the guiding‐center and gyrocenter Hamilton equations, which appear as the characteristics of the gyrokinetic Vlasov equation, is based on the use of Lie‐transform perturbation techniques. Although a general form for u is initially used, attention is later focused on an incompressible toroidal equilibrium flow when considering axisymmetric tokamak geometry.

Variational principle for nonlinear gyrokinetic Vlasov–Maxwell equations

A new variational principle for the nonlinear gyrokinetic Vlasov–Maxwell equations is presented. This Eulerian variational principle uses constrained variations for the gyrocenter Vlasov distribution in eight-dimensional extended phase space and turns out to be simpler than the Lagrangian variational principle recently presented by H. Sugama [Phys. Plasmas 7, 466 (2000)]. A local energy conservation law is then derived explicitly by the Noether method. In future work, this new variational principle will be used to derive self-consistent, nonlinear, low-frequency Vlasov–Maxwell bounce-gyrokinetic equations, in which the fast gyromotion and bounce-motion time scales have been eliminated.

Nonlinear relativistic gyrokinetic Vlasov-Maxwell equations

A set of self-consistent nonlinear gyrokinetic equations is derived for relativistic charged particles in a general nonuniform magnetized plasma. Full electromagnetic-field fluctuations are considered with spatial and temporal scales given by the low-frequency gyrokinetic ordering. Self-consistency is obtained by combining the nonlinear relativistic gyrokinetic Vlasov equation with the low-frequency Maxwell equations in which charge densities and current densities are expressed in terms of moments of the gyrokinetic Vlasov distribution. For these self-consistent gyrokinetic equations, a low-frequency energy conservation law is also derived.

A guiding-center Fokker–Planck collision operator for nonuniform magnetic fields

A formulation for collisional kinetic theory is presented based on the use of Lie-transform methods to eliminate fast orbital time scales from a general bilinear collision operator. As an application of this formalism, a general guiding-center bilinear Fokker–Planck (FP) collision operator is derived following the elimination of the fast gyromotion time scale of a charged particle moving in a nonuniform magnetic field. It is expected that classical transport processes in a strongly magnetized nonuniform plasma can, thus, be described in terms of this reduced guiding-center FP kinetic theory. The present paper introduces the reduced-collision formalism only, while its applications are left to future work.

Relativistic bounce-averaged quasilinear diffusion equation for low-frequency electromagnetic fluctuations

A relativistic bounce-averaged quasilinear diffusion equation is derived to describe stochastic particle transport associated with low-frequency electromagnetic fluctuations in a nonuniform magnetized plasma. Expressions for the relativistic quasilinear diffusion coefficients are calculated explicitly for magnetically-trapped particle distributions in axisymmetric magnetic geometry in terms of drift-bounce resonant contributions associated with low-frequency fluctuations which conserve the first adiabatic invariant.

Hamiltonian theory of adiabatic motion of relativistic charged particles

A general Hamiltonian theory for the adiabatic motion of relativistic charged particles confined by slowly varying background electromagnetic fields is presented based on a unified Lie-transform perturbation analysis in extended phase space (which includes energy and time as independent coordinates) for all three adiabatic invariants. First, the guiding-center equations of motion for a relativistic particle are derived from the particle Lagrangian. Covariant aspects of the resulting relativistic guiding-center equations of motion are discussed and contrasted with previous works. Next, the second and third invariants for the bounce motion and drift motion, respectively, are obtained by successively removing the bounce phase and the drift phase from the guiding-center Lagrangian. First-order corrections to the second and third adiabatic invariants for a relativistic particle are derived. These results simplify and generalize previous works to all three adiabatic motions of relativistic magnetically trapped particles.

Ray-based methods in multidimensional linear wave conversion

A tutorial introduction to the topic of linear wave conversion in multiple spatial dimensions is provided. The emphasis is on physical concepts, particularly those features of multidimensional conversion that are new and different from the more familiar “mode conversion” problem in one spatial dimension. After introductory comments, a brief review of WKB theory for vector wave equations in the absence of conversion is provided in order to introduce notation, terminology, and geometrical ideas. A primary theme of the discussion is that, although WKB (ray-based) methods break down in conversion regions, the ray geometry in the conversion region can be used to develop local wave equations that govern the two coupled wave channels undergoing conversion. These methods can be incorporated into ray-tracing algorithms providing, for the first time, the ability to follow the “ray splitting” associated with linear conversion in multidimensions, including the amplitude and phase changes associated with the conversion.