arXiv subjects
Publications and source records attributed to Alain J. Brizard.
Compact formulas for trapped-particle and passing-particle guiding-center orbits in axisymmetric tokamak geometry are given in terms of the Jacobi elliptic functions and complete elliptic integrals. These formulas can find applications in bounce-center kinetic theory as well as neoclassical transport theory.
The process of dynamical reduction of the Vlasov-Maxwell equations leads to the introduction of classical {\it zitterbewegung} effects in reduced plasma dynamics. These effects manifest themselves in the form of an asymmetric canonical energy-momentum tensor involving the decoupling of the reduced kinetic momentum ${\bf p}$ from the reduced velocity ${\bf u}$ (i.e., ${\bf u}\btimes{\bf p} \neq 0$) as well as reduced polarization and magnetization effects. The reduced intrinsic torque generated by the antisymmetric part of the canonical energy-momentum tensor, which is calculated from the reduced ponderomotive potential, acts as the source for the intrinsic (spin) angular momentum.
The energy-momentum conservation laws for general reduced-fluid (e.g., gyrofluid) models are derived by Noether method from a general reduced variational principle. The reduced canonical energy-momentum tensor (which is explicitly asymmetric and has the Minkowski form) exhibits polarization and magnetization effects associated with dynamical reduction. In particular, the asymmetry in the reduced canonical momentum-stress tensor produces a non-vanishing reduced intrinsic torque that can drive spontaneous toroidal rotation in axisymmetric tokamak plasmas.
The momentum conservation law for general dissipationless reduced-fluid (e.g., gyrofluid) models is derived by Noether method from a variational principle. The reduced-fluid momentum density and the reduced-fluid canonical momentum-stress tensor both exhibit polarization and magnetization effects as well as an internal torque associated with dynamical reduction. As an application, we derive an explicit gyrofluid toroidal angular-momentum conservation law for axisymmetric toroidal magnetized plasmas.
Higher-order guiding-center polarization and magnetization effects are introduced in gyrokinetic theory by keeping first-order terms in background magnetic-field nonuniformity. These results confirm the consistency of the two-step perturbation analysis used in modern gyrokinetic theory.
The exact global energy conservation laws for the full-$f$ and $δf$ formulations of the nonlinear electromagnetic gyrokinetic equations in general magnetic geometry are presented. In both formulations, the relation between polarization and magnetization effects in the gyrokinetic Poisson and Ampère equations and the quadratic ponderomotive gyrocenter Hamiltonian is emphasized.
A proper choice for the sign of the Lorentz-covariant relativistic Lagrangian for a particle moving in an electromagnetic field is presented. This choice is based on which space-time metric is used in the representation of the relativistic particle dynamics.
The Jacobi and Weierstrass elliptic functions used to be part of the standard mathematical arsenal of physics students. They appear as solutions of many important problems in classical mechanics: the motion of a planar pendulum (Jacobi), the motion of a force-free asymmetric top (Jacobi), the motion of a spherical pendulum (Weierstrass), and the motion of a heavy symmetric top with one fixed point (Weierstrass). The problem of the planar pendulum, in fact, can be used to construct the general connection between the Jacobi and Weierstrass elliptic functions. The easy access to mathematical software by physics students suggests that they might reappear as useful tools in the undergraduate curriculum.
The elimination of a fast time scale from the Vlasov equation by Lie-transform methods is an important step in deriving a reduced Vlasov equation such as the drift-kinetic Vlasov equation or the gyrokinetic Vlasov equation. It is shown here that this dynamical reduction also leads to the introduction of polarization and magnetization effects in the reduced Maxwell equations, which ensure that the reduced Vlasov-Maxwell equations possess an exact energy-momentum conservation law.
Nonlinear energy-conserving drift-fluid equations that are suitable to describe self-consistent finite-beta low-frequency electromagnetic (drift-Alfven) turbulent fluctuations in a nonuniform, anisotropic, magnetized plasma are derived from a variational principle. The variational principle is based on a drift-fluid Lagrangian that contains linear and nonlinear E x B velocities derived directly from the corresponding single-particle finite-beta gyrocenter Hamiltonian (in the zero-Larmor-radius limit). Covariant electromagnetic effects introduce a magnetic generalization to the standard ion polarization density as well as introduce a new ion magnetization current, which are both missing from existing gyrofluid and drift-fluid Poisson-Ampere equations. An exact energy conservation law is also derived directly from the drift-fluid Lagrangian by application of the Noether method.
Two applications of the Noether method for fluids and plasmas are presented based on the Euler-Lagrange and Euler-Poincare variational principles, which depend on whether the dynamical fields are to be varied independently or not, respectively. The relativistic cold laser-plasma equations, describing the interaction between an intense laser field with a cold relativistic electron plasma, provide a useful set of equations amenable to both variational formulations. The derivation of conservation laws by Noether method proceeds from the Noether equation, whose form depends on the variational formulation used. As expected, the expressions for the energy-momentum conservation laws are identical in both variational formulations. The connection between the two Lagrangian densities is shown to involve the mass conservation and Lin constraints associated with the cold relativistic electron fluid.
The foundations of gyrokinetic theory are reviewed with an emphasis on the applications of Lagrangian and Hamiltonian methods used in the derivation of nonlinear gyrokinetic Vlasov-Maxwell equations. These reduced dynamical equations describe the turbulent evolution of low-frequency electromagnetic fluctuations in nonuniform magnetized plasmas with arbitrary magnetic geometry.
A relativistic bounce-averaged quasilinear diffusion equation is derived to describe stochastic particle transport associated with arbitrary-frequency electromagnetic fluctuations in a nonuniform magnetized plasma. Expressions for the elements of a relativistic quasilinear diffusion tensor are calculated explicitly for magnetically-trapped particle distributions in axisymmetric magnetic geometry in terms of gyro-drift-bounce wave-particle resonances. The resonances can destroy any one of the three invariants of the unperturbed guiding-center Hamiltonian dynamics.
A new 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 new 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.
The effects of collective neutrino-plasma interactions on the linear wave spectrum supported by a magnetized electron-positron plasma in the presence of a neutrino-antineutrino medium are investigated. When a pair-symmetric background neutrino-plasma medium is perturbed by space-charge waves (electrostatic waves associated with electron-positron charge separation), our analysis shows that the neutrino and antineutrino fluids also separate and the pair symmetry of the background medium is broken. The cosmological implications of this pair-symmetry breaking mechanism are briefly discussed.
The variational formulation for Lie-transform Hamiltonian perturbation theory is presented in terms of an action functional defined on a two-dimensional parameter space. A fundamental equation in Hamiltonian perturbation theory is shown to result from the freedom of choice of the integration path for the action functional.
A new Lagrangian formalism for self-consistent collective neutrino-plasma interactions is presented in which each neutrino species is described as a classical ideal fluid. The neutrino-plasma fluid equations are derived from a covariant relativistic variational principle in which finite-temperature effects are retained. This new formalism is then used to investigate the generation of magnetic fields and the production of magnetic helicity as a result of collective neutrino-plasma interactions.