Alain Brizard - Academia.edu (original) (raw)

Papers by Alain Brizard

Physics of Plasmas, Feb 7, 2011

A higher-order self-consistent energy-conserving gyrokinetic system of equations is derived. It i... more A higher-order self-consistent energy-conserving gyrokinetic system of equations is derived. It is shown that additional terms appear in the quasineutrality condition. These terms are nonlinear in the electric field. The derivation includes higher-order terms in the gyrokinetic Hamiltonian ͑needed for the energy conservation͒ and employs a variational principle that automatically provides all the conservation laws through the Noether theorem. The equations derived here can be applied in certain transition layers such as the stellarator transport barriers caused by the transition between the electron and ion root regimes. The theory may also be of interest for the edge plasma, where the nonlinear terms in the quasineutrality equation could be relevant. The equations derived are simple enough and can readily be used in gyrokinetic codes.

European Journal of Physics, 2009

The Jacobi and Weierstrass elliptic functions used to be part of the standard mathematical arsena... more 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 planar pendulum can, in fact, be used to highlight an important connection between the Jacobi and Weierstrass elliptic functions. The easy access to mathematical software by physics students suggests that they might reappear as useful mathematical tools in the undergraduate curriculum.

Physics of Plasmas, 2011

Compact formulas for trapped-particle and passing-particle guiding-center orbits in axisymmetric ... more 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.

Journal of Plasma Physics, 2005

Two applications of the Noether method for fluids and plasmas are presented based on the Euler-La... more 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.

Physics of Plasmas, 1999

A set of self-consistent nonlinear gyrokinetic equations is derived for relativistic charged part... more 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.

The gyrokinetic reduced description of low-frequency and small-perpendicular-wavelength nonlinear... more The gyrokinetic reduced description of low-frequency and small-perpendicular-wavelength nonlinear tokamak dynamics is presented in three different versions: the reduced dynamical description of test particles moving in electromagnetic fields; the reduced gyrokinetic description of the self -consistent interaction of particles and fields through the Maxwell-Vlasov equations; and the reduced description of nonlinear fluid motion. The unperturbed tokamak plasma is described in terms of a noncanonical Hamiltonian guiding-center theory. The unperturbed guiding-center tokamak plasma is then perturbed by gyrokinetic electromagnetic fields and consequently the perturbed guiding-center dynamical system acquires new gyrophase dependence. The perturbation analysis that follows makes extensive use of Lie-transform perturbation techniques. Because the electromagnetic perturbations affect both the Hamiltonian and the Poisson-bracket structure, the Phase-space Lagrangian Lie perturbation method is used. The description of the reduced test-particle dynamics is given in terms of a non-canonical Hamiltonian gyrocenter theory. The description of the reduced kinetic dynamics is concerned with the self-consistent response of the guiding -center plasma and is described in terms of the nonlinear gyrokinetic Maxwell-Vlasov equations. It is also shown that the gyrokinetic Maxwell-Vlasov system possesses a gyrokinetic energy adiabatic invariant and that, in both the linear and nonlinear (quadratic) approximations, the corresponding energy invariants are exact. The description of the reduced fluid dynamics is concerned with the derivation of a closed set of reduced fluid equations. Three generations of reduced fluid models are presented: the reduced MHD equations; the reduced FLR -MHD equations; and the gyrofluid equations. After a brief review of the derivation of the first two generations of reduced fluid models, we give a thorough presentation of the derivation of the gyrofluid equations, viewed as moments of the gyrokinetic Maxwell-Vlasov system. The moment hierarchy is closed by assuming a small (k_| rho_{i}) ordering, which allows a direct comparison with the other reduced fluid models to be made.

Physics of Plasmas, 2007

A general Hamiltonian theory for the adiabatic motion of relativistic charged particles confined ... more 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.

Physics Letters A, 1993

For the first time, the three-step process (conversion-propagation-conversion) underlying gyrores... more For the first time, the three-step process (conversion-propagation-conversion) underlying gyroresonant reflection is analytically developed for two-dimensional geometry. Phase-space methods are used for the conversion of an incident magnetosonic wave field to a continuum of gyroballistic waves, for their propagation along guiding-center orbits in the poloidal plane of a tokamak, and for their conversion to the two-dimensional magnetosonic reflection field. The wave fronts and amplitude of that field are obtained explicitly.

Physical Review E, 2000

A new Lagrangian formalism for self-consistent collective neutrino-plasma interactions is present... more 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.

Physical Review Letters, 2000

A new Eulerian variational principle is presented for the Vlasov-Maxwell equations. This principl... more A new Eulerian variational principle is presented for the Vlasov-Maxwell equations. This principle uses constrained variations for the Vlasov distribution in eight-dimensional extended phase space. The standard energy-momentum conservation law is then derived explicitly by the Noether method. This new variational principle can be applied to various reduced Vlasov-Maxwell equations in which fast time scales have been asymptotically eliminated (e.g., low-frequency gyrokinetic theory).

A Lagrangian formalism is used to derive coupled nonlinear equations for collective interactions ... more A Lagrangian formalism is used to derive coupled nonlinear equations for collective interactions between an intense neutrino flux and a relativistic cold plasma fluid with multiple particle species. In order to focus on the self-consistent ͑collective͒ treatment of neutrino-plasma interactions, quantum effects are ignored throughout and the ͑spinless͒ neutrinos are represented by a complex-valued Klein-Gordon scalar field. Through the application of Noether's method, the conservation laws for energy, momentum and wave action are derived explicitly. The transfer of energy, momentum and wave action between the neutrinos and the electromagnetic-plasma is discussed in the context of astrophysical applications ͑e.g., type II supernova explosions͒.

Reviews of Modern Physics, 2009

Guiding-center theory provides the reduced dynamical equations for the motion of charged particle... more Guiding-center theory provides the reduced dynamical equations for the motion of charged particles in slowly varying electromagnetic fields, when the fields have weak variations over a gyration radius ͑or gyroradius͒ in space and a gyration period ͑or gyroperiod͒ in time. Canonical and noncanonical Hamiltonian formulations of guiding-center motion offer improvements over non-Hamiltonian formulations: Hamiltonian formulations possess Noether's theorem ͑hence invariants follow from symmetries͒, and they preserve the Poincaré invariants ͑so that spurious attractors are prevented from appearing in simulations of guiding-center dynamics͒. Hamiltonian guiding-center theory is guaranteed to have an energy conservation law for time-independent fields-something that is not true of non-Hamiltonian guiding-center theories. The use of the phase-space Lagrangian approach facilitates this development, as there is no need to transform a priori to canonical coordinates, such as flux coordinates, which have less physical meaning. The theory of Hamiltonian dynamics is reviewed, and is used to derive the noncanonical Hamiltonian theory of guiding-center motion. This theory is further explored within the context of magnetic flux coordinates, including the generic form along with those applicable to systems in which the magnetic fields lie on nested tori. It is shown how to return to canonical coordinates to arbitrary accuracy by the Hazeltine-Meiss method and by a perturbation theory applied to the phase-space Lagrangian. This noncanonical Hamiltonian theory is used to derive the higher-order corrections to the magnetic moment adiabatic invariant and to compute the longitudinal adiabatic invariant. Noncanonical guiding-center theory is also developed for relativistic dynamics, where covariant and noncovariant results are presented. The latter is important for computations in which it is convenient to use the ordinary time as the independent variable rather than the proper time. The final section uses noncanonical guiding-center theory to discuss the dynamics of particles in systems in which the magnetic-field lines lie on nested toroidal flux surfaces. A hierarchy in the extent to which particles move off of flux surfaces is established. This hierarchy extends from no motion off flux surfaces for any particle to no average motion off flux surfaces for particular types of particles. Future work in magnetically confined plasmas may make use of this hierarchy in designing systems that minimize transport losses.

... the electron plasma density and is the electron fluid velocity; and the relativistic (kinetic... more ... the electron plasma density and is the electron fluid velocity; and the relativistic (kinetic) momentum equation where the kinetic momentum is , with , and are the charge and mass of a single electron ... The remaining two Maxwell equations follow directly ... B. Lagrangian formulations. ...

Physics of Plasmas, 2001

A relativistic bounce-averaged quasilinear diffusion equation is derived to describe stochastic p... more 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.

Physics Letters A, 2001

The variational formulation for Lie-transform Hamiltonian perturbation theory is presented in ter... more 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 formulation for collisional kinetic theory is presented based on the use of Lietransform me... more A new formulation for collisional kinetic theory is presented based on the use of Lietransform 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.

Physics of Plasmas, 2000

Nonlinear bounce-gyrocenter Hamilton equations for full electromagnetic field fluctuations in gen... more Nonlinear bounce-gyrocenter Hamilton equations for full electromagnetic field fluctuations in general magnetic geometry are derived by the phase-space Lagrangian Lie-perturbation method. These reduced dynamical equations can be used to follow the orbits of magnetically trapped charged particles in the presence of low-frequency electromagnetic fluctuations in magnetic-field geometries suitable for applications in fusion and space plasma physics.

Saint Michael's College Colchester, VT 05439, USA Nonlinear energy-conserving drift-fluid equatio... more Saint Michael's College Colchester, VT 05439, USA Nonlinear energy-conserving drift-fluid equations that are suitable to describe selfconsistent finite-β low-frequency electromagnetic (drift-Alfvén) 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 × B velocities derived directly from the corresponding singleparticle finite-β 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-Ampère equations. An exact energy conservation law is also derived directly from the drift-fluid Lagrangian by application of the Noether method.