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Papers by Alain Brizard
Physics of Plasmas, Feb 7, 2011
European Journal of Physics, 2009
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.
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 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 Letters, 2000
Reviews of Modern Physics, 2009
... 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, 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.
Physics of Plasmas, Feb 7, 2011
European Journal of Physics, 2009
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.
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 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 Letters, 2000
Reviews of Modern Physics, 2009
... 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, 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.