arXiv subjects

Alain J. Brizard

Publications and source records attributed to Alain J. Brizard.

At least 37 records · Page 2Linked to original sources

arXiv2021

Exact conservation laws for gauge-free electromagnetic gyrokinetic equations

The exact energy and angular-momentum conservation laws are derived by Noether method for the Hamiltonian and symplectic representations of the gauge-free electromagnetic gyrokinetic Vlasov-Maxwell equations. These gyrokinetic equations, which are solely expressed in terms of electromagnetic fields, describe the low-frequency turbulent fluctuations that perturb a time-independent toroidally-axisymmetric magnetized plasma. The explicit proofs presented here provide a complete picture of the transfer of energy and angular momentum between the gyrocenters and the perturbed electromagnetic fields, in which the crucial roles played by gyrocenter polarization and magnetization effects are highlighted. In addition to yielding an exact angular-momentum conservation law, the gyrokinetic Noether equation yields an exact momentum transport equation, which might be useful in more general equilibrium magnetic geometries.

physics.plasm-ph↗

arXiv2021

Asymptotic Limit-cycle Analysis of Oscillating Chemical Reactions

The asymptotic limit-cycle analysis of mathematical models for oscillating chemical reactions is presented. In this work, after a brief presentation of mathematical preliminaries applied to the biased Van der Pol oscillator, we consider a two-dimensional model of the Chlorine dioxide Iodine Malonic-Acid (CIMA) reactions and the three-dimensional and two-dimensional Oregonator models of the Belousov-Zhabotinsky (BZ) reactions. Explicit analytical expressions are given for the relaxation-oscillation periods of these chemical reactions that are accurate within 5\% of their numerical values. In the two-dimensional CIMA and Oregonator models, we also derive critical parameter values leading to canard explosions and implosions in their associated limit cycles.

physics.chem-ph↗

arXiv2020

Hamiltonian formulations for perturbed dissipationless plasma equations

The Hamiltonian formulations for the perturbed Vlasov-Maxwell equations and the perturbed ideal magnetohydrodynamics (MHD) equations are expressed in terms of the perturbation derivative $\partial{\cal F}/\partialε\equiv [{\cal F}, {\cal S}]$ of an arbitrary functional ${\cal F}[\vbψ]$ of the Vlasov-Maxwell fields $\vbψ = ({\sf f},{\bf E},{\bf B})$ or the ideal MHD fields $\vbψ = (ρ,{\bf u},s,{\bf B})$, which are assumed to depend continuously on the (dimensionless) perturbation parameter $ε$. Here, $[\;,\;]$ denotes the functional Poisson bracket for each set of plasma equations and the perturbation {\it action} functional ${\cal S}$ is said to generate dynamically accessible perturbations of the plasma fields. The new Hamiltonian perturbation formulation introduces a framework for functional perturbation methods in plasma physics and highlights the crucial roles played by polarization and magnetization in Vlasov-Maxwell and ideal MHD perturbation theories. One application considered in this paper is a formulation of plasma stability that guarantees dynamical accessibility and leads to a natural generalization to higher-order perturbation theory.

physics.plasm-ph↗

arXiv2020

Symplectic gyrokinetic Vlasov-Maxwell theory

A new representation of electromagnetic gyrokinetic Vlasov-Maxwell theory is presented in which the gyrocenter equations of motion are expressed solely in terms of the perturbed electric and magnetic fields. In this representation, the gyrocenter symplectic (Poisson-bracket) structure and the gyrocenter Jacobian contain electric and magnetic perturbation terms associated with the standard first-order gyrocenter polarization and magnetization terms that traditionally appear in the gyrokinetic Maxwell equations. In addition, the gyrocenter polarization drift (which includes perturbed magnetic-field corrections) now appears explicitly in the gyrocenter velocity. The symplectic gyrokinetic Vlasov-Maxwell equations are self-consistently derived from a constrained Eulerian variational principle, which yields exact energy-momentum conservation laws (through the Noether method) that are verified explicitly. An exact toroidal canonical angular momentum conservation law is also derived explicitly under the assumption of an axisymmetric background magnetic field.

physics.plasm-ph↗

arXiv2020

Energy and momentum conservation in the Euler-Poincaré formulation of local Vlasov-Maxwell-type systems

The action principle by Low [Proc. R. Soc. Lond. A 248, 282--287] for the classic Vlasov-Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler-Poincaré formulation of Vlasov-Maxwell-type systems, effectively starting with Low's action and using constrained variations for the Eulerian description of particle motion, has been known for a while [J. Math. Phys., 39, 6, pp. 3138-3157], it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler-Poincaré machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov-Darwin system [Phys. Plasmas 25, 102506]. The present exposition discusses a generic class of local Vlasov-Maxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler-Poincaré framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear- and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. We also discuss what happens if no symmetries exist. Finally, two explicit examples will be given -- the classic Vlasov-Maxwell and the drift-kinetic Vlasov-Maxwell -- and the results expressed in the language of regular vector calculus for familiarity.

physics.plasm-ph↗

arXiv2018

Perturbative variational formulation of the Vlasov-Maxwell equations

The perturbative variational formulation of the Vlasov-Maxwell equations is presented up to third order in the perturbation analysis. From the second and third-order Lagrangian densities, respectively, the first-order and second-order Vlasov-Maxwell equations are expressed in gauge-invariant and gauge-independent forms. Upon deriving the reduced second-order Vlasov-Maxwell Lagrangian for the linear nonadiabatic gyrokinetic Vlasov-Maxwell equations, the reduced Lagrangian densities for the linear drift-wave equation and the linear hybrid kinetic-magnetohydrodynamic (MHD) equations are derived, with their associated wave-action conservation laws obtained by Noether method. The exact wave-action conservation law for the linear hybrid kinetic-MHD equations is written explicitly. A new form of the third-order Vlasov-Maxwell Lagrangian is derived in which ponderomotive effects play a crucial role.

physics.plasm-ph↗

arXiv2018

Notes on the exact solutions and singularities of an X-point collapse in Hall magnetohydrodynamics

A recent paper by A.~Z.~Janda [J.~Math.~Phys.~{\bf 59}, 061509 (2018)] presented a partial solution of the equations of dissipationless Hall magnetohydrodynamics (MHD) in terms of the Weierstrass elliptic function. Unfortunately, an error crept up in the analysis, where the kinetic and potential energies of the initial Hall MHD state were misidentified. The present comment presents the correct energy analysis and offers a complete solution of dissipationless Hall magnetohydrodynamics in terms of the Jacobi elliptic functions.

math-ph↗

arXiv2018

Equivalent elastica knots

The problem of an elastica knot in three-dimensional space is solved explicitly by expressing the Frenet-Serret curvature and torsion of the knot in terms of the Weierstrass and Jacobi elliptic functions. This solution is obtained by variational methods and is derived by minimizing of the squared-curvature energy integral. In the present work, an equivalency is established between pairs of Jacobi elliptic solutions that are described by the same values for curvature and torsion functionals.

math-ph↗

arXiv2017

Differential formulation of the gyrokinetic Landau operator

Subsequent to the recent rigorous derivation of an energetically consistent gyrokinetic collision operator in the so-called Landau representation, this paper investigates the possibility of finding a differential formulation of the gyrokinetic Landau collision operator. It is observed that, while a differential formulation is possible in the gyrokinetic phase space, reduction of the resulting system of partial differential equations to five dimensions via gyroaveraging poses a challenge. Based on the present work, it is likely that the gyrocentre analogues of the Rosenbluth-MacDonald-Judd potential functions must be kept gyroangle dependent.

physics.plasm-ph↗

arXiv2017

On the validity of the guiding-center approximation in the presence of strong magnetic gradients

The motion of a charged particle in a nonuniform straight magnetic field with a uniform magnetic-field gradient is solved exactly in terms of elliptic functions. The connection between this problem and the guiding-center approximation is discussed. It is shown that, for this problem, the predictions of guiding-center theory agree very well with the orbit-averaged particle motion and hold well beyond the standard guiding-center limit $ε\equiv ρ/L \ll 1$, where $ρ$ is the gyromotion length scale and $L$ is the magnetic-field gradient length scale.

physics.plasm-ph↗

arXiv2017

Variational principle for the parallel-symplectic representation of electromagnetic gyrokinetic theory

The nonlinear (full-$f$) electromagnetic gyrokinetic Vlasov-Maxwell equations are derived in the parallel-symplectic representation from an Eulerian gyrokinetic variational principle. The gyrokinetic Vlasov-Maxwell equations are shown to possess an exact energy conservation law, which is derived by Noether method from the gyrokinetic variational principle. Here, the gyrocenter Poisson bracket and the gyrocenter Jacobian contain contributions from the perturbed magnetic field. In the full-$f$ formulation of the gyrokinetic Vlasov-Maxwell theory presented here, the gyrocenter parallel-Ampère equation contains a second-order contribution to the gyrocenter current density that is derived from the second-order gyrocenter ponderomotive Hamiltonian.

physics.plasm-ph↗

arXiv2016

Motion in an Asymmetric Double Well

The problem of the motion of a particle in an asymmetric double well is solved explicitly in terms of the Weierstrass and Jacobi elliptic functions. While the solution of the orbital motion is expressed simply in terms of the Weierstrass elliptic function, the period of oscillation is more directly expressed in terms of periods of the Jacobi elliptic functions.

math-ph↗

arXiv2015

Notes on the Weierstrass Elliptic Function

A consistent notation for the Weierstrass elliptic function $\wp(z;g_{2},g_{3})$, for $g_{2} > 0$ and arbitrary values of $g_{3}$ and $Δ\equiv g_{2}^{3} - 27 g_{3}^{2}$, is introduced based on the parametric solution for the motion of a particle in a cubic potential. These notes provide a roadmap for the use of {\sf Mathematica} to calculate the half-periods $(ω_{1},ω_{3},ω_{2} \equiv ω_{1} + ω_{3})$ of the Weierstrass elliptic function.

math-ph↗

arXiv2014

Canonical transformation for trapped/passing guiding-center orbits in axisymmetric tokamak geometry

The generating function for the canonical transformation from the parallel canonical coordinates $(p_{\|},s)$ to the action-angle coordinates $(J,ζ)$ for trapped/passing guiding-center orbits in axisymmetric tokamak geometry is presented. Drawing on the analogy between the phase-space portraits of the librating/rotating pendulum and the trapped/passing guiding-center orbits, the generating function is expressed in terms of the Jacobi zeta function, which can then readily be used to obtain an explicit expression for the bounce-center transformation for trapped/passing-particles in axisymmetric tokamak geometry.

physics.plasm-ph↗

arXiv2013

Beyond linear gyrocenter polarization in gyrokinetic theory

The concept of polarization in gyrokinetic theory is clarified and generalized to include contributions from the guiding-center (zeroth-order) polarization as well as the nonlinear (second-order) gyrocenter polarization. The guiding-center polarization, which appears as the antecedent (zeroth-order) of the standard linear (first-order) gyrocenter polarization, is obtained from a modified guiding-center transformation. The nonlinear gyrocenter polarization is derived either variationally from the third-order gyrocenter Hamiltonian or directly by gyrocenter push-forward method.

physics.plasm-ph↗

arXiv2012

Action-angle coordinates for the pendulum problem

The action-angle coordinates for the planar pendulum problem are expressed in terms of the Jacobi elliptic functions and integrals. In particular, we show that the Jacobi zeta function generates the canonical transformation from the pendulum coordinates $\vartheta$ and $p \equiv \partial\vartheta/\partial t$ to the action-angle coordinates $(J,ζ)$ for both the librating pendulum and the rotating pendulum.

physics.class-ph↗

arXiv2012

Equivalent representations of higher-order Hamiltonian guiding-center theory

Two complementary representations of higher-order guiding-center theory are presented, which are distinguished by whether higher-order corrections due to magnetic-field nonuniformity appear in the guiding-center Poisson bracket or the guiding-center Hamiltonian. The equivalence of these two representations implies that the guiding-center gyroaction adiabatic invariant and the guiding-center Hamilton equations of motion are representation-invariants. By using a new perturbative hierarchy in guiding-center theory, second-order corrections are introduced in Hamiltonian guiding-center theory without the need to calculate the guiding-center transformation at second order.

physics.plasm-ph↗

arXiv2011

Exact momentum conservation laws for the gyrokinetic Vlasov-Poisson equations

The exact momentum conservation laws for the nonlinear gyrokinetic Vlasov-Poisson equations are derived by applying the Noether method on the gyrokinetic variational principle [A. J. Brizard, Phys. Plasmas {\bf 7}, 4816 (2000)]. From the gyrokinetic Noether canonical-momentum equation derived by the Noether method, the gyrokinetic parallel momentum equation and other gyrokinetic Vlasov-moment equations are obtained. In addition, an exact gyrokinetic toroidal angular-momentum conservation law is derived in axisymmetric tokamak geometry, where the transport of parallel-toroidal momentum is related to the radial gyrocenter polarization, which includes contributions from the guiding-center and gyrocenter transformations.

physics.plasm-ph↗