Variational principles for reduced plasma physics

Brizard, Alain J.

doi:10.1088/1742-6596/169/1/012003

Cited by 31 publications

(51 citation statements)

References 43 publications

(93 reference statements)

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“…where the guiding-center charge and current densities (̺ gc , J gc ) ≡ e, e d gc X dt F µ dp dµ (16) are expressed as moments of the guiding-center Vlasov phase-space density F µ , and summation over particle species is assumed (wherever appropriate) throughout the text. We note that the guiding-center Maxwell equations (14)- (15) imply that the guiding-center charge and current densities (16) satisfy the guiding-center charge conservation law ∂̺ gc /∂t + ∇ · J gc = 0.…”

Section: B Guiding-center Maxwell Equationsmentioning

confidence: 99%

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Variational formulations of guiding-center Vlasov-Maxwell theory

Brizard

Tronci

2016

Physics of Plasmas

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The Lagrange, Euler, and Euler-Poincaré variational principles for the guiding-center VlasovMaxwell equations are presented. Each variational principle presents a different approach to deriving guiding-center polarization and magnetization effects into the guiding-center Maxwell equations. The conservation laws of energy, momentum, and angular momentum are also derived by Noether method, where the guiding-center stress tensor is now shown to be explicitly symmetric. I. GUIDING-CENTER VLASOV-MAXWELL EQUATIONSThe guiding-center formulation of charged-particle dynamics in nonuniform magnetized plasmas represents one of the most important paradigms in plasma physics. In particular, the Hamiltonian structure of single-particle guiding-center dynamics has proved tremendously useful [1] in the theoretical and numerical analysis of magnetically-confined plasmas. Because of the recent development of variational integration numerical techniques [2][3][4][5][6], it is the primary purpose of the present paper to investigate the variational structures of the guiding-center Vlasov-Maxwell equations.The guiding-center Vlasov-Maxwell equations describe the coupled time evolution of the guiding-center Vlasov distribution function f µ (X, p , t), and the electromagnetic fields E(x, t) and B(x, t). Here, X denotes the guiding-center position, while x denotes the field position, p denotes the parallel guiding-center (kinetic) momentum, µ denotes the guiding-center magnetic moment (which is a guiding-center invariant), and the guidingcenter gyroangle θ (which is canonically conjugate to the guiding-center gyroaction µ B/Ω) is an ignorable coordinate [1] (i.e., ∂f µ /∂θ ≡ 0). The reduced guiding-center phase space is, therefore, four dimensional with coordinates z a ≡ (X, p ), while µ appears as a label on the guiding-center Vlasov distribution function f µ (z, t).In contrast to the drift-kinetic [7] and gyrokinetic [8] Vlasov-Maxwell equations, the electromagnetic fields E(x, t) and B(x, t) appearing in the present work are not separated into a time-independent background magnetic field B 0 (x) and time-dependent electromagnetic field perturbations E 1 (x, t) and B 1 (x, t) that satisfy separate space-time orderings from the background magnetic field B 0 (x). Instead, all Vlasov-Maxwell fields (f µ , E, B) obey the same space-time orderings in which their time dependence is slow compared to the fast gyrofrequency Ω = eB/mc of each particle species (with mass m and charge e) while their weak spatial dependence is used to guarantee the validity of a guiding-center Hamiltonian representation of charged-particle dynamics [1].Moreover, in contrast to previous guiding-center Vlasov-Maxwell theories [9-11], our work does not introduce a transformation to a frame of reference drifting with the E × B velocity. Hence, the guiding-center symplectic structure and the guiding-center Hamiltonian do not acquire an explicit dependence on the electric field E and, thus, the standard polarization does not appear explicitly in our work. As we shall se...

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Section: B Guiding-center Maxwell Equationsmentioning

confidence: 99%

“…The guiding-center magnetization current density c ∇ × M gc in Eq. (15) is defined in terms of the guidingcenter magnetization [15,16] …”

Section: B Guiding-center Maxwell Equationsmentioning

confidence: 99%

Variational formulations of guiding-center Vlasov-Maxwell theory

Brizard

Tronci

2016

Physics of Plasmas

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“…The push-forward representation of particle density is obtained also by another variational principle with constrained variation [21].…”

Section: Reduced Vlasov-poisson Variational Principlementioning

confidence: 99%

“…In this section, we consider push-forward representation of a vector fluid moment, a particle flux, by following Refs. [20,21]. The particle flux is defined in the particle phase space as…”

Section: Push-forward Representation Of Particle Fluxmentioning

confidence: 99%

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Fluid Moments in the Reduced Model for Plasmas with Large Flow Velocity

Miyato

Scott

2011

Plasma and Fusion Research

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Representation of particle fluid moments in terms of fluid moments in the modified guiding-centre model for flowing plasmas with large E × B velocity [N. Miyato et al., J. Phys. Soc. Jpn. 78, 104501 (2009)] is derived from the formal exact representation by a perturbative expansion in the subsonic flow case. It is similar to that in the standard gyrokinetic model in the long wavelength limit, except it has an additional flow term. The flow term has no effect on the representation for particle density, leading to the same representation as the standard one formally. In the conventional guiding-centre models for flowing plasmas, on the other hand, the representation for particle density is different from the standard one. This is due to the difference in the transformation for the guiding-centre position. Although the exact representation usually used in the standard gyrokinetic model has a different form from that in the modified guiding-centre case, correspondence between the two models is shown by considering the alternative form of exact representation in the standard gyrokinetic case. The representation for particle density is also obtained from the single particle Lagrangian by a variational method which is used to derive the representation in the transonic case.

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Stochastic Variational Formulations of Fluid Wave–Current Interaction

Holm

2020

J Nonlinear Sci

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We are modelling multiscale, multi-physics uncertainty in wave–current interaction (WCI). To model uncertainty in WCI, we introduce stochasticity into the wave dynamics of two classic models of WCI, namely the generalised Lagrangian mean (GLM) model and the Craik–Leibovich (CL) model. The key idea for the GLM approach is the separation of the Lagrangian (fluid) and Eulerian (wave) degrees of freedom in Hamilton’s principle. This is done by coupling an Euler–Poincaré reduced Lagrangian for the current flow and a phase-space Lagrangian for the wave field. WCI in the GLM model involves the nonlinear Doppler shift in frequency of the Hamiltonian wave subsystem, which arises because the waves propagate in the frame of motion of the Lagrangian-mean velocity of the current. In contrast, WCI in the CL model arises because the fluid velocity is defined relative to the frame of motion of the Stokes mean drift velocity, which is usually taken to be prescribed, time independent and driven externally. We compare the GLM and CL theories by placing them both into the general framework of a stochastic Hamilton’s principle for a 3D Euler–Boussinesq (EB) fluid in a rotating frame. In other examples, we also apply the GLM and CL methods to add wave physics and stochasticity to the familiar 1D and 2D shallow water flow models. The differences in the types of stochasticity which arise for GLM and CL models can be seen by comparing the Kelvin circulation theorems for the two models. The GLM model acquires stochasticity in its Lagrangian transport velocity for the currents and also in its group velocity for the waves. However, the CL model is based on defining the Eulerian velocity in the integrand of the Kelvin circulation relative to the Stokes drift velocity induced by waves driven externally. Thus, the Kelvin theorem for the stochastic CL model can accept stochasticity in its both its integrand and in the Lagrangian transport velocity of its circulation loop. In an “Appendix”, we also discuss dynamical systems analogues of WCI.

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Variational principles for reduced plasma physics

Cited by 31 publications

References 43 publications

Variational formulations of guiding-center Vlasov-Maxwell theory

Variational formulations of guiding-center Vlasov-Maxwell theory

Fluid Moments in the Reduced Model for Plasmas with Large Flow Velocity

Stochastic Variational Formulations of Fluid Wave–Current Interaction

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