Variational formulation of higher-order guiding-center Vlasov–Maxwell theory

Brizard, Alain J.

doi:10.1063/5.0161171

Physics of Plasmas

2023

DOI: 10.1063/5.0161171

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Variational formulation of higher-order guiding-center Vlasov–Maxwell theory

Alain J. Brizard

Abstract: Extended guiding-center Vlasov–Maxwell equations are derived under the assumption of time-dependent and inhomogeneous electric and magnetic fields that obey the standard guiding-center space-timescale orderings. The guiding-center Vlasov–Maxwell equations are derived up to second order, which contains dipole and quadrupole contributions to the guiding-center polarization and magnetization that include finite-Larmor-radius corrections. Exact energy-momentum conservation laws are derived from the variational for… Show more

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Cited by 3 publications

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Guiding centre motion for particles in a ponderomotive magnetostatic end plug

Rubin,

Rax,

Fisch

2023

J. Plasma Phys.

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The Hamiltonian dynamics of a single particle in a rotating plasma column, interacting with an magnetic multipole is perturbatively solved for up to second order, using the method of Lie transformations. First, the exact Hamiltonian is expressed in terms of canonical action-angle variables, and then an approximate integrable Hamiltonian is introduced, using another set of actions and angles, which describe the centre of oscillation for the particle. The perturbation introduces an effective ponderomotive potential, which to leading order is positive. At the second order, the pseudopotential consists of a sum of terms of the Miller form, and can have either sign. Additionally, at second order, the ponderomotive interaction introduces a modification to the particle effective mass, when considering the motion along the column axis. It is found that particles can be axially confined by the ponderomotive potentials, but acquire radial excursions which scale as the confining potential. The radial excursions of the particle along its trajectory are investigated, and a condition for the minimal rotation frequency for which the particle remains radially confined is derived. Last, we comment on the changes to the aforementioned solution to the pseudopotentials and particle trajectory in the case of resonant motion, that is, a motion which has the same periodicity as the perturbation.

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Guiding centre motion for particles in a ponderomotive magnetostatic end plug

Rubin,

Rax,

Fisch

2023

J. Plasma Phys.

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Polarization effects in higher-order guiding-centre Lagrangian dynamics

Brizard

2024

J. Plasma Phys.

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The extended guiding-centre Lagrangian equations of motion are derived by the Lie-transform perturbation method under the assumption of time-dependent and inhomogeneous electric and magnetic fields that satisfy the standard guiding-centre space–time orderings. Polarization effects are introduced into the Lagrangian dynamics by the inclusion of the polarization drift velocity in the guiding-centre velocity and the appearance of finite-Larmor-radius corrections in the guiding-centre Hamiltonian and guiding-centre Poisson bracket.

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Local momentum balance in electromagnetic gyrokinetic systems

Sugama

2024

Physics of Plasmas

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The Eulerian variational formulation is presented to obtain governing equations of the electromagnetic turbulent gyrokinetic system. A local momentum balance in the system is derived from the invariance of the Lagrangian of the system under an arbitrary spatial coordinate transformation by extending the previous work [H. Sugama et al., Phys. Plasmas 28, 022312 (2021)]. Polarization and magnetization due to finite gyroradii and electromagnetic microturbulence are correctly described by the gyrokinetic Poisson equation and Ampère's law which are derived from the variational principle. Also shown is how the momentum balance is influenced by including collisions and external sources. Momentum transport due to collisions and turbulence is represented by a symmetric pressure tensor, which originates in a variational derivative of the Lagrangian with respect to the metric tensor. The relations of the axisymmetry and quasi-axisymmetry of the toroidal background magnetic field to a conservation form of the local momentum balance equation are clarified. In addition, an ensemble-averaged total momentum balance equation is shown to take the conservation form even in the background field with no symmetry when a constraint condition representing the macroscopic Ampère's law is imposed on the background field. Using the WKB representation, the ensemble-averaged pressure tensor due to the microturbulence is expressed in detail and it is verified to reproduce the toroidal momentum transport derived in previous works for axisymmetric systems. The local momentum balance equation and the pressure tensor obtained in this work present a useful reference for elaborate gyrokinetic simulation studies of momentum transport processes.

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Variational formulation of higher-order guiding-center Vlasov–Maxwell theory

Cited by 3 publications

References 57 publications

Guiding centre motion for particles in a ponderomotive magnetostatic end plug

Guiding centre motion for particles in a ponderomotive magnetostatic end plug

Polarization effects in higher-order guiding-centre Lagrangian dynamics

Local momentum balance in electromagnetic gyrokinetic systems

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