Differential formulation of the gyrokinetic Landau operator

Hirvijoki, Eero; Brizard, Alain J.; Pfefferlé, David

doi:10.1017/s0022377816001203

Cited by 10 publications

(10 citation statements)

References 19 publications

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“…The development of a proper gyrokinetic collision operator has been subject of large analytical (Catto & Tsang 1977;Sugama, Watanabe & Nunami 2009;Li & Ernst 2011;Madsen 2013;Hirvijoki, Brizard & Pfefferlé 2017;Pan & Ernst 2019;Pezzi et al 2019) and numerical (Abel et al 2008;Estève et al 2015) efforts, since collisions provide a transport mechanism and influence turbulence and its associated transport. We provide herein the gyro-moment expansion of a relatively simple nonlinear inter-species collision operator, the Dougherty collision operator (Dougherty 1964).…”

Section: Gyrokinetic Collision Operatormentioning

confidence: 99%

A gyrokinetic model for the plasma periphery of tokamak devices

2020

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A gyrokinetic model is presented that can properly describe strong flows, large and small amplitude electromagnetic fluctuations occurring on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and large deviations from thermal equilibrium. The formulation of the gyrokinetic model is based on a second order description of the single charged particle dynamics, derived from Lie perturbation theory, where the fast particle gyromotion is decoupled from the slow drifts, assuming that the ratio of the ion sound Larmor radius to the perpendicular equilibrium pressure scale length is small. The collective behavior of the plasma is obtained by a gyrokinetic Boltzmann equation that describes the evolution of the gyroaveraged distribution function and includes a non-linear gyrokinetic Dougherty collision operator. The gyrokinetic model is then developed into a set of coupled fluid equations referred to as the gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged distribution function is expanded onto a velocity-space Hermite-Laguerre polynomial basis and the gyrokinetic equation is projected onto the same basis, obtaining the spatial and temporal evolution of the Hermite-Laguerre expansion coefficients. The Hermite-Laguerre projection is performed accurately at arbitrary perpendicular wavenumber values. Finally, the self-consistent evolution of the electromagnetic fields is described by a set of gyrokinetic Maxwell's equations derived from a variational principle, with the velocity integrals of the gyroaveraged distribution function explicitly evaluated. †

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Section: Gyrokinetic Collision Operatormentioning

confidence: 99%

A gyrokinetic model for the plasma periphery of tokamak devices

2020

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“…In continuum kinetic models for plasmas, where small-angle collisions prevail, the effect of collisions is incorporated by the Fokker-Planck operator (FPO) 1 . The gyrokinetic form of this operator also exists [2][3][4][5] and has been shown to produce signficantly different results from 'model' operators in some cases, but agrees closely with model operators in others 6,7 . Nevertheless, exact FPOs often prove to be analytically and numerically challenging for certain applications.…”

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confidence: 58%

Improved multispecies Dougherty collisions

Francisquez,

Juno,

Hakim

et al. 2021

Preprint

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The Dougherty model Fokker-Planck operator is extended to describe nonlinear fullf collisions between multiple species in plasmas. Simple relations for cross-species interactions are developed which obey conservation laws, and reproduce familiar velocity and temperature relaxation rates. This treatment of multispecies Dougherty collisions, valid for arbitrary mass ratios, satisfies the H-Theorem unlike analogous Bhatnagar-Gross-Krook operators.

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“…Finally, to obtain dynamics, an approximate entropy functional is required. This can be done by, e.g., convoluting the delta distribution (29) with some phase-space radial basis function U ðzÞ so that the finite-dimensional entropy is expressed as…”

Section: Particle-in-cell Discretization For the Collisionsmentioning

confidence: 99%

Collisional gyrokinetics teases the existence of metriplectic reduction

Hirvijoki

Burby

2020

Physics of Plasmas

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In purely non-dissipative systems, Lagrangian and Hamiltonian reduction have been proven to be powerful tools for deriving physical models with exact conservation laws. We have discovered a hint that an analogous reduction method exists also for dissipative systems that respect the first and second laws of thermodynamics. In this paper, we show that modern electrostatic gyrokinetics, a reduced plasma turbulence model, exhibits a serendipitous metriplectic structure. Metriplectic dynamics, in general, is a well developed formalism for extending the concept of Poisson brackets to dissipative systems. Better yet, our discovery enables an intuitive particle-in-cell discretization of the collision operator that also satisfies the first and second laws of thermodynamics. These results suggest that collisional gyrokinetics, and other dissipative physical models that obey the laws of thermodynamics, could be obtained using an as-yet undiscovered metriplectic reduction theory and that numerical methods could benefit from such theory significantly. Once uncovered, the theory would generalize Lagrangian and Hamiltonian reduction in a substantial manner.

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Differential formulation of the gyrokinetic Landau operator

Cited by 10 publications

References 19 publications

A gyrokinetic model for the plasma periphery of tokamak devices

A gyrokinetic model for the plasma periphery of tokamak devices

Improved multispecies Dougherty collisions

Collisional gyrokinetics teases the existence of metriplectic reduction

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