Electromagnetic full- gyrokinetics in the tokamak edge with discontinuous Galerkin methods

Mandell, Noah; Hakim, Ammar; Hammett, G. W.; Francisquez, Manaure

doi:10.1017/s0022377820000070

Cited by 50 publications

(72 citation statements)

References 67 publications

(146 reference statements)

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“…2019; Mandell et al. 2019), and fulfils the plasma periphery conditions. We also show that our low- approximation is in fact compatible with the presence of steep equilibrium pressure gradients in § 7.4.…”

Section: Ordering Assumptions For the Plasma Peripherymentioning

confidence: 91%

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: Ordering Assumptions For the Plasma Peripherymentioning

confidence: 91%

A gyrokinetic model for the plasma periphery of tokamak devices

2020

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“…We can test the analytic solution to this equation, f m (t) = f m (t = 0)e −νmt , numerically by using f m=0 = f m=5 = f m=10 = f m=20 = 1 and zero for all other modes, rather than only initializing f m=0 = 1, as in figure (5). The time evolution of the three higher modes is shown in figure (6).…”

Section: Relaxation Tests Of the Gklbomentioning

confidence: 99%

“…This formulation results in a change of variables from the phase-space variables (x, v) to the gyro-averaged particle position, or gyrocenter, phase-space variables (R, v , µ), where R is the gyrocenter coordinate, v is the velocity component parallel to the background magnetic field, and µ = m s v 2 ⊥ /2B is the magnetic moment. Here we focus on the electrostatic gyrokinetic model evolving the gyrocenter distribution function f s (t, R, v , µ), though the collision operator formulation and implementation presented here can also be incorporated in electromagnetic gyrokinetics [5]. The gyrokinetic equation in this case refers to…”

Section: Introductionmentioning

confidence: 99%

“…In this work we limit ourselves to the long-wavelength limit of gyrokinetics, also known as drift-kinetics. Therefore, the true potential φ appears in (5) instead of the gyroaveraged quantity φ α , we use a simple form of the Poisson equation, and we neglect finite Larmor radius (FLR) effects other than the first-order polarization charge density on the left hand side of (6). The Poisson equation is also solved using a spatially constant polarization coefficient.…”

Section: Introductionmentioning

confidence: 99%

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Conservative discontinuous Galerkin scheme of a gyro-averaged Dougherty collision operator

et al. 2020

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A conservative discontinuous Galerkin scheme for a nonlinear Dougherty collision operator in fullf long-wavelength gyrokinetics is presented. Analytically this model operator has the advective-diffusive form of Fokker-Planck operators, it has a non-decreasing entropy functional, and conserves particles, momentum and energy. Discretely these conservative properties are maintained exactly as well, independent of numerical resolution. In this work the phase space discretization is performed using a novel version of the discontinuous Galerkin scheme, carefully constructed using concepts of weak equality and recovery. Discrete time advancement is carried out with an explicit time-stepping algorithm, whose stability limits we explore. The formulation and implementation within the long-wavelength gyrokinetic solver of Gkeyll are validated with relaxation tests, collisional Landau-damping benchmarks and the study of 5D gyrokinetic turbulence on helical, open field lines.

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“…Thus, a direct iterative approach would not normally converge (Belli & Hammett 2005); however, where the time derivative of the fields only appears in 4 A. Y. Sharma and B. F. McMillan the equations of motion and not the field equation, an explicit equation for the time derivative of A may be written by taking the time derivative of the field equation and substituting in the Vlasov equation (Manuilskiy & Lee 2000;Görler et al 2011;Mandell et al 2020). That is, these terms do not actually lead to the conceptual and numerical problems explored in this paper.…”

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confidence: 99%

Solving gyrokinetic systems with higher-order time dependence

Sharma

McMillan

2020

J. Plasma Phys.

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We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most well-known example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the time-dependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in low-frequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For well-behaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may be directly identified. Because we needed to assume that the system is well behaved for small perturbations when deriving gyrokinetic theory, we must continue to assume that when solving it, and the physical solutions are thus the regular ones. The spurious nature of the singular degrees of freedom may also be seen by changing coordinate systems so the varying field appears only in the Hamiltonian. We then describe how methods appropriate for singular perturbation theory may be used to solve these asymptotic equations numerically. We then describe a proof-of-principle implementation of these methods for an electrostatic strong-flow gyrokinetic system; two basic test cases are presented to illustrate code functionality.

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Electromagnetic full- gyrokinetics in the tokamak edge with discontinuous Galerkin methods

Cited by 50 publications

References 67 publications

A gyrokinetic model for the plasma periphery of tokamak devices

A gyrokinetic model for the plasma periphery of tokamak devices

Conservative discontinuous Galerkin scheme of a gyro-averaged Dougherty collision operator

Solving gyrokinetic systems with higher-order time dependence

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