A set of key properties for an ideal dissipation scheme in gyrokinetic simulations is proposed, and implementation of a model collision operator satisfying these properties is described. This operator is based on the exact linearized test-particle collision operator, with approximations to the field-particle terms that preserve conservation laws and an H-Theorem. It includes energy diffusion, pitch-angle scattering, and finite Larmor radius effects corresponding to classical (real-space) diffusion. The numerical implementation in the continuum gyrokinetic code GS2 is fully implicit and guarantees exact satisfaction of conservation properties. Numerical results are presented showing that the correct physics is captured over the entire range of collisionalities, from the collisionless to the strongly collisional regimes, without recourse to artificial dissipation.
The derivation and numerical implementation of a linearized version of the gyrokinetic (GK) Coulomb collision operator (Jorge R. et al., J. Plasma Phys. 85, 905850604 (2019)) and of the widely-used linearized GK Sugama collision operator (Sugama H. et al., Phys. Plasmas 16, 112503 (2009)) is reported. An approach based on a Hermite-Laguerre moment expansion of the perturbed gyrocenter distribution function is used, referred to as gyro-moment expansion. This approach allows considering arbitrary perpendicular wavenumber and expressing the two linearized GK operators as a linear combination of g…
A new analytically and numerically manageable model collision operator is developed specifically for turbulence simulations. The like-particle collision operator includes both pitch-angle scattering and energy diffusion and satisfies the physical constraints required for collision operators: it conserves particles, momentum and energy, obeys Boltzmann's H-theorem (collisions cannot decrease entropy), vanishes on a Maxwellian, and efficiently dissipates small-scale structure in the velocity space. The process of transforming this collision operator into the gyroaveraged form for use in gyrokine…
Recently, the electrostatic gyrokinetic Hamiltonian and change of coordinates have been computed to order $ε^2$ in general magnetic geometry. Here $ε$ is the gyrokinetic expansion parameter, the gyroradius over the macroscopic scale length. Starting from these results, the long-wavelength limit of the gyrokinetic Fokker-Planck and quasineutrality equations is taken for tokamak geometry. Employing the set of equations derived in the present article, it is possible to calculate the long-wavelength components of the distribution functions and of the poloidal electric field to order $ε^2$. These h…
Gyrokinetic simulations have greatly improved our theoretical understanding of turbulent transport in fusion devices. Most gyrokinetic models in use are delta-f simulations in which the slowly varying radial profiles of density and temperature are assumed to be constant for turbulence saturation times, and only the turbulent electromagnetic fluctuations are calculated. New massive simulations are being built to self-consistently determine the radial profiles of density and temperature. However, these new codes have failed to realize that modern gyrokinetic formulations, composed of a gyrokinet…
In this work, we modify the standard flux tube simulation domain to include arbitrary ion gyroradius-scale variation in the radial profile of the safety factor. To determine how to appropriately include such a modification, we add a strong ion gyroradius-scale source (inspired by electron cyclotron current drive) to the Fokker-Planck equation, then perform a multi-scale analysis that distinguishes the fast electrons driven by the source from the slow bulk thermal electrons. This allows us to systematically derive the needed changes to the gyrokinetic model. We find new terms that adjust the io…
We present a study of the linear properties of ion temperature gradient (ITG) modes with collisions modelled by the linearized gyrokinetic (GK) Coulomb collision operator (Frei et al. 2021) in the local limit. The study is based on a Hermite-Laguerre polynomial expansion of the perturbed ion distribution function applied to the linearized GK Boltzmann equation, yielding a hierarchy of coupled equations for the expansion coefficients, referred to as gyro-moments. We explore analytically the collisionless and high-collisional limits of the gyro-moment hierarchy. Parameter scans revealing the dep…
A gyrokinetic simulation of the influence of electron cyclotron current drive and ion kinetic effect on the m/n=2/1 tearing mode (TM) instabilities is presented in HL-2A and DIII-D tokamak configurations. The TM evolution is calculated with a finite mass electron model and the rf current source is obtained by ray-tracing and the Fokker-Planck method. The TMs are found to be perfectly stabilized by a continuous 1MW 68GHz X2-mode in HL-2A tokamak, while instabilities in the DIII-D discharge (with lower value of CR=I_{rf}/I_{0}, where Irf is the wave driven current and I0 is the equilibrium plasm…
A conservative discontinuous Galerkin scheme for a nonlinear Dougherty collision operator in full-f 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 e…
Numerical techniques for discretization of velocity space in continuum kinetic calculations are described. An efficient spectral collocation method is developed for the speed coordinate - the radius in velocity space - employing a novel set of non-classical orthogonal polynomials. For problems in which Fokker-Planck collisions are included, a common situation in plasma physics, a procedure is detailed to accurately and efficiently treat the field term in the collision operator (in the absence of gyrokinetic corrections). When species with disparate masses are included simultaneously, a careful…
Applying gyrokinetic simulations for theoretical turbulence and transport studies to the plasma edge and scrape-off layer (SOL) presents significant challenges. To in particular account for steep density and temperature gradients in the SOL, the "full-f" code PICLS was developed. PICLS is a gyrokinetic particle-in-cell (PIC) code and is based on an electrostatic model with a linearized field equation and uses kinetic electrons. In previously published results we were applying PICLS to the well-studied 1D parallel transport problem during an edge-localized mode (ELM) in the SOL without collisio…
We study the linearized Vlasov equations and the linearized Vlasov-Fokker-Planck equations in the weakly collisional limit in a uniform magnetic field. In both cases, we consider periodic confinement and Maxwellian (or close to Maxwellian) backgrounds. In the collisionless case, for modes transverse to the magnetic field, we provide a precise decomposition into a countably infinite family of standing waves for each spatial mode. These are known as Bernstein modes in the physics literature, though the decomposition is not an obvious consequence of any existing arguments that we are aware of. We…
The linearized model collision operator for multiple species plasmas given by H. Sugama, T.-H. Watanabe, and M. Nunami [Phys.\ Plasmas {\bf 16}, 112503 (2009)] is improved to be properly applicable up to the highly collisional regime. The improved linearized model operator retains conservation laws of particles, momentum, and energy as well as it reproduces the same friction-flow relations as derived by the linearized Landau operator so that this model can be used to correctly evaluate neoclassical transport fluxes in all collisionality regimes. The adjointness relations and Boltzmann's H-theo…
A numerical method to implement a linearized Coulomb collision operator in the two-weight $δf$ Monte Carlo method for multi-ion-species neoclassical transport simulation is developed. The conservation properties and the adjointness property of the operator in the collisions between two particle species with different temperatures are verified. The linearized operator in a $δf$ Monte Carlo code is benchmarked with other two kinetic simulations, a $δf$ continuum gyrokinetic code with the same linearized collision operator and a full-f PIC code with Nanbu collision operator. The benchmark simulat…
In transport barriers, particularly H-mode edge pedestals, radial scale lengths can become comparable to the ion orbit width, causing neoclassical physics to become radially nonlocal. In this work, the resulting changes to neoclassical flow and current are examined both analytically and numerically. Steep density gradients are considered, with scale lengths comparable to the poloidal ion gyroradius, together with strong radial electric fields sufficient to electrostatically confine the ions. Attention is restricted to relatively weak ion temperature gradients (but permitting arbitrary electron…
In this paper, we study Landau damping in the weakly collisional limit of a Vlasov-Fokker-Planck equation with nonlinear collisions in the phase-space $(x,v) \in \mathbb T_x^n \times \mathbb R^n_v$. The goal is four-fold: (A) to understand how collisions suppress plasma echoes and enable Landau damping in agreement with linearized theory in Sobolev spaces, (B) to understand how phase mixing accelerates collisional relaxation, (C) to understand better how the plasma returns to global equilibrium during Landau damping, and (D) to rule out that collision-driven nonlinear instabilities dominate. W…
Upon application of a sufficiently strong electric field, electrons break away from thermal equilibrium and approach relativistic speeds. These highly energetic runaway electrons (MeV) play a significant role in tokamak disruption physics, and therefore their accurate understanding is essential to develop reliable mitigation strategies. For this purpose, we have developed a fully implicit solver for the 0D-2P (i.e., including two momenta coordinates) relativistic nonlinear Fokker-Planck equation (rFP). As in earlier implicit rFP studies (NORSE, CQL3D), electron-ion interactions are modeled usi…
The linearized Vlasov-Poisson system can be exactly solved using the $G$-transform, an integral transform introduced in Refs. 1-3 that removes the electric field term, leaving a simple advection equation. We investigate how this integral transform interacts with the Fokker-Planck collision operator. The commutator of this collision operator with the $G$-transform (the "shielding term") is shown to be negligible. We exactly solve the advection-diffusion equation without the shielding term. This solution determines when collisions dominate and when advection (i.e. Landau damping) dominates. This…
Fusion-born alpha particles in ITER disruption simulations are investigated as a possible drive of Alfvénic instabilities. The ability of these waves to expel runaway electron (RE) seed particles is explored in the pursuit of a passive, inherent RE mitigation scenario. The spatiotemporal evolution of the alpha particle distribution during the disruption is calculated using the linearized Fokker-Planck solver CODION coupled to a fluid disruption simulation. These simulations are done in the limit of no alpha particle transport during the thermal quench, which can be seen as a most pessimistic s…
The numerical implementation of the linearized gyrokinetic (GK) and drift-kinetic (DK) improved Sugama (IS) collision operators, recently introduced by Sugama et al. [Phys. Plasmas 26, 102108 (2019)], is reported. The IS collision operator extends the validity of the widely-used original Sugama (OS) operator [Sugama et al., Phys. Plasmas 16, 112503 (2009)] to the Pfirsch-Schlüter collisionality regime. Using a Hermite-Laguerre velocity-space decomposition of the perturbed gyrocenter distribution function that we refer to as the gyro-moment approach, the IS collision operator is written in a fo…
Collisional effects can play an essential role in the dynamics of plasma waves by setting a minimum damping rate and by interfering with wave-particle resonances. Kinetic simulations of the effects of electron-ion pitch angle scattering on Electron Plasma Waves (EPWs) are presented here. In particular, the effects of such collisions on the frequency and damping of small-amplitude EPWs for a range of collision rates and wave phase velocities are computed and compared with theory. Both the Vlasov simulations and linear kinetic theory find the direct contribution of electron-ion collisions to wav…
In this work, we examine the validity of several common simplifying assumptions used in numerical neoclassical calculations for nonaxisymmetric plasmas, both by using a new continuum drift-kinetic code and by considering analytic properties of the kinetic equation. First, neoclassical phenomena are computed for the LHD and W7-X stellarators using several versions of the drift-kinetic equation, including the commonly used incompressible-ExB-drift approximation and two other variants, corresponding to different effective particle trajectories. It is found that for electric fields below roughly o…
