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Register a new userLinearized model Fokker-Planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests
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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.
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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 Boltzmanns 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 gyrokinetic simulations is detailed. The gyroaveraged model operator is shown to have more suitable behavior at small scales in phase space than previously suggested models. A model operator for electron-ion collisions is also presented.
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 gyro-moments where the expansion coefficients are given by closed analytical expressions that depend on the perpendicular wavenumber and on the temperature and mass ratios of the colliding species. The drift-kinetic (DK) limits of the GK linearized Coulomb and Sugama operators are also obtained. Comparisons between the gyro-moment approach with the GK continuum code GENE are reported focusing on the ion-temperature-gradient (ITG) instability and zonal flow (ZF) damping, finding an excellent agreement. In particular, we demonstrate that the GK linearized Sugama yields a stronger collisional damping of the ZF residual compared to the GK linearized Coulomb. Finally, we show that the numerical efficiency of the gyro-moment approach increases with collisionality, a desired property for boundary plasma applications.
An encoder-decoder neural network has been used to examine the possibility for acceleration of a partial integro-differential equation, the Fokker-Planck-Landau collision operator. This is part of the governing equation in the massively parallel particle-in-cell code, XGC, which is used to study turbulence in fusion energy devices. The neural network emphasizes physics-inspired learning, where it is taught to respect physical conservation constraints of the collision operator by including them in the training loss, along with the L2 loss. In particular, network architectures used for the computer vision task of semantic segmentation have been used for training. A penalization method is used to enforce the soft constraints of the system and integrate error in the conservation properties into the loss function. During training, quantities representing the density, momentum, and energy for all species of the system is calculated at each configuration vertex, mirroring the procedure in XGC. This simple training has produced a median relative loss, across configuration space, on the order of 10E-04, which is low enough if the error is of random nature, but not if it is of drift nature in timesteps. The run time for the Picard iterative solver of the operator scales as order n squared, where n is the number of plasma species. As the XGC1 code begins to attack problems including a larger number of species, the collision operator will become expensive computationally, making the neural network solver even more important, since the training only scales as n. A wide enough range of collisionality is considered in the training data to ensure the full domain of collision physics is captured. An advanced technique to decrease the losses further will be discussed, which will be subject of a subsequent report. Eventual work will include expansion of the network to include multiple plasma species.
A gyrokinetic Coulomb collision operator is derived, which is particularly useful to describe the plasma dynamics at the periphery region of magnetic confinement fusion devices. The derived operator is able to describe collisions occurring in distribution functions arbitrarily far from equilibrium with variations on spatial scales at and below the particle Larmor radius. A multipole expansion of the Rosenbluth potentials is used in order to derive the dependence of the full Coulomb collision operator on the particle gyroangle. The full Coulomb collision operator is then expressed in gyrocentre phase-space coordinates, and a closed formula for its gyroaverage in terms of the moments of the gyrocenter distribution function in a form ready to be numerically implemented is provided. Furthermore, the collision operator is projected onto a Hermite-Laguerre velocity space polynomial basis and expansions in the small electron-to-ion mass ratio are provided.
Microinstabilities exhibit a rich variety of behavior in stellarators due to the many degrees of freedom in the magnetic geometry. It has recently been found that certain stellarators (quasi-isodynamic ones with maximum-$J$ geometry) are partly resilient to trapped-particle instabilities, because fast-bouncing particles tend to extract energy from these modes near marginal stability. In reality, stellarators are never perfectly quasi-isodynamic, and the question thus arises whether they still benefit from enhanced stability. Here the stability properties of Wendelstein 7-X and a more quasi-isodynamic configuration, QIPC, are investigated numerically and compared with the National Compact Stellarator Experiment (NCSX) and the DIII-D tokamak. In gyrokinetic simulations, performed with the gyrokinetic code GENE in the electrostatic and collisionless approximation, ion-temperature-gradient modes, trapped-electron modes and mixed-type instabilities are studied. Wendelstein 7-X and QIPC exhibit significantly reduced growth rates for all simulations that include kinetic electrons, and the latter are indeed found to be stabilizing in the energy budget. These results suggest that imperfectly optimized stellarators can retain most of the stabilizing properties predicted for perfect maximum-$J$ configurations.
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