Moment-Based Approach to the Flux-Tube linear Gyrokinetic Model

Abstract

This work reports on the development and numerical implementation of the linear electromagnetic gyrokinetic (GK) model in a tokamak flux-tube geometry using a moment approach based on the expansion of the perturbed distribution function on a velocity-space Hermite-Laguerre polynomials basis. A hierarchy of equations of the expansion coefficients, referred to as the gyro-moments (GM), is derived. We verify the numerical implementation of the GM hierarchy in the collisionless limit by performing a comparison with the continuum GK code GENE, recovering the linear properties of the ion-temperature gradient, trapped electron, kinetic ballooning, and microtearing modes, as well as the collisionless damping of zonal flows. A careful analysis of the distribution functions and ballooning eigenmode structures is performed. The present investigation reveals the ability of the GM approach to describe fine velocity-space scale structures appearing near the trapped and passing boundary and kinetic effects associated with parallel and perpendicular particle drifts. In addition, the effects of collisions are studied using advanced collision operators, including the GK Coulomb collision operator. The main findings are that the number of GMs necessary for convergence decreases with plasma collisionality and is lower for pressure gradient-driven modes, such as in H-mode pedestal regions, compared to instabilities driven by trapped particles and magnetic gradient drifts often found in the core. The accuracy of approximations often used to model collisions (relative to the GK Coulomb operator) is studied in the case of trapped electron modes, showing differences between collision operator models that increase with collisionality and electron temperature gradient. Such differences are not observed in other edge microinstabilities, such as microtearing modes. The importance of a proper collision operator model is also pointed out by analyzing the collisional damping of geodesic acoustic modes and zonal flows. The present linear analysis demonstrates that the GM approach efficiently describes the plasma dynamics for typical parameters of the tokamak boundary, ranging from the low-collisionality banana H-mode to the highcollisionality Pfirsch-Schlüter conditions.

Gyrokinetic simulations of plasma turbulence in a Z-pinch using a moment-based approach and advanced collision operators

“…Considering the results presented on figure 2, we first note that the gyromoment approach retrieves the converged results obtained with GENE, given a sufficiently large polynomial basis. The convergence properties of the gyromoment model depend on the strength of the gradients and improve at steep gradients, confirming previous results obtained for the slab (Frei et al 2022a) and the toroidal ITG instability (Frei et al 2022b). Second, the results obtained with a number of polynomials below convergence show a stabilization of the high k y tail of the entropy mode and a larger peak growth rate.…”

“…In a more general context, the present study is a first step towards the nonlinear simulation of the tokamak boundary based on the use of the gyromoment approach. Our plan is to consider nonlinear simulations in the s − α flux-tube geometry as a next step, expanding the linear study presented in Frei et al (2022b). Using the cyclone base case as a reference (Dimits et al 2000), we will have a benchmark for evaluating the performance of the gyromoment approach.…”

“…We will then turn to the simulation of the tokamak boundary. According to Frei et al (2022b), we anticipate that the high pressure gradient and level of collisionality present in the tokamak edge will improve the convergence of the nonlinear gyromoment hierarchy with respect to core conditions. However, the δf assumption will need to be relaxed when simulating the SOL.…”

“…This approach, also pursued by Mandell, Dorland & Landreman (2018a) with the GX code (Mandell, Dorland & Landreman 2018b;Mandell et al 2022), yields an infinite set of fluid equations for the basis coefficients, the gyromoments, which describe the deviations of the distribution function from a Maxwellian distribution. The efficiency of the δf Hermite-Laguerre gyromoment approach is demonstrated by Frei, Hoffmann & Ricci (2022a) focusing on the linear properties of the ion temperature gradient (ITG) instability in the slab limit, as well as in a flux-tube geometry (Frei et al 2022b), including the use of the linearized GK Landau form of the Fokker-Planck collision operator. These works demonstrate the improvement of convergence properties of the gyromoment method with collisions, i.e.…”