Elsevier

Journal of Computational Physics

Volume 313, 15 May 2016, Pages 511-531
Journal of Computational Physics

A new hybrid kinetic electron model for full-f gyrokinetic simulations

https://doi.org/10.1016/j.jcp.2016.02.057Get rights and content

Highlights

  • A new hybrid kinetic electron model for electrostatic full-f gyrokinetic simulations.

  • The model keeps ITG-TEM turbulence, neoclassical transport, particle (de-)trapping.

  • Influences of passing and trapped electrons on fixed-flux ITG turbulence are clarified.

Abstract

A new hybrid kinetic electron model is developed for electrostatic full-f gyrokinetic simulations of the ion temperature gradient driven trapped electron mode (ITG-TEM) turbulence at the ion scale. In the model, a full kinetic electron model is applied to the full-f gyrokinetic equation, the multi-species linear Fokker–Planck collision operator, and an axisymmetric part of the gyrokinetic Poisson equation, while in a non-axisymmetric part of the gyrokinetic Poisson equation, turbulent fluctuations are determined only by kinetic trapped electrons responses. By using this approach, the so-called ωH mode is avoided with keeping important physics such as the ITG-TEM, the neoclassical transport, the ambipolar condition, and particle trapping and detrapping processes. The model enables full-f gyrokinetic simulations of ITG-TEM turbulence with a reasonable computational cost. Comparisons between flux driven ITG turbulence simulations with kinetic and adiabatic electrons are presented. Although the similar ion temperature gradients with nonlinear upshift from linear critical gradients are sustained in quasi-steady states, parallel flows and radial electric fields are qualitatively different with kinetic electrons.

Introduction

Turbulent fusion plasmas contain multi-scale phenomena covering wide spatio-temporal scales from the fast Cyclotron motion to slow profile evolutions. The issue of the scale gap between the fast Cyclotron motion and micro-turbulence was successfully resolved by the development of the gyrokinetic theory [1] and gyrokinetic simulations [2], [3], [4], which are standard approaches for analyzing turbulent transport in fusion plasmas. On the other hand, another scale gap between micro-turbulence and slow profile evolutions has been treated by conventional δf approaches, where the total particle distribution f is separated into a macroscopic collisionless or collisional equilibrium distribution f0 and a microscopic turbulent perturbation δf, and only the latter is evolved by assuming a complete scale separation. However, the validity of this scale separation is not so clear, when an interaction between turbulence and plasma profiles plays an essential role as in avalanche like non-local transport phenomena. In order to resolve this fundamental issue, in recent years, the so-called full-f or total-f approaches have been developed. This approach evolves the total particle distribution f following turbulent and collisional transport processes described by the same first principle. Because of this multi-scale and multi-physics properties, full-f simulations [5], [6], [7] disclosed rich physics such as self-organized critical phenomena [8] in avalanche-like non-local transport, turbulence regulation by mean flows or radial electric fields given by the neoclassical radial force balance, and intrinsic rotation induced by non-diffusive momentum transport. With increasing computing power, longer time scale simulations and systematic parameter scans have been enabled with full-f approaches. In Ref. [9], a confinement time scale full-f gyrokinetic simulation was presented, and formations of temperature profiles and intrinsic rotation profiles were discussed based on the power balance and the toroidal angular momentum conservation. Validation studies on transport scalings with respect to the plasma size [10], [11], [12], [13], the collisionality [12], and the heating power [13] were performed, and qualitative features of experimental transport scalings were recovered. In addition to core plasma turbulence, full-f gyrokinetic simulations have been applied to edge plasma turbulence [14] and to energetic particles driven geodesic acoustic modes (EGAMs) [15], [16]. Although applications of full-f gyrokinetic simulations have been greatly expanded, their electron models have been limited to adiabatic electrons, and electron transport in full-f gyrokinetic simulations has been an open issue for a long time.

In order to resolve this critical issue, in this work, we develop a new kinetic electron model for full-f gyrokinetic simulations. In δf flux-tube simulations, an electromagnetic model with full kinetic electrons [17] has been established as a standard model. However, an electromagnetic full-f model is not available yet because of the following difficulties. One is the so-called cancellation problem, where a skin term in the Ampere's law has to be exactly cancelled with an adiabatic part of the guiding-center current. Although this issue was resolved for δf approaches, which is consistent with a linear approximation of the skin term, one may need to compute more accurate pull-back transform in full-f approaches. The other issue is a gyrokinetic MHD equilibrium. Since the total particle distribution f involves equilibrium quantities, one may need to consider a gyrokinetic MHD equilibrium, so that it is consistent with various kinetic effects such as trapped particles, finite orbit widths, finite Larmor radii (FLR), and a bootstrap current. Before starting electromagnetic full-f gyrokinetic simulations, we need further investigations on these issues, and therefore, in this work, we limit ourselves to electrostatic models.

It is well known that if one uses a full kinetic electron model, the so-called ωH mode [2] or a high frequency mode with ωH=(k/k)mi/meΩi appears in the electrostatic limit of the kinetic Alfven wave. Since ωH becomes comparable to the ion Cyclotron frequency Ωi at long wavelength, the ωH mode is unphysical from the viewpoint of the gyrokinetic ordering ω/Ωi1 [1]. This unphysical mode is unfavorable also from the numerical viewpoint, because it limits the time step width extremely short. In order to avoid the ωH mode, a kinetic trapped electron model [18], [19], [20], a bounce-averaged trapped electron model [19], and a fluid-kinetic hybrid electron model [21] are often used. Although these models eliminate the ωH mode from electrostatic ion temperature gradient driven trapped electron mode (ITG-TEM) turbulence simulations by assuming adiabatic passing electrons, the following drawbacks arise. Trapping and detrapping processes due to electron transport, turbulent pitch angle scattering, and collisional pitch angle scattering are not described, and one may need artificial boundary conditions at trapped-passing boundaries. The neoclassical transport and the ambipolar condition are not correctly described without passing electrons, and thus, one cannot determine a relevant kinetic equilibrium in full-f approaches. The ambipolar condition is important also for the toroidal angular momentum conservation.

The above contradicting issues are resolved by a new hybrid kinetic electron model. In the model, we apply a full kinetic electron model to the full-f gyrokinetic equation, the multi-species linear Fokker–Planck collision operator, and an axisymmetric part of the gyrokinetic Poisson equation, while in a non-axisymmetric part of the gyrokinetic Poisson equation, turbulent fluctuations are computed using kinetic trapped electrons responses. By taking this approach, one can avoid the ωH mode with keeping important physics such as the ITG-TEM, the neoclassical transport, the ambipolar condition, and particle trapping and detrapping processes. The model is verified through test calculations on collisional temperature relaxation, neoclassical transport, zonal flow damping, and linear stability of ITG-TEMs. After these verification tests, the model is applied to a full-f gyrokinetic simulation, and influences of kinetic electrons are discussed from comparisons of flux driven ITG turbulence simulations with kinetic electrons and with adiabatic electrons.

The reminder of the paper is organized as follows. In Sec. 2, the multi-species full-f gyrokinetic equations with the new hybrid kinetic electron model is presented. In Sec. 3, a series of verification tests on the hybrid kinetic electron model are shown. In Sec. 4, flux driven ITG turbulence simulations with kinetic electrons and with adiabatic electrons are compared, and influences of kinetic electrons are discussed. In this section, the numerical convergence is also shown. Finally, a summary is given in Sec. 5.

Section snippets

Gyrokinetic equation

We consider the electrostatic ITG-TEM turbulence in an axisymmetric tokamak configuration. GT5D is based on the modern gyrokinetic theory [1], in which the gyrokinetic equation is simply given as the Liouville equation,DfsDtfst+{fs,Hs}=fst+R˙fsR+v˙fsv=0, using the gyro-center Hamiltonian,Hs=12msv2+μB+qsϕα, and the gyrokinetic Poisson bracket operator,{F,G}ΩsB(FαGμFμGα)+BmsB(FGvFvG)cqsBbF×G, in the gyro-center coordinates Z=(t;R,v,μ,α). Here, fs

Benchmark parameters

In this work, we consider deuterium plasmas in a circular concentric tokamak configuration with R0/a=2.79, a/ρti=150, and q(r)=0.85+2.18(r/a)2, which has the following Cyclone like parameters [23] at rs=0.5a: ϵ=rs/R0.18, q(rs)1.4, sˆ(rs)=[(r/q)dq/dr]r=rs0.78, ne4.6×1019m3, R0/Ln=2.22, TeTi2keV, R0/Lte=R0/Lti=6.92. Here, q is the safety factor, ϵ is the local inverse aspect ratio, ne is the electron density, Te and Ti are the electron and ion temperature, Ln, Lte, and Lti are the

Nonlinear convergence tests

In this section, we present an ITG turbulence simulation using the new hybrid kinetic electron model (kinetic electron case), and compare its physics properties against an ITG turbulence simulation with the conventional adiabatic electron model (adiabatic electron case). Simulation parameters are chosen based on the Cyclone like parameters. In both cases, on-axis ion heating with Pin=4MW and without particle and momentum sources is imposed for r/a<0.4. The edge temperatures in kinetic and

Summary

In this work, we have developed a full-f gyrokinetic model with kinetic electrons for simulating the ITG-TEM turbulence at the ion scale. The model consists of the multi-species full-f gyrokinetic equations with the new hybrid kinetic electron model and the multi-species linear Fokker–Planck collision operator. In the new hybrid kinetic electron model, turbulent fluctuations are determined by the non-axisymmetric part of the gyrokinetic Poisson equation with kinetic trapped electrons responses,

Acknowledgements

The computation in this work was performed on the BX900 at the JAEA, the Helios at the IFERC, the FX10 at the Univ. Tokyo, and the K-computer (hp150027) at the Riken. The author would like to thank Dr. S. Jolliet for implementing the multi-species linear Fokker–Planck collision operator on GT5D. He is also grateful to Drs. Y. Asahi, S. Maeyama, and M. Nakata for providing useful information on kinetic electrons transport in the GKV code and to Dr. N. Hayashi for his support to this work. This

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