ITG–TEM turbulence simulation with bounce-averaged kinetic electrons in tokamak geometry

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Abstract

We develop a novel numerical scheme to simulate electrostatic turbulence with kinetic electron responses in magnetically confined toroidal plasmas. Focusing on ion gyro-radius scale turbulences with slower frequencies than the time scales for electron parallel motions, we employ and adapt the bounce-averaged kinetic equation to model trapped electrons for nonlinear turbulence simulation with Coulomb collisions. Ions are modeled by employing the gyrokinetic equation. The newly developed scheme is implemented on a global δf particle in cell code gKPSP. By performing linear and nonlinear simulations, it is demonstrated that the new scheme can reproduce key physical properties of Ion Temperature Gradient (ITG) and Trapped Electron Mode (TEM) instabilities, and resulting turbulent transport. The overall computational cost of kinetic electrons using this novel scheme is limited to 200%–300% of the cost for simulations with adiabatic electrons. Therefore the new scheme allows us to perform kinetic simulations with trapped electrons very efficiently in magnetized plasmas.

Introduction

High temperature plasmas confined by strong magnetic fields often exhibit turbulent behaviors. Accurate modeling and simulation of plasma turbulence are highly critical for a reliable prediction of the thermodynamic properties of strongly magnetized plasmas, such as fusion plasmas in tokamak devices. In general, magnetically confined high temperature plasma is an extremely complex system with a wide range of spatio-temporal scales, and its complete modeling in realistic configurations requires impractically huge computing resources. However, it is experimentally known that turbulences causing the anomalous transport of heat and particle have much slower time scales than the ion gyro-frequency Ωi=qimiBc in many cases. Also, the observed spatial scales of these turbulences are in a comparable range with the ion gyro-radius ρi=vtiΩi. If we denote the characteristic frequency and wavenumber as ω and k, these properties can be written as ωΩi and kρi1. In present day tokamak experiments, electrostatic fluctuations in these ion scales are routinely measured with various diagnostic systems, and firm links have been established between the anomalous transports and the observed ion scale electrostatic fluctuations.

The nonlinear gyrokinetic theory was developed based on these properties of turbulence in strongly magnetized plasma [1]. Firm theoretical foundations including relevant conservation laws and Hamiltonian structure have been established consequently using the systematic phase-space Lagrangian Lei-perturbation theory [2], [3]. The theory provides a systematic procedure to remove unnecessary spatio-temporal scales, and reduces the 6 dimensional (6D) Boltzmann equation to a 5 dimensional one, which allows a highly efficient way to model and simulate plasma turbulence in magnetically confined system. There are many different types of simulation codes employing the gyrokinetic model, and they are used to study various aspects of turbulence and transport phenomena in magnetic fusion plasma.

If we want to simulate ion scale turbulences with ωΩi, the necessary size of time step can be thought as Δt1ωρivti. However, this does not give us a great freedom in the choice of time step size, because the restriction in the time step mainly comes from the Courant condition for parallel motions of charged particles. Since ion and electron temperature are in a similar range TiTe in fusion plasma, this time step restriction comes from the parallel electron motions due to the large mass ratio of ion and electron mime103, which is very demanding when we deal with ion scale turbulences. Also, in electrostatic turbulence simulations, the presence of so called omega-H mode complicates this time step issue further [4]. If we employ explicit time integration algorithms, time steps small enough to resolve the mode should be chosen to avoid numerical instabilities, which becomes unnecessarily demanding for practical simulation.

Physically, if fluctuations have much slower time scales than the time scale for electron motions, the response of freely moving electrons against the fluctuations can be approximately modeled as adiabatic. This physical property is often used to model electron responses in ion turbulence simulations. However, in a plasma confined by inhomogeneous magnetic fields, there always exists a finite fraction of electrons, which are trapped in certain limited lower magnetic field regions of the system. These trapped electrons show highly non-trivial behaviors under the slow fluctuations, which can result in qualitatively different phenomena even in slow ion scale turbulences. It is known that the trapped electrons can enhance micro-instabilities originating from ion dynamics such as ion temperature gradient (ITG) mode [5], [6] or cause new types of instabilities called trapped particle mode (TEM) [7], [8], [9].

The nonlinear bounce-averaged kinetic theory can provide a useful alternative to model the trapped electrons [10]. Similar to the gyrokinetic theory, which removes physics faster than the ion gyro-frequencies, the bounce-averaged kinetic theory explicitly removes physics with time scales faster than the bounce motions of trapped particles from the original kinetic equation, and keeps relevant dynamics slower than the bounce periods of trapped particles. As the magnetic moment μ=mv22B becomes an adiabatic invariant of particle motions in the gyrokinetic model, the so called second adiabatic invariant of trapped particle motion Ivdl now becomes an invariant in time scales assumed for the derivation of the bounce-averaged kinetic model.

We note that there have been previous efforts to employ the bounce-averaged kinetic electron model for simulations of ion scale micro-instabilities [11], [12]. Recently, it has also been shown that the bounce-averaged model can be used for global simulations of micro-instabilities in shaped tokamak plasma equilibrium [13]. Through highly efficient simulations employing time step sizes comparable to the sizes for gyrokinetic simulation with the adiabatic electron model, this work successfully demonstrated the stabilizing effects of key global equilibrium parameters on ITG–TEM instabilities. All these are highly encouraging as they open a new way to develop efficient numerical models for nonlinear kinetic simulation of ion scale turbulence with important electron physics.

However, we also want to emphasize that it is non-trivial to extend the previously proposed model for linear simulation to one for nonlinear turbulence simulation. In linear simulations, electrons can be simply divided into two separate groups, i.e. freely passing electrons and trapped electrons, which can be treated independently by employing the adiabatic model for the former and the bounce-averaged model for the latter. In nonlinear simulations, turbulence scattering can mix electrons in the two groups. Also, if we want to include Coulomb collisions, which can significantly modify ITG–TEM instabilities and thereby resulting turbulences, an accurate description of the two different groups of electron fluxes becomes critical. These considerations urge us to employ numerical means to model the evolution of passing electrons beyond the simple adiabatic responses. Depending on the level of sophistication of the passing electron model, computing costs will increase inevitably. So, we need a careful consideration for the balance between the accuracy of physics and the computational costs.

In this paper, we report our recent progresses in developing an efficient numerical scheme for nonlinear simulation of ion scale turbulence with bounce-averaged kinetic electrons. Focusing on electrostatic ITG–TEM turbulences in the core of tokamak plasma, we integrate and extend the previously proposed numerical algorithms. In this aim, a global gyrokinetic particle-in-cell (PIC) code gKPSP is employed to implement and test the developed scheme. The code is based on δf method solving perturbed parts of distribution functions in general tokamak geometry, and has been used to study various subjects on ITG turbulence and related transport physics [14], [15]. Through the present work, gKPSP is extended for global nonlinear ITG–TEM turbulence in general tokamak geometry including Coulomb collisions.

The remaining part of this paper is organized as follows. In Section 2, we present simulation models used in this work. The gyrokinetic and bounce-averaged kinetic equations are introduced in toroidal geometry aiming to model the evolutions of the distribution functions for fusion plasma in tokamak device. In Section 3, we explain numerical methods to solve these models and discuss their implementations on the global δf PIC code. In Section 4, we present various linear and nonlinear benchmark simulation results to verify the proposed numerical scheme and its implementation. Finally, summary and conclusion are given Section 5.

Section snippets

Simulation model

In this section, we explain the simulation models used in this work. We numerically solve an electrostatic version of the gyrokinetic and bounce-averaged kinetic equations in toroidal geometry. Axisymmetric equilibrium magnetic field is assumed as B=gφ+φ×ψ, where ψ and φ denote the poloidal magnetic flux and toroidal angle respectively. Magnetically confined toroidal plasma is assumed, which is composed of hydrogen ions and electrons. Each species is represented by the subscript s=i,e.

The

Simulation method

In this section, we present detailed numerical algorithms for the models explained in the previous section. Especially, the algorithms are designed to be implemented on PIC code. Each subsection in the following paragraphs deals with a separate component of the new scheme. The final subsection summarizes how they are organized and arranged to make the new scheme work.

Simulation results

In this section, we present various simulation results to verify the physical validness of the proposed numerical scheme.

Summary and conclusion

In this work, we have developed a novel numerical scheme for turbulence simulation of strongly magnetized plasma in toroidal geometry. In the new scheme, we employed the gyrokinetic and bounce-averaged kinetic equation to model ions and trapped electrons in realistic tokamak geometry. Passing electrons were also modeled to balance the particle fluxes crossing the trapped-passing boundary in the phase space for the electrons, which are caused by Coulomb collisions and turbulent scatterings of

Acknowledgments

This work was supported by R & D Program through National Fusion Research Institute (NFRI) funded by the Ministry of Science, ICT and Future Planning of the Republic of Korea (NFRI-EN1641-2).

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