A new hybrid kinetic electron model for full-f gyrokinetic simulations
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 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 mode [2] or a high frequency mode with appears in the electrostatic limit of the kinetic Alfven wave. Since becomes comparable to the ion Cyclotron frequency at long wavelength, the mode is unphysical from the viewpoint of the gyrokinetic ordering [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 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 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 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, using the gyro-center Hamiltonian, and the gyrokinetic Poisson bracket operator, in the gyro-center coordinates . Here,
Benchmark parameters
In this work, we consider deuterium plasmas in a circular concentric tokamak configuration with , , and , which has the following Cyclone like parameters [23] at : , , , , , , . Here, q is the safety factor, ϵ is the local inverse aspect ratio, is the electron density, and are the electron and ion temperature, , , and 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 and without particle and momentum sources is imposed for . 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
References (49)
Gyrokinetic particle simulation model
J. Comput. Phys.
(1987)- et al.
Kinetic simulations of turbulent fusion plasmas
C. R. Phys.
(2006) Massively parallel Vlasov simulation of electromagnetic drift-wave turbulence
Comput. Phys. Commun.
(2000)- et al.
Linear comparison of gyrokinetic codes with trapped electrons
Comput. Phys. Commun.
(2007) - et al.
New conservative gyrokinetic full-f Vlasov code and its comparison to gyrokinetic δf particle-in-cell code
J. Comput. Phys.
(2007) - et al.
Conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation
Comput. Phys. Commun.
(2008) - et al.
Fully conservative higher order finite difference schemes for incompressible flow
J. Comput. Phys.
(1998) - et al.
Parallel filtering in global gyrokinetic simulations
J. Comput. Phys.
(2012) - et al.
Foundations of nonlinear gyrokinetic theory
Rev. Mod. Phys.
(2007) - et al.
Gyrokinetic simulations of turbulent transport
Nucl. Fusion
(2010)
Study of ion turbulent transport and profile formations using global gyrokinetic full-f Vlasov simulation
Nucl. Fusion
Interplay between gyrokinetic turbulence, flows, and collisions: perspectives on transport and poloidal rotation
Phys. Rev. Lett.
Physics of intrinsic rotation in flux-driven itg turbulences
Nucl. Fusion
On the dynamics of turbulent transport near marginal stability on the dynamics of turbulent transport near marginal stability on the dynamics of turbulent transport near marginal stability
Phys. Plasmas
Full-f gyrokinetic simulation over a confinement time
Phys. Plasmas
Predictions on heat transport and plasma rotation from global gyrokinetic simulations
Nucl. Fusion
Plasma size scaling of avalanche-like heat transport in tokamaks
Nucl. Fusion
Plasma size and collisionality scaling of ion temperature gradient driven turbulence
Nucl. Fusion
Plasma size and power scaling of ion temperature gradient driven turbulence
Phys. Plasmas
Compressed ion temperature gradient turbulence in diverted tokamak edge
Phys. Plasmas
Impact of energetic-particle-driven geodesic acoustic modes on turbulence
Phys. Rev. Lett.
Finite-orbit-width effects on energetic-particle-induced geodesic acoustic mode
J. Plasma Fusion Res.
Simulations of global electrostatic microinstabilities in ASDEX upgrade discharges
Phys. Plasmas
Gyrokinetic simulations of tokamak micro-turbulence including kinetic electron effects
J. Plasma Fusion Res.
Cited by (43)
Time diffusion method for gyrokinetic simulation of electrostatic turbulence with kinetic electrons
2021, Computer Physics CommunicationsBenchmark of a new multi-ion-species collision operator for δf Monte Carlo neoclassical simulation
2020, Computer Physics CommunicationsORB5: A global electromagnetic gyrokinetic code using the PIC approach in toroidal geometry
2020, Computer Physics CommunicationsQuadratic conservative scheme for relativistic Vlasov–Maxwell system
2019, Journal of Computational PhysicsCitation Excerpt :In one work, the Vlasov equations of the conservative form were discretized with the conservative form of the interpolated differential operator (IDO-CF) method [14], but the errors of energy conservation were much larger than the round-off level [15]. In gyrokinetic simulations, there is a charge-conserving algorithm based on finite-difference methods, although the momentum and energy cannot be conserved [16–18]. To perform long time-scale simulations stably, these codes employ the Morinishi scheme [19] to maintain the conservation laws of charge and L2-norm.
High-order discretization of a gyrokinetic Vlasov model in edge plasma geometry
2018, Journal of Computational PhysicsTransport from electron-scale turbulence in toroidal magnetic confinement devices
2024, Reviews of Modern Plasma Physics