# Fully Kinetic Simulation of Ion-Temperature-Gradient Instabilities in Tokamaks

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## Abstract

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## 1. Introduction

## 2. Fully Kinetic Ion Model of ITG Instabilities

#### Electrostatic Limit with Adiabatic Electrons

## 3. Implicit $\delta f$ Particle-in-Cell Method and Full Orbit Integrator

## 4. Magnetic Field Specification and Field-Line-Following Coordinates

## 5. Simulation Results of ITG Instabilities

#### 5.1. Linear Results of ITG Instabilities and Benchmarking with Other Codes

#### 5.2. Nonlinear Results of ITG Instabilities and Analysis of Saturation due to $E\times B$ Trapping

## 6. Summary

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Guzdar, P.N.; Chen, L.; Tang, W.M.; Rutherford, P.H. Ion-temperature-gradient instability in toroidal plasmas. Phys. Fluids
**1983**, 26, 673–677. [Google Scholar] [CrossRef] - Rettig, C.L.; Rhodes, T.L.; Leboeuf, J.N.; Peebles, W.A.; Doyle, E.J.; Staebler, G.M.; Burrell, K.H.; Moyer, R.A. Search for the ion temperature gradient mode in a tokamak plasma and comparison with theoretical predictions. Phys. Plasmas
**2001**, 8, 2232–2237. [Google Scholar] [CrossRef] - Lin, Z.; Hahm, T.S.; Lee, W.W.; Tang, W.M.; White, R.B. Turbulent Transport Reduction by Zonal Flows: Massively Parallel Simulations. Science
**1998**, 281, 1835–1837. [Google Scholar] [CrossRef] [PubMed] - Xie, H.S.; Xiao, Y.; Lin, Z. New Paradigm for Turbulent Transport Across a Steep Gradient in Toroidal Plasmas. Phys. Rev. Lett.
**2017**, 118, 095001. [Google Scholar] [CrossRef] [PubMed] - Parker, S.E.; Lee, W.W.; Santoro, R.A. Gyrokinetic simulation of ion temperature gradient driven turbulence in 3D toroidal geometry. Phys. Rev. Lett.
**1993**, 71, 2042–2045. [Google Scholar] [CrossRef] [PubMed] - Idomura, Y.; Tokuda, S.; Kishimoto, Y. Global gyrokinetic simulation of ion temperature gradient driven turbulence in plasmas using a canonical Maxwellian distribution. Nucl. Fusion
**2003**, 43, 234. [Google Scholar] [CrossRef] - Gao, Z.; Sanuki, H.; Itoh, K.; Dong, J.Q. Short wavelength ion temperature gradient instability in toroidal plasmas. Phys. Plasmas
**2005**, 12, 022502. [Google Scholar] [CrossRef] - Chen, Y.; Parker, S.E. Electromagnetic gyrokinetic δf particle-in-cell turbulence simulation with realistic equilibrium profiles and geometry. J. Comput. Phys.
**2007**, 220, 839–855. [Google Scholar] [CrossRef] - Ye, L.; Xu, Y.; Xiao, X.; Dai, Z.; Wang, S. A gyrokinetic continuum code based on the numerical Lie transform (NLT) method. J.Comput. Phys.
**2016**, 316, 180–192. [Google Scholar] [CrossRef] - Görler, T.; Lapillonne, X.; Brunner, S.; Dannert, T.; Jenko, F.; Merz, F.; Told, D. The global version of the gyrokinetic turbulence code GENE. J. Comput. Phys.
**2011**, 230, 7053–7071. [Google Scholar] [CrossRef] - Ku, S.; Chang, C.; Diamond, P. Full-f gyrokinetic particle simulation of centrally heated global ITG turbulence from magnetic axis to edge pedestal top in a realistic tokamak geometry. Nucl. Fusion
**2009**, 49, 115021. [Google Scholar] [CrossRef] - Chang, C.S.; Ku, S.; Tynan, G.R.; Hager, R.; Churchill, R.M.; Cziegler, I.; Greenwald, M.; Hubbard, A.E.; Hughes, J.W. Fast Low-to-High Confinement Mode Bifurcation Dynamics in a Tokamak Edge Plasma Gyrokinetic Simulation. Phys. Rev. Lett.
**2017**, 118, 175001. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.; Parker, S.E. Particle-in-cell simulation with Vlasov ions and drift kinetic electrons. Phys. Plasmas
**2009**, 16, 052305. [Google Scholar] [CrossRef] - Belova, E.V.; Gorelenkov, N.N.; Cheng, C.Z. Self-consistent equilibrium model of low aspect-ratio toroidal plasma with energetic beam ions. Physics of Plasmas
**2003**, 10, 3240–3251. [Google Scholar] [CrossRef] - Lin, Y.; Wang, X.Y.; Chen, L.; Lu, X.; Kong, W. An improved gyrokinetic electron and fully kinetic ion particle simulation scheme: Benchmark with a linear tearing mode. Plasma Phys. Controll. Fusion
**2011**, 53, 054013. [Google Scholar] [CrossRef] - Waltz, R.E.; Deng, Z. Nonlinear theory of drift-cyclotron kinetics and the possible breakdown of gyro-kinetics. Phys. Plasmas
**2013**, 20, 012507. [Google Scholar] [CrossRef] - Kramer, G.J.; Budny, R.V.; Bortolon, A.; Fredrickson, E.D.; Fu, G.Y.; Heidbrink, W.W.; Nazikian, R.; Valeo, E.; Zeeland, M.A.V. A description of the full-particle-orbit-following SPIRAL code for simulating fast-ion experiments in tokamaks. Plasma Phys. Controll. Fusion
**2013**, 55, 025013. [Google Scholar] [CrossRef] - Kuley, A.; Lin, Z.; Bao, J.; Wei, X.S.; Xiao, Y.; Zhang, W.; Sun, G.Y.; Fisch, N.J. Verification of nonlinear particle simulation of radio frequency waves in tokamak. Phys. Plasmas
**2015**, 22, 102515. [Google Scholar] [CrossRef] - Sturdevant, B.J.; Parker, S.E.; Chen, Y.; Hause, B.B. An implicit δf particle-in-cell method with sub-cycling and orbit averaging for Lorentz ions. J. Comput. Phys.
**2016**, 316, 519–533. [Google Scholar] [CrossRef] - Miecnikowski, M.T.; Sturdevant, B.J.; Chen, Y.; Parker, S.E. Nonlinear saturation of the slab ITG instability and zonal flow generation with fully kinetic ions. Phys. Plasmas
**2018**, 25, 055901. [Google Scholar] [CrossRef] - Sturdevant, B.J.; Chen, Y.; Parker, S.E. Low frequency fully kinetic simulation of the toroidal ion temperature gradient instability. Phys. Plasmas
**2017**, 24, 081207. [Google Scholar] [CrossRef] - Lapillonne, X.; McMillan, B.F.; Görler, T.; Brunner, S.; Dannert, T.; Jenko, F.; Merz, F.; Villard, L. Nonlinear quasisteady state benchmark of global gyrokinetic codes. Phys. Plasmas
**2010**, 17, 112321. [Google Scholar] [CrossRef] - Parker, S.E.; Lee, W.W. A fully nonlinear characteristic method for gyrokinetic simulation. Phys. Fluid. B Plasma Phys.
**1993**, 5, 77–86. [Google Scholar] [CrossRef] - Aydemir, A.Y. A unified Monte Carlo interpretation of particle simulations and applications to non-neutral plasmas. Phys. Plasmas
**1994**, 1, 822–831. [Google Scholar] [CrossRef] - Süli, E.; Mayers, D.F. An Introduction to Numerical Analysis; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Birdsall, C.; Langdon, A. Plasma Physics via Computer Simulation; CRC Press: Bocaton, FL, USA, 2004. [Google Scholar]
- Qin, H.; Zhang, S.; Xiao, J.; Liu, J.; Sun, Y.; Tang, W.M. Why is Boris algorithm so good? Phys. Plasmas
**2013**, 20, 084503. [Google Scholar] [CrossRef] - Parker, S.; Birdsall, C. Numerical error in electron orbits with large ω
_{ce}Δt. J. Comput. Phys.**1991**, 97, 91–102. [Google Scholar] [CrossRef] - Dimits, A.M.; Bateman, G.; Beer, M.A.; Cohen, B.I.; Dorland, W.; Hammett, G.W.; Kim, C.; Kinsey, J.E.; Kotschenreuther, M.; Kritz, A.H.; et al. Comparisons and physics basis of tokamak transport models and turbulence simulations. Phys. Plasmas
**2000**, 7, 969–983. [Google Scholar] [CrossRef] - Lao, L.; John, H.S.; Stambaugh, R.; Kellman, A.; Pfeiffer, W. Reconstruction of current profile parameters and plasma shapes in tokamaks. Nucl. Fusion
**1985**, 25, 1611. [Google Scholar] [CrossRef] - Beer, M.A.; Cowley, S.C.; Hammett, G.W. Field-aligned coordinates for nonlinear simulations of tokamak turbulence. Phys. Plasmas
**1995**, 2, 2687–2700. [Google Scholar] [CrossRef] - Chen, Y.; Parker, S.E. A delta-f particle method for gyrokinetic simulations with kinetic electrons and electromagnetic perturbations. J. Comput. Phys.
**2003**, 189, 463–475. [Google Scholar] [CrossRef]

**Figure 1.**Comparison between the full orbits calculated by the Boris scheme with different time step sizes: $\Delta t=T/16$, $\Delta t=T/8$, $\Delta t=T/4$, $\Delta t=T/2$, $\Delta t=T$, and $\Delta t=2T$, where T is the ion (Deuteron) gyro-period at its initial location ($R=2.1$ m, $Z=0$ m, $\varphi =0$). The results show that the full orbits agrees with the guiding-center orbit for the cases with time-step $\Delta t<T/4$. When $\Delta t$ is further increased, the computed full orbits deviate from the guiding-center orbit. Further note that the gyro-radius obtained remains nearly the same when the time-step $\Delta t<T/4$. When $\Delta t$ is further increased, the gyro-radius becomes larger than the correct value. The magnetic configuration is from EAST tokamak discharge#[email protected]. The initial velocity is given by ${v}_{R}={v}_{Z}=1.0\times {10}^{6}$ m/s, and ${v}_{\varphi}=5\times {10}^{5}$ m/s, where $({v}_{R},{v}_{\varphi},{v}_{Z})$ are the velocity components in the cylindrical coordinate system. This corresponds to a kinetic energy of $23keV$. For $\Delta t=T/16$, the orbit is advanced by 23,250 time-steps, in which the particle finishes one banana period.

**Figure 2.**The relative variation of the kinetic energy (

**a**) and toroidal angular momentum (

**b**) given by the Boris full orbit integrator over a time period of $t{\mathsf{\Omega}}_{i}=2.5\times {10}^{4}$ for different time step sizes in a tokamak equilibrium magnetic field. The initial conditions of the orbit are $R=1.176{R}_{0}$, $Z=3.912\times {10}^{-3}{R}_{0}$, ${v}_{R}=3.371\times {10}^{-3}{R}_{0}{\mathsf{\Omega}}_{i}$, ${v}_{Z}=3.371\times {10}^{-3}{R}_{0}{\mathsf{\Omega}}_{i}$, and ${v}_{\varphi}=-1.798\times {10}^{-3}{R}_{0}{\mathsf{\Omega}}_{i}$, where ${R}_{0}=1.32$ m and the magnetic configuration is the DIII-D cyclone base case, which is specified in Table 1. The toroidal angular momentum is defined by ${P}_{\varphi}={m}_{i}R{v}_{\varphi}+{q}_{i}\mathsf{\Psi}$, where $\mathsf{\Psi}$ is the poloidal magnetic flux function.

**Figure 3.**Time evolution of an $n=29$ linear ITG instability for DIII-D cyclone base case parameters. In the simulation, the perturbed electric potential $\delta \mathsf{\Phi}$ is Fourier filtered along the toroidal direction to retain only the $n=29$ harmonic, which is further sine transformed along the radial direction and only low radial harmonics are retained. Shown here is the fundamental radial sine harmonic of $\delta \mathsf{\Phi}$ near the low-field-side midplane, which corresponds to ${k}_{r}{\rho}_{i}=0.095$. $\delta \mathsf{\Phi}$ is normalized by ${T}_{e}/e$. The frequency and growth rate in this case are ${\omega}_{r}/{\mathsf{\Omega}}_{i}=2.388\times {10}^{-3}$ and $\gamma /{\mathsf{\Omega}}_{i}=5.8\times {10}^{-4}$, which correspond to the fifth data point in Figure 6, where the corresponding results from Ref. [21] are ${\omega}_{r}/{\mathsf{\Omega}}_{i}=2.423\times {10}^{-3}$ and $\gamma /{\mathsf{\Omega}}_{i}=6.0\times {10}^{-4}$.

**Figure 4.**Time evolution (

**a**) of the electric potential during $t{\mathsf{\Omega}}_{i}=[0:200]$ and the corresponding frequency spectrum (

**b**), which shows a clear peak near the ion gyro-frequency. Here ${f}_{cyc}={\mathsf{\Omega}}_{i}/2\pi $.

**Figure 5.**Comparison of ITG instabilities computed with different marker numbers ${N}_{p}=6.4\times {10}^{6}$ and ${N}_{p}=9.6\times {10}^{6}$. Other parameters are the same as those in Figure 3.

**Figure 6.**Dependence of ITG mode frequency and growth rate on the ion temperature gradient ${\kappa}_{{T}_{i}}$ for the DIII-D cyclone base case. The upper horizontal axis shows the ion temperature gradient normalized by the thermal ion gyro-radius. Also plotted are the results from Ref. [21], which are in good agreement with our new results.

**Figure 8.**A re-plot of the results in Figure 7. The only modification is that the value of ${k}_{\theta}{\rho}_{i}$ used in plotting the fully kinetic results is scaled by a factor of $450.5/500.0$ to take into account the different values of ${\rho}_{i}$ used in the simulation. In this case, the only significant discrepancy between the fully kinetic results and GENE 2010 is at the last data point, which has a small growth rate and may be subject to a larger numerical error.

**Figure 9.**Mode structure of the $n=29$ ITG instability in the poloidal plane. Plotted here is the perturbed electric potential $\delta \mathsf{\Phi}$ at $t{\mathsf{\Omega}}_{i}$ = 12,000.

**Figure 10.**Nonlinear saturation of the $n=29$ ITG instability in the DIII-D cyclone base case. The perturbed potential $\delta \mathsf{\Phi}$ (normalized by ${T}_{e}/e$) is measured on the low-field-side midplane and averaged over the radial domain. The blue line shows the saturation predicted by the $E\times B$ trapping formula (26), which gives $e\delta {\mathsf{\Phi}}_{s}/{T}_{e}=6.16\times {10}^{-3}$ for this case.

**Table 1.**DIII-D cyclone base case parameters [29]. The safety factor profile is given by $q\left(r\right)={q}_{0}+(r-{r}_{0}){q}^{\prime}\left({r}_{0}\right)$ with ${q}^{\prime}\left({r}_{0}\right)=\widehat{s}{q}_{0}/{r}_{0}$, where $\widehat{s}$ is the magnetic shear at $r={r}_{0}$ (the radial center of the simulation box). In this case, ${R}_{0}/{\rho}_{i}=450.5$, where ${\rho}_{i}={v}_{ti}/{\mathsf{\Omega}}_{i}$ is the thermal ion gyro-radius at the magnetic axis, ${v}_{ti}=\sqrt{{T}_{i0}/{m}_{i}}$, ${\mathsf{\Omega}}_{i}={B}_{axis}{q}_{i}/{m}_{i}$ is the ion cyclotron angular frequency at the magnetic axis. Deuterium plasma is assumed in our simulation.

${\mathit{R}}_{0}$ | a | ${\mathit{B}}_{axis}$ | ${\mathit{q}}_{0}$ | $\widehat{\mathit{s}}$ | ${\mathit{r}}_{0}$ | ${\mathit{\kappa}}_{{\mathit{T}}_{\mathit{i}}}{\mathit{R}}_{0}$ | ${\mathit{\kappa}}_{{\mathit{n}}_{\mathit{i}}}{\mathit{R}}_{0}$ | ${\mathit{T}}_{\mathit{i}0}$ | ${\mathit{q}}_{\mathit{i}}{\mathit{T}}_{\mathit{i}0}/\left({\mathit{eT}}_{\mathit{e}0}\right)$ |
---|---|---|---|---|---|---|---|---|---|

1.32 m | 0.48 m | $1.91T$ | 1.40 | 0.78 | 0.24 m | 6.9 | 2.2 | 1.5 keV | 1 |

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**MDPI and ACS Style**

Hu, Y.; Miecnikowski, M.T.; Chen, Y.; Parker, S.E.
Fully Kinetic Simulation of Ion-Temperature-Gradient Instabilities in Tokamaks. *Plasma* **2018**, *1*, 105-118.
https://doi.org/10.3390/plasma1010010

**AMA Style**

Hu Y, Miecnikowski MT, Chen Y, Parker SE.
Fully Kinetic Simulation of Ion-Temperature-Gradient Instabilities in Tokamaks. *Plasma*. 2018; 1(1):105-118.
https://doi.org/10.3390/plasma1010010

**Chicago/Turabian Style**

Hu, Youjun, Matthew T. Miecnikowski, Yang Chen, and Scott E. Parker.
2018. "Fully Kinetic Simulation of Ion-Temperature-Gradient Instabilities in Tokamaks" *Plasma* 1, no. 1: 105-118.
https://doi.org/10.3390/plasma1010010