# Presentation: 553247

## Particle-in-Cell Simulations of Electron Transport from Plasma Turbulence: Recent Progress inGyrokinetic Particle Simulations of Turbulent Plasmas

- Z. Lin,1 _G. Rewoldt_,2 S. Ethier,2
- T.S. Hahm,2 W.W. Lee,2
- J.L.V. Lewandowski,2 Y. Nishimura,1
- W.X. Wang2
*1**University of California, Irvine *
*2**Princeton Plasma Physics Laboratory, Princeton University*

## Global Gyrokinetic Toroidal Code (GTC)

- Gyrokinetic particle simulation
- Efficient sampling of *5D* phase space
- GTC global field-aligned mesh:
- Respects physical periodicity
- Efficient for toroidal eigenmode
- # of computation ~ *(a/****)**2*
- Reduces computation by *n~10**3 *
- Massively parallel computing
- Reactor scale plasmas
- Keeps all toroidal modes *n~10**3*
- Resources: US DOE SciDAC

*GTC mesh*

**Notes:**
These contradictions are presumably consequences of different representations of geometry in flux-tube and global codes. With regard to ETG simulation, the global code respects physical periodicity, rigorously treats radial variations of safety factor q(r), and keeps all toroidal mode number n from 0, 1, 2, up to thousands. This sounds too good to be true. So what is the catch? Obviously, the global simulation of re scale ETG turbulence is computationally extremely challenging. Our realistic ETG simulation using the gyrokinetic toroidal code GTC was enabled by a global field-aligned mesh, which reduces the computational requirement by three orders of magnitude. The idea of using field-aligned coordinates to describe the toroidal eigenmode has a long history. Nevertheless, the GTC code is a unique global code in the world fusion program that achieves the optimal performance that the number of computation scales with the square, rather than cubic, of the device size without sacrifying any physics. The particle-in-cell approach and effective utilization of massively parallel computing allow simulation device size up to DIIID tokamak. Still, a single simulation used near one million hours of computer CPU time. Fortunately, such large scale computational resources have been provided by the US Department of Energy’s SciDAC initiative, or Scientific Discovery through Advanced Computing, which funded the Gyrokinetic Particle Simulation Center. The GTC code is being developed by a multi-institutions team.

7min

## Linear Frequency Comparison: GTC, GT3D,FULL

- R / LTi (i) Scan with Trapped Electrons

-  FULL: local only, GTC fixed density and temperature values but varying gradients; GT3D varying density and temperature values and gradients (different profile shapes)
- Vary R/LTi (and i) at fixed R/LTe = 6.92, R/Ln = 2.22, and k i = 0.335 (on reference surface) with trapped electrons

- G. Rewoldt, Z. Lin, Y. Idomura (JAERI)

## Electron Transport Insensitive to ETG Streamer Length

- At *t=20/***** after saturation
- Streamer length scales with device size
- Eddy turnover time *** ~ 16/*****
***nl** << ****0*
- Electron does not rotate with streamers
- Transport driven by wave-particle interaction
- Mixing length estimate inaccurate

- Z. Lin

**Notes:**
The structure of fully developed ETG turbulence is shown in this poloidal contour plot of electrostatic potential, or density perturbation, at 20 linear growth times after saturation. These seemingly coherent streamers actually contain hundreds of toroidal modes as shown in previous slide. By varying the simulation device size, we find that streamer size scales with the device size and that the eddy turn over time is much longer than the linear growth time. So the turbulence structures evolve slowly and long simulation time is needed to reach a steady state. You may wonder if you should expect to find such a long streamer in the actual tokamak experiments. Well it is not so obvious. Since the nonlinear decorrelation rate is much smaller than the linear growth rate, the streamers could be prone to shearing effects of strong E X B flows, which are neglected in these simulations. In this sense, our simulation provides a “rigorous upper bound” on the streamer size and transport level. Within this simplified model, the key question now is whether these long streamers drive large electron transport. Before answering this question, it is instructive to examine the particle dynamics since it is the particle radial diffusion that carries the heat flux. Superimposed here is an electron orbit during roughly an eddy turn over time. The electron orbit is slightly perturbed from the equilibrium orbit and the radial excursion is much smaller than the streamer size. During the same period, the streamers would have completed a rotation from inside out. Therefore, the streamers only cause a small perturbation to electron parallel motion and that the electron does not rotate with the streamers. Note that this electron orbit is projected on to a poloidal plane, it does not cut across the eddies in the poloidal direction. The point here is that electron does not rotate with streamers in the radial direction. Of course, single particle orbit could be misleading. We need to examine the radial motions of all particles.

## Nonlinear Toroidal Couplings Regulate ETG Turbulence

- 1st step: generation of low-*n* quasi-mode
- “Meso-scale: optimal mode number  n ~ *n**1**1/2*
- No ballooning structure:* ****||**~qRn**1**1/2*
- 2nd step: energy transfer to nonlinear mode
- Streamers nonlinearly generated
- Spectral transfer facilitated by quasi-modes
- Nonlocal in *n*-space, “Compton Scattering”
- Streamer coupling: toroidal geometry-specific
- Need to keep all toroidal modes
- Sufficient channels for spectral transfer

- Z. Lin

- n

**Notes:**
In this controlled simulation to illustrate ETG saturation, we first apply artificial damping to all modes except for two unstable eigenmodes with toroidal mode number 95 and 110. When the amplitude of these two modes are much higher than any other modes, we remove the artificial damping. The first step of the nonlinear toroidal coupling is the generation of a low-n quasi-mode with a mode number 15 by the beating of the two pump modes. The next step is the coupling of pump mode 95 with the quasi-mode 15 to drive a nonlinear mode 80. Subsequently, pump mode 110 couples with nonlinear mode 80 to generate another quasi-mode 30. Now the coupling of pump mode 95 and the quasi-mode 30 drive another nonlinear mode 65. These two-step process proceeds until all n-matching modes are populated, while other modes still have small amplitudes. This successive transferring of energy from unstable to stable modes lead to ETG  saturation and nonlinear streamer formation. Direct transfer from pump modes to nonlinear modes are not efficient because of frequency mismatch. Instead, the spectral transfer is through the scattering of high-n unstable modes on the low-n quasi-mode. This is a nonlocal interaction in k-space, similar to Compton Scattering with the quasi-mode playing a crucial role of quasi-particle. Note that the spectral transfer occurs before the unstable modes reach a high amplitude to excite secondary instability. All toroidal modes need to be treated since they all participate in nonlinear toroidal coupling and provide channels for spectral transfer. Again, only a few modes are treated in flux-tube simulations. 

## W.W. Lee & J.L.V. Lewandowski

- Particle Diffusion Due To Toroidal ITG Modes With/Without Parallel Velocity-Space Nonlinearity

- • (q/m) E|| (∂f /∂v||) term in GTC
- • Additional channel to reach steady state
- • Different (test particle) diffusion pattern (and scaling)?

- with                      without

- adiabatic electrons

- adiabatic electrons

## Turbulence Spreading from Edge to Stable Core

- T.S. Hahm, et al., to appear, Phys. Plasmas ‘05

- • Nonlinear GTC Simulations of Ion Temperature Gradient Turbulence
-          = 5.3 at core
-           (within Dimits shift regime)
-          = 10.6 at edge
- • Initial growth at edge
-         Penetration into stable core
-      (Lin, Hahm, Diamond, PRL ‘02, PPCF, PoP ‘04)
- • Saturation level at core:

## cf. poster, Weixing Wang, Wed. evening

**Gyrokinetic Simulation of Microturbulence for Shaped Plasmas**
- • General Geometry GTC developed with generalized  and extended features: realistic plasma profiles and MHD equilibrium(ESC, JSOLVER...); systematic treatment of plasma rotation and equilibrium **E**x**B** flow (calculated from GTC-Neo); nonuniform mesh in correlation with local gyroradius; accurate gyrokinetic transformation; ES with adiabatic electrons (tested); trapped electrons via higher order correction (to be tested).

**To do:** update field solver; incorporate split-weight scheme for electron dynamics; fully develop and deliver EM general geometry capability for turbulence simulations;
- Physics: TEM, Alfvenic ITG (KBM), micro-tearing, ITB dynamics ...

- W.X. Wang

*cf. poster,** **Weixing Wang, Wed. evening*

## Finite beta effects in the massless limit are investigated

- • A finite element method (FEM) elliptic solver has enabled kinetic electron studies using the split-weight scheme (Lee, et al ‘01) and also using the hybrid model (mass-ratio expansion) (Lin & Chen ‘01)
- • Electromagnetic effects currently being studied. AITG/KBM and TAE modes expected to be unstable for sufficient beta and gradient of energetic particles.  Two steps: (1) massless electron limit (solve fluid equations for electrons), (2) kinetic electrons
- • Initial results: Left: 2D plot of    (electrostatic potential) exhibiting the eigenmode for the ITG instability; Right: Corresponding A|| eigenfunction for ITG

- Y. Nishimura

## J.L.V. Lewandowski

- Split-weight Scheme for Toroidal, Kinetic PIC Simulations with Kinetic Electrons

- • Remove the adiabatic electron response analytically, and solve for non-adiabatic response numerically - currently ES but later EM (solve GK Poisson equation & Ampere’s law)
- • I-D simulations showed: (1) more accurate linear growth rate, (2) cleaner power spectrum, and(3) better conservation properties even for few electrons, Ne << Ni.
- • Splitting scheme for toroidal plasmas: Fe = FM exp(e/Te) + h, and solve for non-adiabatic weight w = h/Fe.
- • Split-weight scheme for non-adiabatic electron response only (allows for turbulent & collisional friction between trapped & untrapped electrons).
- • Current density and other scalar quantities deposited on structured (but not logically rectangular) grid every timestep, and inversion of field equation carried out using finite-element method, with triangular elements.
- • Global finite-element Poisson solver
- used to invert A ∂t = S (32 or 64
- different Stiffness matrices A, on
- different poloidal planes)
- • Numerical method is stable for
- large time step t = (5 - 10) / ci

## GTC-Neo for Neoclassical fluxes with finite-orbit effects

- Near-neoclassical(?) shot

- ITB shot

- • Global particle code GTC-Neo [W.X. Wang, *et al.,* Comp. Phys. Comm.**164**, 178 (2004)] calculates neoclassical fluxes of particles, momentum, and energy, as well as
- associated quantities such as radial electric field, bootstrap
- current, and poloidal velocity
- • Intrinsically non-local over
- the scale-length of the ion banana width due to included finite
- orbit effects (implies smoothing)
- • Interfaced with MHD codes for the numerical MHD equilibrium and with TRANSP for experimental plasma profiles
- • Runs currently on the massively-parallel IBM-SP computer at NERSC
- - 9 cases run for NSTX now
- • Impurity and hot beam species to be added in future

- W.X. Wang, G. Rewoldt

## Conclusions

- Progress on many fronts for GTC code!
- GTC working now in ES limit, with circular concentric magnetic surfaces, including trapped electrons, producing physics results:
- Linear and nonlinear benchmarking
- ETG modes
- Parallel nonlinearity
- Turbulence spreading
- Non-circular-cross-section generalization beginning to produce results
- Two complementary approaches for EM generalization being investigated
- GTC-Neo code for neoclassical fluxes & Er
- Still need to put everything together!