Edge Gyrokinetic Theory and Continuum Simulations

1. Edge Gyrokinetic Theory and Continuum Simulations X. Q. Xu1,6, K. Bodi2, J. Candy3, B. I. Cohen1, R. H. Cohen1, P. Colella4, M. R. Dorr1, J. A. Hittinger1, G. D. Kerbel1, S. Krasheninnikov2, W. M. Nevins1, H. Qin5, T. D. Rognlien1, P. B. Snyder3, M. V. Umansky1, Z. Xiong1 1Lawrence Livermore National Laboratory, Livermore, CA 94550 USA, 2University of California, San Diego, La Jolla, CA 92093 USA 3General Atomics, San Diego, CA 92186 USA 4Lawrence Berkeley National Laboratory, Berkeley, CA 94720 USA 5Princeton Plasma Physics Laboratory, Princeton, NJ 08543 USA 6Zhejiang University Presented at International Symposium onFusion Energy Science & The 5th Workshop on Nonlinear Plasmas Sciences October 23, 2006, Hangzhou, China

2. Tempest is a gyro-kinetic code to bridge the range of collisionalities

3. The Edge Transport Barrier isCritical to ITER’s Performance Projection of ITER’s Fusion Gain • Transport barriers form spontaneously at plasma edge • Studies of core turbulence show • Turbulent transport constrains gradient scale lengths • Tcentral ~ proportional to Tped • Tped is the largest source of uncertainty in projecting ITER’s performance • Fusion gain = Pfusion/Paux TEMPEST is aimed at reducing uncertainty in projections of ITER’s fusion gain After R. Waltz et al, 2002 SnowMass Mtg.

4. Orbit width A kinetic edge code is required to model both today’s tokamaks and ITER DIII-D Edge Barrier • Fluid approximation requires: • Not satisfied on DIII-D todayWon’t be satisfied on ITER • Need to move beyond fluid codes or  Describe each species with akinetic distribution function, F(a)(y, , ,E0, ,)

5. Impurities in the edge plasma are important for power balance Output Multi-parameter table for C, CH4 sputtering yield Output Multi-parameter table for transfer from CH4 to C Output C-ion charge-states in edge coupled to GK code Impurities coupled through tables Carbon plate MD and kinetic MC simulations for sputtering Widely varying length scales ~ nm Plasma flux ~10 cm Near-surface carbon plasma transport and chemistry for C transfer rate DT plasma ~10 m Multi-charge state C-ion fluid transport in whole edge region

6. Fully nonlinear ion gyrokinetic equation has been cast in E0-m coordinates Kinetic simulations • The field is split into two parts: F0 and df; • E0=mv2/2+qF0, a constant of motion if df~0 and a coordinate aligned with flow; • E0xB flow terms and the slow variation F0 from Qin’s formulation will be added.

7. + + Fully nonlinear gyrokinetic Poisson equation---in collaboration with Hong Qin Orbit squeezing by large Er shear Full FLR effect Qin, Cohen, Nevins, and Xu, Contrib. Plasma Phys. 46, 7-9, 477 (2006)

8. Fully nonlinear gyro-kinetic Poisson equation in the long wavelength limit • Diamagnetic density is included; • It is fully nonlinear since the Na and Pa are calculated from Fa; • If the first-order Pade approximation to G0 = 1/(1 + b) for the modified Bessel function is used, then the same field solve will be used in the arbitrary wavelength regime.

9. Self-consistent sheath boundary conditions in SOL and private flux regions • Sheath boundary conditions for potential f • Sheath boundary conditions for distribution function F

10. TEMPEST is a fully nonlinear gyrokinetic continuum code in a divertor geometry • 5D ( y,q,z,E0, μ) • Realistic X-point divertor geometry • Open + closed flux surfaces • Implicit backward-differencing scheme in time • using a Newton-Krylov iteration • Higher order accuracy in phase space: • Finite difference in spatial space • 4th order upwinding & 5th Weno scheme • finite volume method in velocity space (E0, μ) • Uses Hypre library of parallel linear algebra solvers and preconditioners

11. Spatial convection using 4th order upwinding and 5th order Weno scheme t=0, t=1st cycle and t=14thcycles are overlapped, Nz=64 Ni z(index)

12. Velocity space finite volume are used in TEMPEST for conservative flux difference---Requires treatment of cut cells at turning point boundary E0 Vll=0 m • Yields very accurate moment calculation Bt ~ 1/R After R 3% Density r Before Spurious density variation due to B field change Poloidal Angle

13. Endloss tests demonstrate viability of continuum code with real collision operator (CQL)

14. TEMPEST reproduces neoclassical parallel flow in circular geometry inaccessible <U||> inaccessible Simplest case: circular geometry

15. V|| V|| V|| V|| Ion distribution function F(R,Z,E0,m) in DIII-D geometry with endloss at plates in the SOL looks as expected

16. TEMPEST recovers theoretical U|| inside separatrix and increases as expected in SOL (eV/m2/s) 1023 0 -1023

17. TEMPEST exhibits collisionless damping of GAMs and zonal Flow f(t)/f(t=0) • Axis-symmetric mode (no toroidal variation) • Parallel ion dynamics • Magnetic curvature • Acceleration  TEMPEST should see GAMs • TEMPEST model • Drift kinetic ions with radial drift, streaming, and acceleration • Boltzmann electron • Gyrokinetic Poisson equation in limit small rs/Lx • Periodic radial boundary conditions • GAMs provide opportunity to “verify” TEMPEST physics

18. TEMPEST shows that damping of GAMs follows theory with large banana orbits • Sugama & Watanabe show damping sensitive to kri at large q (large banana orbit) • TEMPEST show reasonable agreement (~38%) with theory.

19. TEMPEST yields a self-consistent neoclassical Er, which agrees with theory F(t)(eV) • In Circular geometry • Large aspect ratio • Boltzmann electrons • Small collision • Radial boundary buffer zones used ---- theory ---- simulation Buffer zone Buffer zone

20. TEMPEST shows Er generation by neoclassical polarization in a steep gradient region • Relative maximum charge separation is only 0.4% • Lq >> Lr • A plausible mechanism for pedestal Er

21. Going from 4D to 5D Turbulence and Transport Neoclassical Transport    (,,,0, (,,0,

22. Summary • TEMPEST is a fully nonlinear (full-f) five dimensional (3d2v) gyrokinetic continuum edge-plasma code. • As a test of the interaction of collisions and parallel streaming, TEMPEST is compared with published analytic and numerical results for endloss of particles confined by combined electrostatic and magnetic wells. Good agreement is found over a wide range of collisionality, confining potential, and mirror ratio; and the required velocity space resolution is modest. • In a large-aspect-ratio circular geometry, excellent agreement is found for a neoclassical equilibrium with parallel ion flow in the banana regime. • The four dimensional (2d2v) version of the code produces the first self-consistent simulation results of collisionless damping of geodesic acoustic modes and zonal flow with Boltzmann electrons using a full-f code. In divertor geometry, it is found that the endloss of particles and energy induces parallel flow stronger than the core neoclassical predictions in the SOL. • Our 5D gyrokinetic formulation yields a set of nonlinear electrostatic gyrokinetic equations that are for both neoclassical and turbulence simulations.