X. Q. Xu Lawrence Livermore National Laboratory, Livermore, CA 94550 USA

1. Dynamics of kinetic Geodesic-Acoustic modes and the radial electric field in tokamak neoclassical plasmas X. Q. Xu Lawrence Livermore National Laboratory, Livermore, CA 94550 USA Institute of Fusion Theory and Simulations Zhejiang University, Hangzhou, China In collaboration with E. Belli, K. Bodi, J. Candy, C. S. Chang, B. I. Cohen, R. H. Cohen, P. Colella, A. M. Dimits, M. R. Dorr, Z. Gao, J. A. Hittinger, S. Ko, S. Krasheninnikov, G. R. McKee, W. M. Nevins, T. D. Rognlien, P. B. Snyder, J. Suh, M. V. Umansky 2nd International West Lake Workshop on Fusion Theory and Simulation December 25–27, 2008; Hangzhou, China

2. Acknowledgments • We thank Drs. A. J. Brizard, L. Chen, T. S. Hahm, R. D. Hazeltine, F. L. Hinton, G. Hammet, H. Qin, E. J. Synakowski, R. E. Waltz, W. X. Wang, Z. Xiong, C. X. Yu, and F. Zonca for fruitful physics discussions. • This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory in part under Contract DE-AC52-07NA27344, Grant No. DE-FG02-04ER54739 at UCSD, and grants DE-FG03-95ER54309 at general Atomics.

3. Summary • TEMPEST correctly calculates the frequency and collisionless damping of geodesic acoustic modes and zonal flow (Rosenbluth-Hinton residual) with Boltzmann electrons using a full-f code in small e=r/R0limit. • TEMPEST shows GAM radial propagation in the pedestal plasmas, as theories predicted and expts measured. • TEMPEST shows that the long-time collisional decay of zonal flow with Lorentz collisions is consistent with theories • The enhanced GAM damping may explain experimental beam emission spectroscopy measurements on DIII-D and Doppler reflectometry measurements on ASDEX Upgrade on the edge q scaling of the GAM amplitude

4. TEMPEST: fully nonlinear (full-f) continuum gyrokinetic turbulence code • Solve for the particle distribution functionf(y,q,x,E,m,t)(avg. over gyration: 6D  5D) • 2D or 3D configuration space • 2D – velocity space (E0,m) • Solving GK field equations for F(y,q,x,t) using HYPRE • Realistic toroidal geometry, kinetic ions & electrons, electrostatic fluctuations, collisions, sophisticated algorithms. In this talk, we focus on 4D and circular geometry X. Q. Xu, Z. Xiong, M. R. Dorr, J. A. Hittinger, et al., Nuclear Fusion 47, 809-816(2007).

5. GAM is prominent in the edge plasmas • GAM and zonal flow has been clearly identified experimentally in tokamak and stellarator plasmas • GAM is a coherent, radially localized poloidal flow oscillation that is dominant in the outer regions of the confined plasmas • GAM and zonal flows are driven by the turbulence and act to regulate it via time-varying E×B flow shear de-correlation. DIII-D BES GAM expt. Mckee, PPCF, 48, s123(2006) ASDEX-U Conway, IAEA-CN-149/EX/2-1

6. TEMPEST resolves GAM and zonal flow active in barrier formation f(t)/f(t=0) • GAM is a coherent, radially localized poloidal flow oscillation that is dominant in the outer regions of confined plasmas • 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 • Simulation setup: • Homogeneous plasma with initial dni dni=dn0sin(2pr/Lr) Rosenbluth-Hinton Residual zonal flow e=r/R=0.2 q =3 ne=30,nm=60 ny=32,nq=16 ne=50,nm=100 ny=32,nq=64 Time(vti/R) Rosenbluth and Hinton, PRL 80, 724 (1998)

7. TEMPEST shows that the higher order resonances are keys to explain the large damping of GAM at large q Landau damping ( ) g=1-k2ri2/4 z=qRw/vti Gao, K.Itoh, et al, PoP 08 Sugama &Watanabe, JPP 72, 825(06) Gao, Itoh, et al, PoP,06 • Agree with theory for t=Te/Ti=0, where f is uniform in q • Previous calculations retain only up to 2nd order (n=2) Xu, Xiong, Gao, Nevins, Mckee, PRL, 100, 215001(2008)

8. Gyrokinetic codes show consistent higher order orbit resonances to explain large damping of GAM at large q * * (tokamak saftey factor) * X.Q. Xu, W.M. Nevins, Z. Gao, G.R. McKee, A.M. Dimits, E. Belli, J. Candy, P. Snyder, C.S. Chang, S. Ko, and J. Suh,TTF & FEC 2008

9. The change of ion’s parallel kinetic energy is the result of the work done by Er during the radial guiding center motion over its orbit, which is then damped by wave-particle resonances. wGAM<<wti wGAM>>wti Hinton & Robertson, Phys.Fluids, 27 1243(1984)

10. Derived analytical expression for GAM damping rate over a broad range of q Valid for Zhiyong Qiu, Liu Chen and Fulvio Zonca, PPCF 51, 012001(2008)

11. Xu, Xiong, Gao, Nevins, Mckee, PRL, May 12 (2008) Our GAM results are qualitatively consistent with relative DIII-D BES measured GAM amplitude If Sturb is a weak function of q, then a strong drop in  should correlate with an increase in GAM intensity I:

12. Our GAM results are qualitatively consistent with a relative GAM amplitude measured using Doppler reflectometry in ASDEX Upgrade with nearly circular cross section G D Conway et al, Plasma Phys. Control. Fusion 50 (2008) 085005

13. Simple analytical model for radial electric field dynamics From Gyrokinetic Poisson equation, we have Taking time derivative and assuming fi≈Fm+df in the integral with Ti=const. The solution gives the GAM oscillation, damping, and neoclassical residual Er(t)=Erneo+[ErHR+ ErGAMe-(iwGAM+gGAM)t]e-nct nc≈nii

14. TEMPEST simulation shows that steady state electric potential reaches a Boltzmann relation and agrees with Rosenbluth theory for the case of flat ion temperature profile and zero flow boundary conditions ef/Ti+ln(Ni/Nimid)=0 Rosenbluth, Rutherford, Taylor, Frieman, & Kovrizhnikh, IAEA 1971

15. TEMPEST shows the development of a neoclassical electric field with Lorentz collisions through different phaseswith ion temperature gradient F(y,q) ny=32,nq=64 nE0=30,nm=60 KEmax=20Ti q y

16. TEMPEST shows that the long-time collisional decay of zonal flow with Lorentz collisions is consistent with theories Hinton & Rosenbluth, PPCF 1999 Xiao, Catto & Molvig, PoP 14, 2007

17. During the relaxation of the electric field after the initial GAM phase and Rosenbluth-Hinton residual zonal flow, the neoclassical radial electric field from TEMPEST simulations follows with the standard neoclassical expression for parallel flow in the plateau regime. Hinton and Hazeltine, Rev. Mod. Phys. 48, 239 (1976)

18. Within one ion-ion collision time, the maximum ion heat flux reaches a steady state while the ion parallel velocity still evolves. The further development of U|| (or a toroidal component of ion velocity Uti) on the transport time scale may require a careful formulation of the gyrokinetic equation and the gyrokinetic Poisson equation, including sources and sinks, as well as higher order in ri/LB corrections to the first-order gyrokinetic equation

19. TEMPEST solves Gyrokinetic Poisson Eq in a steep gradient region • In circular geometry • Boltzmann e- • Lorentz collision • plateau regime Ni(y,q) F(y,q) e=0.2 q=3 q q y y

20. TEMPEST simulation shows that the neoclassical polarization is the spreading of the charge cloud over the thickness of particle banana orbits Z(m) dNi R(m)

21. Using the relaxation approach to solve the gyrokinetic Poisson equation, TEMPEST simulations yield the standard neoclassical relationship between Er and U||, and the 1st order (poloidal) correction to Er. Hinton-Rosenbluth analytical F. L. Hinton and M. N. Rosenbluth, Phys. Fluids 16, 836 (1973) E. A. Belli and J.Candy, Plasma Phys. Control. Fusion 50 (2008) 095010.

22. TEMPEST shows the radial propagation of GAM when there is a gradient in Ti Ti is inhomogeneous with radial propagation Ti is homogeneous, no radial propagation Xu, Phys. Rev. E, 78, 016406(2008)

23. TEMPEST shows the radial propagation of GAM when there is a gradient in Ti

24. TEMPEST shows that the kinetic GAM radially propagates outward in edge pedestal plasma as in expt. Evidence of outward propagating GAM In HL-2A [ Lan et al. PoP 2008] TEMPEST simulation Ti n*=0.2 e=0.3 Lorentz collision r-a(cm) (radial position) lGAMItoh~pri(LTi/ri)1/3 VprItoh~wGAM/lGAMItoh=15.95vTiri/R VprSim=12.08vTiri/R w2=w2GAM(y)+C(y,w)k2yri2 Itoh, Itoh, Diamond, et al, Plasma and Fusion Res., 1, 037 (2006) L Chen & Zonca, Europ. Phys. Lett, 83, 35001 (2008)

25. TEMPEST shows that the GAM radial propagation velocity is consistent with BES expt. measurement Tempest simulations yield Vpr TEMPEST simulation Ti n*=0.2 e=0.3 • The measured and calculated radial propagation velocity are vprDIII-D =2p/kr≈(879 – 1000)m/s and vprSim≈869m/s with LTi ≈10ri and t= 1, showing agreement to within 15%. • The GAM velocity is determined by the parameters at the location where the GAM is excited, not where it is measured. • By combining with Ti, Te, and q profiles, the measured velocity can be used to determine the location where the GAM is excited Lorentz collision (radial position)

26. In TEMPEST simulations, radial propagating GAMs are damped by collisionless Landau wave-particle resonances TEMPEST simulation For typical edge parameters, q≈2-4, gGAM≈0.1wGAM, the attenuation length due to the ion Landau damping can be estimated by Ti n*=0.2 e=0.3 Lorentz collision (radial position) Which is on the order of the edge ion temperature gradient scale length,

27. Summary • TEMPEST correctly calculates the frequency and rates of collisionless damping of geodesic acoustic modes and zonal flow (Rosenbluth-Hinton residual) with Boltzmann electrons using a full-f code in small e=r/R0 limit. • TEMPEST shows GAM radial propagation in pedestal plasmas, as theories predicted and expts measured. • TEMPEST shows that the long-time collisional decay of zonal flow with Lorentz collisions is consistent with theories • The enhanced GAM damping may explain experimental beam emission spectroscopy measurements on DIII-D and Doppler reflectometry measurements on ASDEX Upgrade on the edge q scaling of the GAM amplitude.