1. Zonal flow suppression of ETG turbulence S. Parker, J. Kohut, Y. Chen, Univ. of Colorado W. Lee, PPPL Z. Lin, UC-Irvine Disclosure: Views expressed are my own, and I am a particle simulator

2. Summary ETG zonal flow is weak for early times ETG zonal flow grows via the Rosenbluth-Hinton random walk process Eventually, the algebraically growing zonal flow shear suppresses the turbulence ETG flux drops to low values for the collisionless problem no matter what

3. Outline - ETG controversy - Particle convergence studies - Zonal flow behavior - Shear flow suppression - Zonal flow dynamics explained

4. ETG CAN BE EXPERIMENTALLY RELAVENT! -- Dorland, Jenko, Kotschenreuther, Rogers PRL 2000 GTC SEES ETG TRANSPORT IS SMALLER BY 10 OR MORE! -- Lin, Chen, Zonca, PoP 2005 (APS 2004 Invited talk) GTC IS NOISE DOMINATED! -- Nevins, Hammett, Dimits, Dorland, Shumaker, PoP 2005 Thesis Advisor: Ned Birdsall Thesis Advisor: Wei-li Lee

5. Nevins, et al. PoP 2005 -- Discrete particle noise causes the flux to drop. The fewer the particles, the more the noise and the flux will drop sooner. Effects both ETG and ITG particle simulations. Abstract: “… Discrete particle noise is a numerical artifact so both these PG3EQ simulations and by inference, the GTC simulations that they reproduced have little to say about steady-state ETG turbulence..” Nevins, et al. never produced a numerically converged result. 16 million particle runs are quite modest. Physics analysis of nonlinear numerical solution gone astray. What might be the governing equations? It is not the N-body problem.  f method is a numerical scheme to solve the smooth Vlasov problem.

6. GEM should see similar (not converged) behavior as PG3EQ Noise is always an issue MUST ALWAYS CHECK CONVERGENCE WRT PARTICLE NUMBER! Monitor (  f/f ) rms

7. GEM shows convergence wrt particle number: R/L T =6.9 Qualitatively similar to Dorlan/Jenko and Nevins result, disagrees with GTC

8. GEM shows convergence wrt particle number: R/L T =5.3 Now agrees with GTC! What is causing the drop? It’s not noise… Nevin’s noise argument would predict 512 ppc would drop latest?? Cyclone base case parameters except R/L T =5.3

9. Zonal flows dominate when flux drops -- cause or effect?

10. “The mean square potential fluctuation increases linearly in time, neglecting collisions …” Rosenbluth and Hinton PRL 2000 -- BUT that’s for ITG! Rosenbluth and Hinton ITG model equation  s = 

11. Explanation of ETG flux drop: Zonal flow suppression 1) Zonal flow continues to grow due to turbulent kicks via random walk process a la Rosenbluth and Hinton 2) Zonal flow eventually suppresses turbulence when  sh ≥  T Zonal flow can grow via random kicks by either turbulence (R/LT=5.3 case) or numerical noise (not converged R/LT=6.9 case) -- we really need enough particles to show the time of the drop is not sensitive to particle number BUT, if we are collisionless and the turbulent viscosity term is small, the concept of a turbulent stationary-state is ill-posed!!

12. Shear flow suppression of turbulence Biglari, Diamond, Terry PFB 1990 Hahm PoP 1994  sh ≥  T For a crude estimate take:  T ≈   sh = (k  / k r ) k r 2 |  | k / B ≥  Inverse correlation lengths of turbulence in r and  k r of the zonal flow Zonal flow mode

13. Suppression when shearing rate equals growth rate (approximate)

14. Flux drops when zonal flow removed

15. Zonal flow suppresses turbulence in R/L T =6.9 case too -- fewer particles causes zonal flow to build up at a faster rate 64 particle per cell case R/L T =6.9 Flux drops out early

16. Slab gyrofluid equations to describe zonal flow dynamics ITG and ETG are nearly isomorphic

17. Evolution of zonal flows -- ETG drive is k r 2 (= 0.002) smaller!

18. Summary Zonal flow continues to grow in late times due to ExB nonlinearity via random walk (Rosenbluth-Hinton mechanism) When  sh ≥  T, the turbulence is suppressed and the flux drops out The random kicks can come from both turbulent fluctuations or numerical error The same mechanism may explain why there is a secondary flux drop at later times in ITG simulations Numerical convergence studies are essential (no surprise here) Careful monitoring of (  f/f ) rms is another practical approach