James Kniep, Jean-Noel Leboeuf and Viktor Decyk UCLA

1. Gyrokinetic Particle-In-Cell Calculations of Ion Temperature Gradient Driven Turbulence with Parallel Nonlinearity and Strong Flow Corrections James Kniep, Jean-Noel Leboeuf and Viktor Decyk UCLA Presentation for Univ. of South Alabama October 25, 2007, Mobile, AL

2. Outline • Motivation • ITG Model and Numerics: Strong flow corrections and Parallel nonlinearity • Effect of strong sheared poloidal flow corrections • Effect of parallel nonlinearity • Summary

3. First, a Little Background on Gyrokinetics & PIC Code Gyrokinetics treats particle orbits as an average around the center particle Center of Orbit

4. First, a Little Background on Gyrokinetics & PIC Code While PIC codes treat the orbit as a particle for purposes of interactions Particle-in-cell particle Center of Orbit

5. Motivation • Extensions of gyrokinetics, such as for instance finite r*= ri /a effects, are receiving renewed attention: • We (re-)examine two extensions here: - Strong sheared poloidal flow corrections (ITB relevant) - Parallel nonlinearity (Intrinsic relevance)

6. Gyrokinetics with Parallel Nonlinearity and Strong Poloidal Flows - Formulated by T. S. Hahm, “Nonlinear gyrokinetic equations for turbulence in core transport barriers”, Phys. Plasmas 3, 4658-4664 (1996) Strong Flow Terms in Red

7. Gyrokinetics with Parallel Nonlinearity and Strong Poloidal Flows Parallel Nonlinearity in Blue Strong Flow Terms in Red External potential drives external ExB flow uE New formulation implemented in UCAN gyrokinetic code

8. UCAN Gyrokinetic PIC Code - 3D - Toroidal geometry - Cartesian coordinates - Circular cross section - Nonlinear f - Adiabatic electrons - Massively parallel implementation using 1D domain decomposition the long way around the torus (z-direction) and MPI for message passing - Parallel calculations performed on our own Dawson cluster of Macintosh G4s and on the IBM-SPs at ORNL (Eagle) and at NERSC (Seaborg)

9. Model Equilibrium Profiles Temperature Ti Safety Factor q r*= ri /a=1/90 Flat Density

10. cosine sine External Poloidal Flow Profiles tanh Flow Velocity in Red Flow Shear in Blue

11. Strong Flow Effects on ITG Turbulence Externally imposed poloidal sheared flow retards growth rate and reduces saturation level; similar to previously obtained results, e.g. Kissick et al, Phys. Plasmas 6, 4722-7 (1999).

12. Strong Flow Effects on ITG Turbulence Sine poloidal sheared flow profile is more effective at retarding growth rate and saturation level

13. Strong Flow Terms vs. Regular Flow Terms

14. Strong Flow Terms vs. Regular Flow Terms Sine Profile: Er=2x10-6 Electrostatic potential contours in late nonlinear phase (wcit=1.5x105) With Regular Flow Terms With Strong Flow Terms

15. Strong Flow Terms vs. Regular Flow Terms

16. Strong Flow Terms vs. Regular Flow Terms

17. Strong Flow Terms vs. Regular Flow Terms Tanh Profile: Er=1x10-6 Electrostatic potential contours in late nonlinear phase (wcit=1x105) With Regular Flow Terms With Strong Flow Terms

18. Strong Flow Terms are dependent upon Flow Gradient As gradient is changed, noticeable effects are seen in the saturation level of the Electrostatic Energy

19. Summary: Strong Flow Effects • Gyrokinetic simulations of ITG turbulence with externally imposed sheared poloidal flows show that: - Inclusion of flow strongly reduces growth rate and saturation level - Addition of strong flow terms has little effect on linear phase & slightly more noticeable effect on nonlinear phase - Strong flow terms have more noticeable effect when flow gradient is altered

20. Parallel Nonlinearity Effect: No Flow No difference in linear growth; very little difference in saturation level

21. Parallel Nonlinearity Effect: External (Tanh) Flow Only Little change seen in saturation level with parallel nonlinearity

22. Parallel Nonlinearity Effect: Zonal (Fluctuation-Generated) Flow Only Combination of parallel nonlinearity and zonal flow leads to significantly lower saturation level

23. Parallel Nonlinearity On and Zonal Flows On wcit=5.5x104

24. Parallel Nonlinearity Off and Zonal Flows On wcit=5.5x104

25. Parallel Nonlinearity Effect Parallel Nonlinearity On and Zonal Flows On wcit=5.5x104 Parallel Nonlinearity Off and Zonal Flows On wcit=5.5x104

26. Scaling with System Size or r*= ri /a Equation of Motion: Parallel Nonlinearity Mirror Force

27. Factoring out common terms: 1 Mixing Length << 1 Parallel nonlinearity should become less of a factor with increase in system size

28. Observed Scaling with System Size: R and a doubled ri fixed, LT fixed=> a/ ri doubled, R/ LT doubled Reduction of saturation level and flux by parallel nonlinearity when combined with zonal flows persists as a function of system size, i.e. ri/a Double system size => r*= ri /a=1/180

29. Observed Scaling with System Size: R and a doubled ri fixed, LT doubled => a/ ri doubled, R/ LT fixed Reduction in saturation level suppressed when system size is doubled but R/ LT is keptfixed

30. Overall Summary • Gyrokinetic simulations of ITG turbulence with externally imposed sheared poloidal flows show that: - Inclusion of flow strongly reduces growth rate and saturation level - Addition of strong flow terms has little effect on linear phase & minute quantitative effects on nonlinear phase - Strong flow terms have more noticeable effect on changes of the flow gradient • Parallel nonlinearity has: - No effect in the absence of zonal flows - Little effect in the presence of externally imposed sheared flow - Quantitatively reduces saturation level and flux when zonal flows are allowed to evolve (in agreement with Villard et al) - Effect much diminished as system size is increased and ion Larmor radius and instability drive kept fixed

31. Future Research Directions • With the construction of ITER, further research in higher-order terms is important: • Scaling up to ITER sized dimensions • Run w/ proper plasma shaping (D cross section) • Inclusion of electromagnetic interactions • Inclusion of non-adiabatic electrons