1. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Implicit Particle Closure IMP D. C. Barnes NIMROD Meeting April 21, 2007

2. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Outline The IMP algorithm –Implicit fluid equations –Closure moments from particles –  f with evolving background –Constraint moments Symmetry – Energy conservation theorem –Conservation for discrete system – absolutely bounded, no growing weight problem! Present restricted implementation G-mode tests Future directions

3. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ IMP Algorithm Fluid equations –Quasineutral, no displacement current –Electrons are massless fluid (extensions possible)

4. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ IMP Algorithm Fluid equations –Ions are massive, collisionless, kinetic species –Use ion (actually total) fluid equations w. kinetic closure

5. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ IMP Algorithm  f particle closure algorithm –Background is fixed T with n, u evolving –Particle advance uses particular velocity w (very important)

6. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ IMP Algorithm Note: (perturbation) E does not enter closure directly There is some kind of symmetry between advance and

7. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Constraint Moments With infinite precision and particles, should have …

8. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Constraint Moments Satisfy constraints by shaping particle in both x and w

9. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Constraint Moments Using Hermite polynomials, find Projection of weight equation is then

10. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Constraint Moments …and, closure moment has symmetric form

11. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Symmetry Leads to Energy Integral Usual fluid w. isoT ionsInterchange w. closure

12. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Symmetry Leads to Energy Integral Usual fluid w. isoT ionsClosure energy

13. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Symmetry Leads to Energy Integral

14. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Symmetry Leads to Energy Integral r.m.s. of particle weights absolutely bounded Stability comparison theorem –Kinetic system more stable than isoT ion fluid system –But only for marginal mode at zero frequency This is absolutely the most important point!

15. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ IMP2 Implementation 2D, Cartesian TE polarization –B normal to simulation plane –E in plane Linearized, 1D equilibrium Uniform T

16. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Time Centering Moments use Sovinec’s time-centered implicit leap-frog –Direct solve Particles use simple predictor-corrector –Present, use full Lorentz orbits w. orbit averaging (Anticipating Harris Sheet or FRC calculations) –Iteration required (3 – 5 typical count)

17. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Time Centering Particles use average of u (depends on A, so need iterate)

18. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Space Differencing Use Yee mesh, w. velocity with B

19. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ G-Mode Tests 1 y x 0 00.1 Contours of u x for Roberts- Taylor G-mode g Following Roberts, Taylor, Schnack, Ferraro, Jardin, …

20. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ G-Mode Tests Two series –Low  Hall stabilized – with and w/o closure –High  gyro-viscous stabilized (Hall turned off) Numerical parameters –N x x N y = 30 x 16 –9 – 25 particles/cell –Typically 100 particle steps/fluid step

21. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Perturbed density G-Mode Tests Low  –  0.02 –B = 6.0 T –n = 2. x 10 20 m -3 –g = 1. x 10 12 m/s 2 –L n = 120 m –T = 8.94 keV –  = 2.28 mm Arrow marks k  = 0.15 Fluid only Closure

22. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ G-Mode Tests High  –  1.0 –B = 0.482 T –n = 5.78 x 10 19 m -3 –g = 2.7 x 10 8 m/s 2 –L n = 10 m –T = 10 keV –  /  i = 2.25 x 10 -4 –  = 2.99 cm Arrow marks k  = 0.15

23. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ G-Mode Tests Hall and gyro-viscous stabilization of fundamental G-mode observed –Stability seems consistent with Roberts- Taylor, modified by Schnack & Ferraro, Jardin New, higher k x mode observed at k  > 0.2 or so –Drift wave? –Present in fluid (+ Braginskii) also?

24. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Temperature Variation? Vlasov equation is linear, so superposition allowed –Superimpose number of uniform T solutions –Slightly different  equation Mild restriction on equilibrium T –Density must depend only on potential

25. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Future Directions Short-term –Understand high k x mode –Finish manuscript Medium-term –Add T variation –Add full polarization, check Landau damping Long-term –Gyrokinetic version –Parallel implementation

26. University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies http://cips.colorado.edu/ Preprints