Implicit Particle Closure IMP

1. Implicit Particle ClosureIMP D. C. Barnes NIMROD Meeting April 21, 2007

2. Outline • The IMP algorithm • Implicit fluid equations • Closure moments from particles • df 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. IMP Algorithm • Fluid equations • Quasineutral, no displacement current • Electrons are massless fluid (extensions possible)

4. IMP Algorithm • Fluid equations • Ions are massive, collisionless, kinetic species • Use ion (actually total) fluid equations w. kinetic closure

5. IMP Algorithm • df particle closure algorithm • Background is fixed T with n, u evolving • Particle advance uses particular velocity w (very important)

6. IMP Algorithm • Note: (perturbation) E does not enter closure directly • There is some kind of symmetry between advance and

7. Constraint Moments • With infinite precision and particles, should have …

8. Constraint Moments • Satisfy constraints by shaping particle in both x and w

9. Constraint Moments • Using Hermite polynomials, find • Projection of weight equation is then

10. Constraint Moments • …and, closure moment has symmetric form

11. Symmetry Leads to Energy Integral Usual fluid w. isoT ions Interchange w. closure

12. Symmetry Leads to Energy Integral Usual fluid w. isoT ions Closure energy

13. Symmetry Leads to Energy Integral

14. 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. IMP2 Implementation • 2D, Cartesian • TE polarization • B normal to simulation plane • E in plane • Linearized, 1D equilibrium • Uniform T

16. 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. Time Centering Particles use average of u (depends on A, so need iterate)

18. Space Differencing • Use Yee mesh, w. velocity with B

19. 1 x g 0 0 0.1 y Contours of ux for Roberts-Taylor G-mode G-Mode Tests • Following Roberts, Taylor, Schnack, Ferraro, Jardin, …

20. G-Mode Tests • Two series • Low b Hall stabilized – with and w/o closure • High b gyro-viscous stabilized (Hall turned off) • Numerical parameters • Nx x Ny = 30 x 16 • 9 – 25 particles/cell • Typically 100 particle steps/fluid step

21. Perturbed density G-Mode Tests • Low b • b = 0.02 • B = 6.0 T • n = 2. x 1020 m-3 • g = 1. x 1012 m/s2 • Ln = 120 m • T = 8.94 keV • r = 2.28 mm • Arrow marks kr = 0.15 Fluid only Closure

22. G-Mode Tests • High b • b = 1.0 • B = 0.482 T • n = 5.78 x 1019 m-3 • g = 2.7 x 108 m/s2 • Ln = 10 m • T = 10 keV • g/Wi = 2.25 x 10-4 • r = 2.99 cm • Arrow marks kr = 0.15

23. 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 kx mode observed at kr > 0.2 or so • Drift wave? • Present in fluid (+ Braginskii) also?

24. Temperature Variation? • Vlasov equation is linear, so superposition allowed • Superimpose number of uniform T solutions • Slightly different w equation • Mild restriction on equilibrium T • Density must depend only on potential

25. Future Directions • Short-term • Understand high kx mode • Finish manuscript • Medium-term • Add T variation • Add full polarization, check Landau damping • Long-term • Gyrokinetic version • Parallel implementation

26. Preprints