University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Implicit Particle Closure IMP D. C. Barnes NIMROD Meeting April 21, 2007

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies 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

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies IMP Algorithm Fluid equations –Quasineutral, no displacement current –Electrons are massless fluid (extensions possible)

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

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

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

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Constraint Moments With infinite precision and particles, should have …

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Constraint Moments Satisfy constraints by shaping particle in both x and w

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Constraint Moments Using Hermite polynomials, find Projection of weight equation is then

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Constraint Moments …and, closure moment has symmetric form

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Symmetry Leads to Energy Integral Usual fluid w. isoT ionsInterchange w. closure

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Symmetry Leads to Energy Integral Usual fluid w. isoT ionsClosure energy

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Symmetry Leads to Energy Integral

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies 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!

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

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies 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)

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Time Centering Particles use average of u (depends on A, so need iterate)

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Space Differencing Use Yee mesh, w. velocity with B

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies G-Mode Tests 1 y x Contours of u x for Roberts- Taylor G-mode g Following Roberts, Taylor, Schnack, Ferraro, Jardin, …

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies 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

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

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

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies 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?

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies 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

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies 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

University of Colorado at Boulder Santa Fe Campus Center for Integrated Plasma Studies Preprints