Hybrid Kinetic-MHD simulations Status and Updates

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1 in NIMROD simulations Status and Updates Charlson C. Kim 1,2 Yasushi Todo 2 and the NIMROD Team 1. University of Washington, Seattle 2. National Institute for Fusion Science NIMROD Team Meeting Austin, Texas April 30, 2011 university-logo

2 Outline in NIMROD 1 in NIMROD Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D 2 Components of δf 3 components configuration space velocity space 3 Antenna Simulations Energetic Particle Simulations

3 in NIMROD Summary of PIC Capabilities Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D tracers, linear, (nonlinear) two equations of motion drift kinetic (v,µ), Lorentz force ( v) multiple spatial profiles - loading in x proportional to MHD profile, uniform, peaked gaussian multiple distribution functions - loading in v slowing down distribution, Maxwellian, monoenergetic room for growth improved parallelization efficient high order implementation develop multispecies option, e.g. drift+lorentz full f(z) PIC numeric representation of f eq ( x, v) e.g. load experimental phase space profiles for evolution of δf kinetic closure university-logo

4 in NIMROD Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D Upgrades to hybrid kinetic-mhd implementation improved B evaluation previous implementation calculated operator locally for all particles new implementation computes B as grid quantity eliminates need for additional PIC gradient shape function improves accuracy (same order accuracy as other grid quantities) improves performance 40% (primarily through elimination of extra corrector step) added new particle diagnostics

5 in NIMROD Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D Benchmark of Drift Kinetic (1,1) Kink With M3D C. C. Kim, PoP (2008) circular, monotonic q, q 0 =.6, q a = 2.5, β 0 =.08, R/a = 2.76, E hmax = 10KeV, dt=1e-7, τ A = 1.e6

6 in NIMROD n = 1 Contour of Ideal Kink Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D

7 in NIMROD n = 1 Contour of Ideal Kink Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D δv normal shows (downward) tilt of q 1 volume δv φ unipolar flow (into screen) at q = 1 surface tangential flow similar (m=1) δj normal B 0φ consistent with δv normal B 0φ is out of screen B 0pol counter clockwise δp build-up consistent with δv normal

8 in NIMROD n = 1 Contour β frac =.25 Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D

9 in NIMROD n = 1 Contour β frac =.75 Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D

10 in NIMROD Evolution of n = 1 δp hot Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D δp hot t = 0 t.2ω h t.4ω h δ p hot t = 0 t.2ω h t.4ω h

11 in NIMROD Evolution of n = 1 δf(v,v ) Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D t = 0 t.1ω h t.2ω h t.25ω h t.35ω h t.4ω h

12 in NIMROD Summary of PIC Capabilities Upgrades and improvements (1,1) benchmark with M3D Observations of Moments of δf n=1 energetic particles cause real frequency response δ p concentrated on outboard side topology of toroidal flow changes? particular to δv φ? signature of fishbone does not (dramatically) change topology of other field quantities does change phase relation, e.g. δv norm and δp concentration of δf n=1 (v,v ) activity to trapped cone? what is origin and role of the asymmetry? what is role of trapped vs. passing particles? what is the structure in the trapped cone university-logo

13 in NIMROD 3 Components of δf 1 f eq δf = v D = 3 components configuration space velocity space [( ) ] mg v 2 eψ 0B 3 + v2 δb B µ 0v J δe (1) 2 + δv ψp (2)+ 3 ψ 0 2 ( m v 2 eb 3 + v2 2 δv = δe B +v B 2 δb B δf has 3 components 1 gρ - kinetic term 2 v E B ψ - radial particle flux 3 v D E - energy exchange eε 1/2 ε 3/2 +ε 3/2 c v D δe(3) ) (B B)+ µ0mv2 eb 2 J examine δf moments of components, e.g. (gρ )δfdv convolution is axisymmetric, i.e. n = 0 and static n = 2 also exists but have not examined yet university-logo

14 0D Comparison in NIMROD 3 components configuration space velocity space normalized amplitude (wrt v E B ψ - dominant component) %(passing/trapped) contribution β frac gρ v E B ψ v D E (-13/113) (69/31) (45/55) (ε/100) (58/42) (27/73) (0/100) (52/48) (18/82) v E B ψ equal passing and trapped (destabilizing) gρ &v D E small, negative, and mostly trapped (stabilizing) gρ term interesting at lower β frac (destabilizing?)

15 in NIMROD Configuration Space β frac =.25 3 components configuration space velocity space

16 in NIMROD Configuration Space β frac =.75 3 components configuration space velocity space

17 in NIMROD Configuration Comparison 3 components configuration space velocity space β frac gρ v E B ψ v D E (-13/113) (69/31) (45/55) (ε/100) (58/42) (27/73) (0/100) (52/48) (18/82) v D E increases tilt v E B ψ tail grows subtle differences in gρ peak at q 1 surface shifts outboard all trapped particle for β frac >.25 worth examining passing vs. trapped population university-logo

18 in NIMROD Velocity Space β frac =.25 3 components configuration space velocity space

19 in NIMROD Velocity Space β frac =.75 3 components configuration space velocity space

20 in NIMROD Velocity Space Comparison 3 components configuration space velocity space β frac gρ v E B ψ v D E (-13/113) (69/31) (45/55) (ε/100) (58/42) (27/73) (0/100) (52/48) (18/82) v D E most dramatic change as trapped particles dominate peak in v E B ψ shifts to larger v as expected from increasing ω (1,1) interesting peaks in passing particles at v 0 barely passing potatoe orbits strong asymmetries are evident particles have orbits in both x and v space trapped particles dominate outboard passing particles dominate inboard plot in alternate coordinates, e.g. (P ζ,µ) university-logo

21 in NIMROD ITPA TAE benchmark Antenna Simulations Energetic Particle Simulations q = ( ) r 2 a R = 10m, a = 1m, B 0 = 3T, n = H + /m 3 τ A =.67µs [ n h = n 0f exp L f tanh ( s s f ) ] n 0f = D + /m 3, =.2, L f =.3, s f =.5 f h (v) is Maxwellian, T h = KeV TAE gap at s =.5m, q = 1.75 n = 6, m = [10,11] dominant modes expect ω rad/s, γ = O(10 5 )s 1

22 q profile in NIMROD Antenna Simulations Energetic Particle Simulations not exactly ITPA benchmark profile erroneously thought I had better control of q profile q wall a little large uncertain of curvature will need better q-profile control university-logo

23 in NIMROD Antenna drive looks promising Antenna Simulations Energetic Particle Simulations drive with imposed external source in δp hot ω = rad/s, m = 11, r =.55, =.05 university-logo

24 in NIMROD Antenna drive time trace Antenna Simulations Energetic Particle Simulations response ω rad/s after shutoff university-logo

25 in NIMROD Antenna response contours Antenna Simulations Energetic Particle Simulations mix of m = 10,11 university-logo

26 in NIMROD Vnorm response contours Antenna Simulations Energetic Particle Simulations dominant m = 10, r =.45 university-logo

27 in NIMROD Antenna Simulations Energetic Particle Simulations Timetrace of TAE shows real frequency ω = rad/s, m = 10, r =.45, γ rough agreement with ITPA benchmark V φ amplitude larger than expected! university-logo

28 in NIMROD Energetic Particles drive TAE Antenna Simulations Energetic Particle Simulations [ n h = n 0f exp L f tanh ( s s f ) ] n 0f = D + /m 3, =.2, L f =.3, s f =.5 university-logo

29 Contour of Vnorm in NIMROD Antenna Simulations Energetic Particle Simulations poloidal mode coupling anti-ballooning university-logo

30 in NIMROD Antenna Simulations Energetic Particle Simulations δf diagnostic shows expected symmetry in response original motivation for implementation of diagnostic v D E about 2% of v E B ψ

31 in NIMROD Banded Structure in δf diagnostic Antenna Simulations Energetic Particle Simulations banded structure indicates TAE is resonant wrt ε not v not sure if nodule structure is real

32 Summary in NIMROD Antenna Simulations Energetic Particle Simulations improved accuracy and performance of hybrid kinetic-mhd implemented new diagnostics identify interesting phenomenological features both trapped and passing particles is important barely passing particles are crucial from phenomenology, move to quantification, then theory are promising large V φ amplitude troubling δf diagnostic shows anti-symmetry between v D E and v E B ψ work continues on energetic particle driven TAE will need better q-profile control

33 Drift Kinetic Equation of Motion follows gyrocenter in limit of zero Larmour radius [ ] 1 2 reduces 6D to 4D +1 x(t),v (t),µ = mv2 B drift kinetic equations of motion ẋ = v ˆb+v D +v E B v D = m eb 4 ( v 2 + v2 2 v E B = E B B 2 m v = ˆb (µ B ee) ) )(B B2 + µ 0mv 2 2 eb 2 J university-logo

34 Slowing Down Distribution for Hot Particles slowing down distribution function f eq = P 0exp( P ζ ψ 0 ) ε 3/2 +ε 3/2 c P ζ = gρ ψ canonical toroidal momentum, ε energy, ψ p poloidal flux, ψ 0 gradient scale length, ε c critical energy the evolution equation for δf { δf mg = f eq eψ 0 B 3 [( v 2 + v2 2 eε 1/2 ) ] δb B µ 0 v J δe } + δv ψ p + 3 ψ 0 2ε 3/2 +εc 3/2 v D δe v D = m ( ) eb 3 v 2 + v2 (B B)+ µ 0mv 2 2 eb 2 J δv = δe B B 2 +v δb B university-logo

35 n = 1 Contour β frac =.50

36 Configuration Space β frac =.50

37 Velocity Space β frac =.50

38 Contour of Phot peaks inboard! university-logo