Hybrid Kinetic-MHD simulations with NIMROD

Transcription

1 simulations with NIMROD 1 Yasushi Todo 2, Dylan P. Brennan 3, Kwang-Il You 4, Jae-Chun Seol 4 and the NIMROD Team 1 University of Washington, Seattle 2 NIFS, Toki-Japan 3 University of Tulsa 4 NFRI, Daejeon-Korea CEMM Meeting June 12, 2012

2 Outline ITPA TAE benchmark 1 ITPA TAE benchmark Velocity Space Diagnostics 2 3

3 Summary of Capabilities 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 multispecies capability HKMHD visualization uses self-consistent+tracers e.g. - Maxwellian+slowing down, drift+lorentz multiple coupling models pressure coupling current coupling

4 more on YouTube, search:charlsonification visualizations generated with VisIt

5 ITPA TAE benchmark A. Mishchenko, PoP (2009) Velocity Space Diagnostics Toroidicity induced shear Alfven Eigenmodes (TAE) toroidicity produces gaps in Alfven continuum discrete modes can be excited in these gaps energetic particles with ω gap k v can excite TAEs circular cross section R = 10, a = 1, B 0 = 3 q = (r/a) 2 single gap bulk ions are H + at n = T = 1KeV energetic particles are D + T h = (100KeV,800KeV) ( n h = n 0 exp ( )) L tanh ρ ρ0 n 0 = , =.2, L =.2, ρ 0 =.5 ω gap

6 Velocity Space Diagnostics n = 6 eigenmode shows good agreement current coupling model δv normal shows clear m = 10,m = 11 ω good agreement with gap γ small by 2

7 Velocity Space Diagnostics 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 { [( ) ] δf mg = f eq eψ 0 B 3 v 2 + v2 δb B µ 0 v 2 J δe } + δv ψ p + 3 ψ 0 2 eε 1/2 ε 3/2 +ε 3/2 c v D δe δv = δe B B 2 +v δb B v D = m ( ) eb 3 v 2 + v2 (B B)+ µ 0mv 2 2 eb 2 J

8 Velocity Space Diagnostics 3 Components of δf for phase space analysis 1 f eq δf = [( ) ] mg v 2 eψ 0B 3 + v2 δb B µ 0v J δe (1) 2 + δv ψp (2)+ 3 ψ 0 2 eε 1/2 ε 3/2 +ε 3/2 c v D δe(3) δv = δe B +v B 2 δb B ( ) m v D = v 2 eb 3 + v2 (B B)+ µ0mv2 J 2 eb 2 δf has 3 components 1 gρ - kinetic correction term 2 δv E B ψ - radial particle flux 3 v D δe - energy exchange examine δf moments of components, e.g. (v D δe)δfdv convolution is axisymmetric, i.e. n = 0 and static

9 Components of δf show symmetry Velocity Space Diagnostics clear v resonance symmetry between spatial and velocity term dominantly passing particles!

10 n = 1 DIII-D linear simulations D. P. Brennan, C. C. Kim, and R. J. LaHay, Nuclear Fusion 52, , (2012) 6 a) x b) c) P 0 e -2ψ/ψ 0 10 q P (pa) Z (m) 0 q=1.5 Z (m) q= ψ (Wb) R (m) 1 2 R (m) understand the stability properties of D3D hybrid discharge hybrid discharge sensitive to small changes in β N explore range of q = [.9,1.4] β N = [2,4]

11 β N DIII-D experiments indicate sensitivity to increases in δβ N La Haye - Physics of Plasmas, 17 (2010), Nuclear Fusion, 51 (2011) n = 1 MHD Stability Map in (q min,β N ) Ideal MHD Stability Boundary Reconstruction Flattop γτ A MHD Only (n=1) S=τ R /τ A ~ Pr=100 Experimental Trajectory q min contours of MHD growth rate (NIMROD) - γτ A as q min Ideal MHD Stability Boundary in red (PEST) - stable to right Experimental Trajectory primarily in stable region of MHD stability map β N trigger 2/1 in experiments

12 q scan at fixed q-profile highlights abrupt transitions increase in q accompanied by decrease in β N (γ,ω)τ A q min γ and ω a) NIMROD Pressure 1.2 x Experiment γτ A MHD Only γτ A with particles ωτ A with particles β N 4 3 β N energetic particles stabilizing in ideal region map divides into 3 regions ideal MHD unstable high frequency low frequency abrupt transition at q 1.25 m = 1 m = 2

13 β N Energetic particles may help explain sensitivity to δβ N Kinetic-MHD Stability Map in (q min,β N ) Ideal MHD Stability Boundary Flattop Reconstruction γτ A Energetic Particle Driven MHD Modes (n=1) q min 0.01 Experimental Trajectory contours of γτ A have dependence on β N Ideal MHD Stability Boundary in red EP s drive instabilities in MHD stable region

14 Typical sawtooth activity in L-mode KSTAR K.I. You et al. moderate frequency sub-signal, T O(10 4 )sec ECI indicates a coherent mode undetermined poloidal mode number m = 1,2? EP driven nonresonant mode?

15 Linear n = 1 HKMHD simulation of KSTAR EFIT equilibrium shot#6017 at t = 3502 B 0 = 1.96T, β 10 3 q 0 = 1.85,q 99 = 5.7 NIMROD - scale q-profile to perform HKMHD scan energetic D +1 10% of β f 0 (v) 1/(v 3 +v 3 c), v max = m/s should be considered Proof of Concept

16 Scaled profiles for g , q 0 = [.98,1.2,1.4]

17 γ and ω vs q 0 shot#6017 shows abrupt transition energetic particles destabilizing for q < 1! abrupt transition in γ and ω differs from D3D hybrid shot results not necessarily converged

18 Ideal n = 1 contours of T i shows dominant m = 1 q 0 = [.98,1.0] - difference in radius and profile

19 HKMHD n = 1 contours show m = 1 2 transition q 0 = [.98,1.3] - topology differs from ideal mode r mode 10cm

20 Velocity space contours are an orthogonal view in (x, v) q 0 = [.98,1.3] - more activity in q 0 = 1.3 broader cone suggests interactions at larger radii

21 q 0 =.98 shows dominantly passing particles +v barely passing particles (interacting on inboard side) have strongest contribution

22 q 0 =.98 radial subslice shows q < 1 dominant total q > 1 r <.2m q < 1 r >.2m

23 q 0 = 1.3 shows Alfven-like signature symmetric δv E B ψ and v D δe is Alfven-like ω too small for TAE, maybe BAE?

24 q 0 = 1.3 radial subslice shows q > 1 dominant total q > 1 r <.2m q < 1 r >.2m

25 n = 1 HKMHD offers plausible explanation 1 core heats, moves left in q 0 2 start high ω, low γ plateau 3 at q 0 1 high γ (1,1) kink 4 kink drives crash

26 Conclusion ITPA TAE benchmark ITPA TAE benchmark shows reasonable agreement more model work required DIII-D hybrid discharge published - NF(2012) preliminary promising nonresonant n = 1 EP driven activity similar to observations topology and features reminiscent ECI and inversion radius BUT results are preliminary and not necessarily converged velocity space diagnostics useful in inferring HKMHD interaction

27 drift kinetic equations of motion and δf δf PIC 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

28 drift kinetic equations of motion and δf δf PIC Drift Kinetic requires CGL-like δp h PIC moment for drift kinetic equations, assume CGL-like δp 0 0 δp h = 0 δp δp evaluate pressure moment at x δp(x) = m v V v V δf(x,v)d 3 v δf is perturbed phase space density, m mass of particle, and V is COM velocity, usually substitute COM of particles implemented as δp = δp I+δ pbb, δ p = δp δp

29 drift kinetic equations of motion and δf δf PIC δf PIC reduces 1/ N noise S. E. Parker and W. W. Lee, PFB 5, (1993) δf PIC reduces the discrete 1/ N particle noise Vlasov Equation f(z) t +ż f(z) z = 0 split phase space f = f eq (z)+δf(z,t) δf evolves along the characteristics ż (equations of motion) δf = δz f eq z using ż = z eq + δz and ż eq feq z = 0 moments of distribution function are A n = v n δf dv