Hybrid Kinetic-MHD simulations with NIMROD

Transcription

1 in NIMROD simulations with NIMROD Charlson C. Kim 1 Dylan P. Brennan 2 Yasushi Todo 3 and the NIMROD Team 1 University of Washington, Seattle 2 University of Tulsa 3 NIFS, Toki-Japan December 2&3, 2011

2 Outline in NIMROD 1 in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 2 3 Components of δf 3 3 regions of stability map

3 Outline in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 1 in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 2 3 Components of δf 3 3 regions of stability map

4 in NIMROD NIMROD C.R. Sovinec, JCP, 195, 2004 intro to NIMROD hybrid kinetic-mhd the equations of motion and δf parallel, 3-D, initial value extended MHD code 2D high order finite elements + Fourier in symmetric direction linear and nonlinear simulations semi-implicit and implicit time advance operators model fusion relevant geometry and physical parameters magnetic Lundquist number - S = τ η /τ A large thermal anisotropy - χ χ 10 8 active developer and user base continually expanding capabilities large and growing Verification&Validation

5 in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf Example NIMROD grid - D3D with conformal vacuum

6 in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf NIMROD s Extended MHD Equations B t = E+κ divb ( B) J = 1 µ 0 B E = V B+ηJ [ + me e (J B p e) n ee 2 m e + J ] t + (JV+VJ) n α + (nv) α = D n α ( t ) V ρ t +V V = J B p + ρν V Π p h ( ) n α Tα +V α T α = p α V α Γ 1 t q α+q α Π α : V α resistive MHD Hall and 2-fluid Braginski and beyond closures energetic particles

7 in NIMROD Equations intro to NIMROD hybrid kinetic-mhd the equations of motion and δf momentum equation modified by hot particle pressure tensor: ( ) U ρ t +U U = J B p b p h b,h denote bulk plasma and hot particles ρ,u for entire plasma, both bulk and hot particle quasi-neutrality n e = n i +n h steady state equation J 0 B 0 = p 0 = p b0 + p h0 p b0 is scaled to accomodate hot particles scalar pressure requires isotropic velocity distribution alternative J h current coupling possible

8 in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf Schematic of Hybrid δf PIC-MHD model advance particles and δf 1 deposit δp(η) = space z n+1 i δf n+1 i = z n i +ż(z i ) t = δf n i + δf(z i ) t N δf i m(v i V h ) 2 S(η η i ) on FE logical i=1 advance NIMROD hybrid kinetic-mhd with modified momentum equation ρ s δu t = J s δb+δj B s δp b δp h 1 S. E. Parker and W. W. Lee, A fully nonlinear characteristic method for gyro-kinetic simulation, Physics of Fluids B, 5, 1993

9 in NIMROD Summary of PIC Capabilities intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 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 initial implementation as self-consistent tracers e.g. - Maxwellian+slowing down, drift+lorentz

10 in NIMROD Drift Kinetic Equation of Motion intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 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

11 in NIMROD Drift Kinetic δp h PIC Moment intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 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 h v V h δf(x,v)d 3 v δf is perturbed phase space density, m mass of particle, and V h is COM velocity of particles implemented as δp = δp I+δ pbb, δ p = δp δp

12 in NIMROD δf PIC reduces 1/ N noise S. E. Parker and W. W. Lee, PFB 5 (1993) intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 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

13 in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 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

14 Outline in NIMROD 3 Components of δf 1 in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 2 3 Components of δf 3 3 regions of stability map

15 in NIMROD 3 Components of δf 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

16 in NIMROD 3 Components of δf movie1

17 in NIMROD 3 Components of δf Evolution of n = 1 δp hot δp hot t = 0 t.2ω h t.4ω h δ p hot tr = 0 t.2ω h t.4ω h

18 in NIMROD 3 Components of δf Orthogonal view of n = 1, δf(v,v ) t = 0 t.1ω h t.2ω h t.25ω h t.35ω h t.4ω h

19 in NIMROD 3 Components of δf Summary of Observations of Moments of δf n=1 energetic particles cause real frequency response phase lag between particles and fluid δ p concentrated on outboard side trapped particles 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

20 in NIMROD 3 Components of δf 3 Components of δf 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 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

21 in NIMROD 3 Components of δf 0D Comparison of (1, 1) benchmark normalized amplitude (wrt δv E B ψ) %(passing/trapped) contribution β frac gρ δv E B ψ v D δe (-13/113) (69/31) (45/55) (ε/100) (58/42) (27/73) (0/100) (52/48) (18/82) δv E B ψ comparable passing trapped (destabilizing) gρ &v D δe small, negative, and trapped (stabilizing)

22 in NIMROD 3 Components of δf 3 Components in Velocity Space (β frac =.50)

23 Outline in NIMROD 3 regions of stability map 1 in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf 2 3 Components of δf 3 3 regions of stability map

24 in NIMROD 3 regions of stability map 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

25 in NIMROD n = 1 DIII-D linear simulations 3 regions of stability map 6 a) x b) c) P 0 e -2ψ/ψ 0 10 q P (pa) (m) 0 q=1.5 (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]

26 in NIMROD q scan at fixed q-profile increase in q accompanied by decrease in β N 3 regions of stability map (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

27 in NIMROD 3 regions of stability map 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

28 a) in NIMROD 3 regions of stability map Spatial Eigenmodes show distinct topologies Br Vr P P per P ani b) c) R B r R Br R V r R Vr R R R P Pper Pani P R R Pper R Pani starts as dominantly core m = 1 at higher q more m = 2 abrupt shift to m = 2 dominant at q > 1.25 even some m = 3 d) R R Br Vr R P R Pper R Pani R R R R R

29 in NIMROD 3 regions of stability map Velocity space eigenmodes have more distinct features v (m/s) v (m/s) v (m/s) x x x10 δf total δf total δf total a) (m/s) x v δf r<r(q min ) δf r<r(q min ) δf r<r(q min ) (m/s) v x δf r>r(q min ) v 1 (m/s) 2 x (m/s) 2 x (m/s) 2 x10 6 b) (m/s) x v v x10 6 c) (m/s) x v v v (m/s) v x δf r>r(q min ) v x v 1 2 x10 6 δf r>r(q min ) (m/s) v x v x10 6 v x10 6 v x10 6 δf(v,v ) = r 2 r 1 δf(r,v)dr velocity eigenmodes have distinct structures r qmin fixed for q-scan divide into inner and outer region q min =[1.04,1.21,1.36] transition from inner to outer dominance for q > 1.25 transition from co-direction to counter-direction dominance

30 in NIMROD q = D diagnostic vs. time 3 regions of stability map phase volume integrated vs time, total(green), passing(blue), trapped(red) comparable amplitudes from δv E B ψ and v D δe passing particles have larger contribution!

31 in NIMROD 3 regions of stability map Velocity space plots detail 0D diagnostic significant v resonance

32 in NIMROD q = D diagnostic vs. time 3 regions of stability map δv E B ψ > v D δe, 5 larger passing particles have larger contribution!

33 in NIMROD 3 regions of stability map Velocity space plots detail 0D diagnostic δv E B ψ shows (v,v ) localized activity

34 Conclusion in NIMROD 3 regions of stability map new phase space diagnostics have been presented identifying interesting phenomenological features eigenmode in (v,v ) often more distinctive than spatial eigenmode distinct resonances features in trapped cone co-/counter- passing dominance both trapped and passing particles is important passing particles are crucial and often a dominant contribution from phenomenology, move to quantification, then theory

35 in NIMROD 3 regions of stability map movie2