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
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
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
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 10 7 8 large thermal anisotropy - χ χ 10 8 active developer and user base continually expanding capabilities large and growing Verification&Validation
in NIMROD intro to NIMROD hybrid kinetic-mhd the equations of motion and δf Example NIMROD grid - D3D with conformal vacuum
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
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
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
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
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
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 0 0 0 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
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
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
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
in NIMROD 3 Components of δf Benchmark of Drift Kinetic (1,1) Kink With M3D C. C. Kim, PoP 15 072507 (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
in NIMROD 3 Components of δf movie1
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
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
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
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
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.25-0.031(-13/113) (69/31) -0.010 (45/55).50-0.055(ε/100) (58/42) -0.015 (27/73).75-0.057(0/100) (52/48) -0.020 (18/82) δv E B ψ comparable passing trapped (destabilizing) gρ &v D δe small, negative, and trapped (stabilizing)
in NIMROD 3 Components of δf 3 Components in Velocity Space (β frac =.50)
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
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 ) 4 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 0.04 0.03 0.02 0.01 0.005 0 Ideal MHD Stability Boundary 0.01 0.005 Reconstruction Flattop γτ A MHD Only (n=1) S=τ R /τ A ~10 7-10 8 Pr=100 Experimental Trajectory 2 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 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
in NIMROD n = 1 DIII-D linear simulations 3 regions of stability map 6 a) x 10 4 15 b) c) 1 1 4 P 0 e -2ψ/ψ 0 10 q P (pa) (m) 0 q=1.5 (m) 0 2 5 q=2 1 1 0 0.3 0.2 0.1 ψ (Wb) 0 1 2 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]
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 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 1 1.1 1.2 1.3 1.4 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 2 1 1.5 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
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 ) 4 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 Ideal MHD Stability Boundary 0.005 0.0025 Flattop 0 0.005 Reconstruction 0.01 0.0025 γτ A 0.015 Energetic Particle Driven MHD Modes (n=1) 0.005 0.01 0.015 0.005 2 1 1.1 1.2 q 1.3 1.4 1.5 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
a) in NIMROD 3 regions of stability map Spatial Eigenmodes show distinct topologies Br Vr P P per P ani b) c) 1.0 1.5 2.0 2.5 R B r 1.0 1.5 2.0 2.5 R Br 1.0 1.5 2.0 2.5 R V r 1.0 1.5 2.0 2.5 R Vr 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 R R R P Pper Pani P 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 R R Pper 1.0 1.5 2.0 2.5 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) 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 R R Br Vr 1.0 1.5 2.0 2.5 R P 1.0 1.5 2.0 2.5 R Pper 1.0 1.5 2.0 2.5 R Pani 1.0 1.5 2.0 2.5 R 1.0 1.5 2.0 R 2.5 1.0 1.5 2.0 2.5 R 1.0 1.5 2.0 2.5 R 1.0 1.5 2.0 2.5 R
in NIMROD 3 regions of stability map Velocity space eigenmodes have more distinct features v (m/s) v (m/s) v (m/s) 0.0 1.0 2.0 3.0 x10 0.0 1.0 2.0 3.0 x10 0.0 1.0 2.0 3.0 x10 δf total δf total δf total a) (m/s) x10 6 0.0 1.0 2.0 3.0 v δf r<r(q min ) δf r<r(q min ) δf r<r(q min ) (m/s) v x10 6 0.0 1.0 2.0 3.0 δf r>r(q min ) -2-1 0v 1 (m/s) 2 x10 6-2 -1 0 1 (m/s) 2 x10 6-2 -1 0 1 (m/s) 2 x10 6 b) (m/s) x10 6 0.0 1.0 2.0 3.0 v -2-1 0v 1 2-2 -1 0 1 2 x10 6 c) (m/s) x10 6 0.0 1.0 2.0 3.0 v v v (m/s) v x10 6 0.0 1.0 2.0 3.0 δf r>r(q min ) v x10 6-2 -1 0v 1 2 x10 6 δf r>r(q min ) (m/s) v x10 6 0.0 1.0 2.0 3.0-2 -1 0v 1 2-2 -1 0 1 2-2 -1 0 1 2 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
in NIMROD q = 1.24 0D 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!
in NIMROD 3 regions of stability map Velocity space plots detail 0D diagnostic significant v resonance
in NIMROD q = 1.36 0D diagnostic vs. time 3 regions of stability map δv E B ψ > v D δe, 5 larger passing particles have larger contribution!
in NIMROD 3 regions of stability map Velocity space plots detail 0D diagnostic δv E B ψ shows (v,v ) localized activity
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
in NIMROD 3 regions of stability map movie2