Database of Gyrokinetic Transport Simulations and Comparison to a

Database of Gyrokinetic Transport Simulations and Comparison to a New Comprehensive Theory-based Transport Model J. Kinsey Lehigh University PERSISTENT SURVEILLANCE FOR PIPELINE PROTECTION AND THREAT INTERDICTION in collaboration with G. Staebler, R. Waltz, J. Candy General Atomics 11th EU-US Transport Task Force Workshop September 4-7, 2006 Marseille, France Outline • Overview of GYRO transport database – A nonlinear database for testing & developing theory-based transport models and benchmarking is needed • Effect of shaped geometry (Miller) on turbulence and ExB shear quenching – – – – – GYRO simulations show the normalized diffusivities scaling like κ-1 for fixed midplane minor radius, gradients defined using minor radius Less ExB shear is needed to quench transport for high elongation and for low aspect ratio New quench rule found for shaped geometry, arbitrary aspect ratio • Comparisons between TGLF model and GYRO simulations TGLF shows significantly better agreement electron energy transport from GYRO simulations compared to GLF23 model TGLF valid for shaped geometry ! J. Kinsey - EU/US TTF06 A Database of Over 350 Nonlinear Gyrokinetic Simulations Has Been Created Using The GYRO Code • A nonlinear simulation database has been created for benchmarking and transport model development http://fusion.gat.com/comp/parallel • Scans in R/a, r/a, q, s, α, a/Ln, a/LT, ν, β, Ti/Te, κ, δ, dilution, and ExB shear (most simulations w/ kinetic electrons) – ExB shear scans: Kinsey, Waltz, Candy, Phys. Plasmas 12, 062302 (2005) – q, shat scans: Kinsey, Waltz, Candy, Phys. Plasmas 13, 022305 (2006) ^ • Simulations around several reference cases assuming s-α geometry, electrostatic (except for β scan), and flat profiles across annulus, zero boundary conditions ^ – GA Standard Case (STD): R/a=3, r/a=0.5, q=2, s=1, α=0, a/LT=3, a/Ln=1, Ti/Te=1, ν=0, β=0 ^ – TEM1 Case: STD w/ a/Ln=2, a/LT=2 TEM2 Case: STD w/ a/Ln=3, a/LT=1 • Miller geometry used to study effect of κ and δ on χ and ExB shear – Nine parameters are required to describe the local equilibrium using Miller geometry : κ, δ, q, s, α, A=R0/r, ∂rR0, along with gradient factors of κ and δ (sκ and sδ) J. Kinsey - EU/US TTF06 Simulations Show the GYRO Normalized Energy Diffusivities Decreasing Linearly with Increasing Elongation in Agreement with Experimental Data • ^ ^ Linear decrease in both χe and χi robust for κ scans using GYRO – Equal to 2/[κ(1+ κ2)] at fixed B0 in ITER units ! ! DIII-D κ scans (Luce, EPS01): ^ χITER(@fixed B0) ∝ κ-4 (H-mode) GB 20 GYRO STD Case, Kinetic Electrons Miller geometry !i !e D 15 ∝ κ-3 (L-mode) !/ ! ! MM95 model has empirical factor multiplying χITER κ-4 10 !#" 5 $1.0 – κ varied for a variety of q, magnetic shear, and δ values using the STD parameters 0 1.0 1.5 2.0 2.5 – For κ scans, we also varied sκ=(r/κ)∂rκ ≈ (κ-1)/κ – Particle transport shows little or no κ dependence if D is negative " ^ χ=χ/χGB ! Bunit held fixed J. Kinsey - EU/US TTF06 Miller Geometry Simulations Show The ExB Shear Quench Point Varies Systematically with Elongation & Aspect Ratio • Quench point scales like γE /γmax ∝ (1/κ)A0.65 for Waltz γE over the range of 1 < κ < 2 and 2 < A < 5 – Hahm-Burrell γE quench pt approximately 2x higher, slightly stronger κ dependence 4.0 3.5 Waltz ExB shear rate STD Case Kinetic Electrons Real Geometry !=1.0 !=1.5 !=2.0 ) Quench • • 72 nonlinear kinetic electron simulations w/ Miller equil. model New ExB shear quench rule: 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 χ ∝ [1 - αE (κ,A) (γE/γmax)] αE = κ/A0.65 flux-surface-avg Waltz γE αE = 0.5κ1.25/A0.65 outboard Hahm-Burrell γE For A=3, κ=1.5 : αE (Waltz) = 0.73, αE (HB) = 0.41 (" /" E max * 4.5 5.0 A (R/a) Relevant DIII-D and JET parameters * s-α result (γE=2γmax) • Effect of δ on quench pt appears to be weak (< 10%) based on STD case runs w/ A=3, κ=1.5 J. Kinsey - EU/US TTF06 The TGLF Trapped Gyro-Landau-Fluid Transport Model • TGLF is the next generation GLF model – Model valid continuously from low-k ITG/TEM to high-k ETG (GLF23 contained a limited ITG/TEM spectrum w/ max kθρs=0.5) – Extended validity to NCS and pedestal relevant parameters – Valid for shaped geometry via Miller equilibrium model (GLF23 is an shifted circle model) • TGLF solves for the eigenvalues using a new set of 6-moment gyro-fluid equations for linear drift-wave instabilities in tokamaks using a Hermite basis function approach • TGLF has been systematically tested against a database of about 1800 linear growth rates and frequencies created using the GKS gyrokinetic code (Staebler, Kinsey, Waltz, PoP 12, 102508 (2005)) Avg σ (γ) = 0.11 for TGLF model Avg σ (γ) = 0.38 for 1997 GLF23 model • Mixing length rule for saturation levels being finalized J. Kinsey - EU/US TTF06 TGLF Mixing Length Rule With Quasilinear Weights Fit to GYRO Nonlinear Simulations • Transport computed w/ fluxes of the form 2 2 2 ˆ ˆ = e# /T /( $ /a) Qi = n e Te c s ( " s /a) # QiQL " e s 2 ˆ ˆ ˆ2 " = # D 0 F /k $ where F is a combination of • Quasilinear fit to be of the form ^ ^ growth rate and curvature drift frequency (ωd0=ky/R0) c5 ! ˆ ˆ ˆ ˆ " i = c i (c15 " iib + " ieb! ) Fi = (" net /# d 0 ) /(1+ (c 3" net /# d 0 ) c 5 ) ib eb " e = c e (c16 " e + " e ) ! D = c d (c17 Diib + Dieb ) • Coefficients & exponents in mixing length formula found be minimizing ! error (w/ zero offset) between TGLF and GYRO diffusivity spectrum for 87 ! nonlinear simulations (1305 spectral pts) ! ˆ ˆ ˆ ˆ Fe = (" net /# d 0 ) /(1+ (c 4 " net /# d 0 ) c 6 ) c7 c8 ˆ ˆ k",i = k y shear k",e = k y shear c6 ! ! – Used same spectrum for TGLF as used for GYRO (16 modes) • Fit confirms QL theory ! – ci,ce,cd about equal and c15-17 about equal (QL constraint) – c7,c8 near 4 and c5,c6 near 2 J. Kinsey - EU/US TTF06 TGLF Saturation Rule Fits the Energy and Particle Transport Spectrums from 87 Nonlinear GYRO Simulations Very Well • • Comparisons for: Shifted circle geometry, electrostatic, collisionless A low-k spectral cutoff is applied to each branch – Cutoff acts like a Dimits shift; constant value times (1/q) applied at all ky’s seems to suffice • Best fit yields RMS errors of 19%, 21%, 35% for ion, electron, particle fluxes J. Kinsey - EU/US TTF06 RMS Errors in Electron Energy Diffusivity Significantly Smaller for TGLF Model in Comparison With GLF23 Model • RMS errors computed between model and GYRO for scans in shat, q, a/Lt, a/Ln, Ti/Te, νei, r/a, R/a, κ, δ, ExB shear 1.0 TGLF GLF23 v1.61 χi STD Case 3.0 TGLF GLF23 v1.61 χe STD Case 3.0 TGLF GLF23 v1.61 D STD Case 0.8 2.5 2.5 2.0 0.6 2.0 "e "i ! 0.4 ! 1.0 0.2 ! 1 2 3 4 5 6 7 8 9 1011121314151617181920 1.5 D 1.5 1.0 0.5 0.5 0.0 1 2 3 4 5 6 7 8 9 1011121314151617181920 0.0 0.0 1 2 3 4 5 6 7 8 9 1011121314151617181920 Scan No. Scan No. Scan No. #16=γE-scan, #17=κ-scan, #18=δ-scan, #19=A-scan, #20 = shat-scan w/ pedestal parameters σx=[ ∑i (XiGYRO-XiTGLF)2 / ∑i (XiGYRO )2]1/2 where X = χ or D J. Kinsey - EU/US TTF06 TGLF Model Valid for Real Geometry & Reproduces Stabilizing Effect of Elongation Seen in GYRO Simulations • • • TGLF compared to GYRO for STD case w/ kinetic electrons varying κ and sκ using Miller geometry, β=0, ν=0, δ=0 Used same QL formula found in fitting s-α simulations Electron energy transport increases from 3.0 to 5.3 χe/χGB in GYRO going from s-α to Miller geometry w/ κ=1.0 0.5 STD Case Kinetic Electrons 0.4 Miller geometry, "=0 GYRO - ky=0.30 TGLF - ky=0.30 GYRO - ky=0.15 TGLF - ky=0.15 20 15 GB STD Case Kinetic Electrons Miller geometry, "=0 ! ! i e ! ! i e D GYRO D TGLF ! / (c /a) 0.3 0.2 !/! 10 5 0 s 0.1 s =(#-1)/# 0 1.0 # 1.5 2.0 2.5 1.0 1.5 2.0 2.5 # J. Kinsey - EU/US TTF06 # TGLF Shows Good Agreement With GKS Growth Rates for DIII-D ITB Discharge Including Real Geometry, Collisions • TGLF compared to GKS for #84736 for the case w/ β=0 and ν=0 and for the case w/ full physics (except for parallel velocity shear) • Reduction in growth rates w/ full physics due to finite β in the inner plasma and collision in the outer plasma 1.2 1.0 DIII-D #84736 t=1.30 s GKS (!=0) GKS (Full physics) TGLF (!=0) TGLF (Full physics) " / (c /a) 0.8 0.6 0.4 0.2 0.0 0.0 s 0.20 0.40 0.60 0.80 1.0 # J. Kinsey - EU/US TTF06 TGLF Development Path • Completed tasks – Real geometry effects (Miller equilibrium model) – Collisions and finite beta – Mixing length formula found using 87 GYRO nonlinear simulations w/ kinetic electrons. QL theory works quite well ! • Near future work – Explore methods of improving speed (e.g. reduce number of modes and/or spectrum) – More comparisons to GYRO nonlinear simulations: cases w/ pedestal parameters, real geometry, collisions, finite beta – Test ExB shear quench rule for real geometry using GYRO fits – Include parallel velocity shear using GLF23 recipe • Longer term future work – Replace ExB shear rule with rotational ballooning mode net linear growth rate model – Add nonlocal transport effects, broken gyro-Bohm scaling – Begin testing in XPTOR code against DIII-D data, optimize code performance J. Kinsey - EU/US TTF06 EXTRA SLIDES J. Kinsey - EU/US TTF06 An Excellent Fit to the GKS Linear Gyrokinetic Database Was Obtained for the TGLF Model • Model tested around 3 reference cases: – – – STD case: R/a=3, r/a=0.5, q=2, s=1, a/LT=3, a/Ln=1, Ti/Te=1, α=0 and β=0 PED case: STD Case + r/a=0.75, a/LT=10, a/Ln=3, q=4, s=3, α=5 NCS case: STD Case + a/LTi=10, a/LTe=4, s=-0.5 Avg σγ (STD) = 0.13 (TGLF), 0.21 (ori GLF23); Avg σγ (PED) = 0.14 (TGLF), 0.54 (ori GLF23) 1.0 Ori. GLF23 Model New GLF Model 2.0 STD Case 1.5 Ori GLF23 Model New GLF Model PED Case 0.8 α=12 0.6 " " ! ! 1.0 0.4 0.5 0.2 α=9 α=6 α=3 22 23 24 25 26 27 28 29 0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 0.0 Scan No. Scan No. σx=[ ∑i (XiGKS-XiTGLF)2 / ∑i (XiGKS )2]1/2 where X = γ or ω J. Kinsey - EU/US TTF06 TGLF Model Shows Improved Agreement With GYRO Electron Energy Diffusivity Compared to GLF23 • • TGLF reproduces q-scaling observed in GYRO simulations Electron energy transport from GLF23 much too large at all q values 30 25 20 STD Case ! ! i e ! ! i e ! ! i e D GYRO TGLF D GLF23 D v1.61 (“retuned”) !/! GB 15 10 5 0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 q scan @ fixed shear, etc q R/a=3, r/a=0.5, q=2, s=1, a/LT=3, a/Ln=1, Ti/Te=1, α=0 and β=0 J. Kinsey - EU/US TTF06 TGLF-GYRO Comparison for a Temperature Gradient Scan • TGLF shows significant improvement in a/LT dependence and accurately reproduces critical gradient 30 STD Case 25 20 GB ! ! i e ! ! i e ! ! i e D GYRO D TGLF D GLF23 !/! 15 10 5 0 -5 1.0 1.5 2.0 2.5 3.0 T 3.5 4.0 a/L J. Kinsey - EU/US TTF06 ExB Shear Quench Rule • Effect of ExB flow shear on ITG/TEM transport implemented in GLF23 model is originally based on adiabatic electron simulations • Gyrofluid simulations by Waltz, et al found that ITG transport was quenched when γE=γmax – Quench rule: χ ∝ [1 - αE (γE/γmax)] where γE = ExB shear rate, γmax = max linear growth rate, and αE = 1.0 +- 0.5 • Recent GYRO simulations with kinetic electrons show the ExB shear quench point is at γE=2γmax for ions and electrons (αE=0.5) – Quench point robust for both ITG and TEM dominated cases but is only valid for shifted circle geometry • Effects of elongation and aspect ratio on ExB shear quenching have recently been investigated in GYRO simulations using the Miller equilibrium model with kinetic electrons included J. Kinsey - EU/US TTF06 GYRO Diffusivities In Real Geometry • Elongation scans performed at fixed midplane minor radius, r, and gradient scale lengths (defined in terms of r) ^ ^ ^ • Translation of χGYRO to χITER where χ= χ / χGB = χ / (csρs2/a) χGYRO =<|" r|2> χITER ! For concentric ellipses where “r” is the midplane minor radius: ! So, ! we have χITER ! <|" r|2>=(1+κ2)/(2κ2) /χ =[2κ2/(1+κ2)] χ^ GB_B0 GYRO χGB_Bunit /χGB_B0 =[2κ2/(1+κ2)] χGYRO )(B0/Bunit)2 ! Where χGB_B0 and χGB_Bunit are the GB χ’s at fixed B0 and Bunit We have a κ dependence that enters thru Bunit in ρs Bunit=(ρ/r)(dρ/dr)B0 ≈ κB0 since ρ ≈ ( κ )0.5 r Finally, we have ! ^ ^ χITER ≈2/(1+κ2) χGYRO ! ! J. Kinsey - EU/US TTF06 Near a Null Flow Point Some Modes Drive a Flow Up the Density Gradient While Other Modes Drive a Flow Down the Density Gradient • Null pt in ky space NOT correlated with ITG-TEM transition – Linear stability shows transition from ITG to TEM above kθρs=0.65 for s=0.5 • Particle diffusivity can change sign as magnetic shear is varied at fixed q 16 STD Case Kinetic Electrons q=2, "=0 !i !e D GB 12 8 !/ ! 4 0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 s ^ Γ=-D dn/dr J. Kinsey - EU/US TTF06 TGLF Shows Remarkable Agreement With GKS Growth Rates & Frequencies in Vicinity of DIII-D H-mode Pedestal • Ranges of data (time-averaged): ρ = 0.84 - 0.94 s = 2.2-2.5 α = 0.4-1.6 a/Lne = 1.1-6.1 a/LTe = 3.3 - 14.8 a/LTi = 1.5 - 9.3 3.0 DIII-D #98889 2.5 t=4.50 s GKS New GLF Model ! / (c s/a) q = 3.0-3.9 2.0 1.5 1.0 0.5 0.0 0.80 1.0 0.5 DIII-D #98889 t=4.50 s TEM ITG 0.85 0.90 ^ 0.95 1.0 ! / (c s/a) • Assumed s-α geometry, no collisions, electrostatic, Zeff=1, kqρs=0.3 • TGLF follows transition from TEM to ITG well near ρ=0.86 " GKS New GLF Model 0.0 -0.5 -1.0 -1.5 0.80 0.85 0.90 ^ 0.95 1.0 " J. Kinsey - EU/US TTF06 Dependence of Transport on Triangularity Weak For Circular Plasmas, Somewhat Stronger For Elongated Plasmas • • δ varied for κ=1.0, 1.5, and 2.0 using STD parameters – Miller geometry, delta gradient factor sδ varied along with δ Transport increases with δ for high elongation – Stronger dependence for κ=1.5, 2.0 cases compared to κ=1.0 case 24 20 16 GB κ=1.0 STD Case Kinetic Electrons "'="/(1-" ) 2 0.5 κ=1.5 !e D !i 16 STD Case Kinetic Electrons #=1.5,"'="/(1-" ) 2 0.5 !i !e D 12 !/ ! 12 8 4 0 -4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 8 4 0 0.0 0.1 0.2 0.3 sκ=0.33 0.4 0.5 " " Bunit held fixed J. Kinsey - EU/US TTF06 Triangularity Strongly Impacts Particle Transport Spectrum for Elongated Plasmas Near a Null Flow Point • • δ varied from 0.0 to 0.5 for STD case w/ κ=2.0 Particle transport changes from D/DGB=-1.73 to D/DGB=+0.51 – Transport from low k modes changes sign – Less of an effect at κ=1.0 (D/DGB=-0.8 -> D/DGB=-0.1 when δ=0.0 -> 0.5) δ=0.0 δ=0.5 J. Kinsey - EU/US TTF06 GAM Frequency Decreases With Increasing κ • • • Elongation varied in GYRO simulations holding all other quantities fixed GAMs evident at n=0 at ω(a/cs) = ±0.75 for k=1.0 GAM frequency decreases with increasing elongation – Also consistent with BES measurements on DIII-D (McKee, et al PPCF, 2006) ITG J. Kinsey - EU/US TTF06 TEM