GYRO gyrokinetic simulations and TGLF gyrofluid transport models: from core to the edge. R.E. Waltz* General Atomics

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1 GYRO gyrokinetic simulations and TGLF gyrofluid transport models: from core to the edge R.E. Waltz* PERSISTENT SURVEILLANCE FOR PIPELINE PROTECTION AND THREAT INTERDICTION General Atomics * Work with J. Candy, G.M. Staebler, and J.E. Kinsey ITPA Workshop Oct 23-25, 2006 Chengdu, China

2 OUTLINE GYRO gyrokinetic status and capability: what s next for gyrokinetic codes? Newest GYRO simulations of coupled ITG/TEM-ETG with ExB shear. GYRO work started on DIII-D L- and H-mode edge simulations Status and plans for the new TGLF [17] transport code model APPENDIX: Published (Phys. Plasmas) DIII-D core experimental analysis with GYRO: Bohm scaled L-modes, gyrobohm scaled H-mode, and the problem of perfect profile similarity in ρ*-scaling pairs[18] Discovery of large profile corrugations at q-min = 2/1 triggering an ITB[19]

3 GYRO gyrokinetic status and capability Minimal code capabilities for required for realistic tokamak core simulations: _ kernel ITG-adiabatic electron (and ETG-adiabatic ion) _ kinetic trapped and passing electron low-k ITG/TEM _ electron-ion pitch angle r collisions _ finite beta (at least B ) fluctuations and magnetic flutter transport added to ExB _ shaped geometry " _ equilibrium ExB shear stabilization _ finite- ρ* global needs adaptive source & benign 0-value boundary conditions _ input of actual experimental profiles and strong post-run diagnostics (vugyro) _ energy, particle, momentum flow diagnostics GYRO completed early 2003 and now has a large user base beyond Candy& Waltz developers: Kinsey (ULeigh-GA), Estrada-Mila (UCSD-GA), Holland (UCSD), Bundy and Mikkelsen (PPPL), Bravennec (UTx), Hallatschek(IPP) students at UCLA, UWis GYRO is both a local flux-tube [ρ* =0] and global [finite-ρ* ~ O(1%)]code. Visit the GYRO website:

4 GYRO gyrokinetic status and capability cont d Additional features used for special physics & computational studies: [>2004 ITPA] _ impurity and hot particle flow added to plasma thermal pinch studies[20a,20b,21]: D-V impurity flow verified, low-k ITG/TEM can transport hot-alphas _ finite- ρ* n=0 neoclassical flow driver + conserving Krook ion-ion collisions[12]: neoclassical and turbulent flows: little cross-talk ; needed for large orbit neoclass. _ profile feedback adjustments for transport at fixed experimental flow rather than usual fixed experimental gradient [11,12]: best comparison with experiment for stiff and pinched plasma flow in core _ diagnostic for currant-voltage relation looking of turbulent dynamo EMF[22]: found low-m/n current density corrugation may affect tearing instability _ partial (Δn >>1) to full torus (Δn=1) resolving profile corrugation at low order m/n[19]: not needed for accurate transport flows in monotonic q-profiles _ very large profile corrugation at q-min = 2/1 can trigger ITB [19] _ velocity space grid resolution and entropy conservation-dissipation[23]: 8 energies x 8 pitch angles x 2 sign(v ) = 128 grids/space cell OK: no v-grad s!!!! _ O(ρ*) parallel (and drift) nonlinearity (PNL) do not effect transport flows [24] _ diagnostic for gyrokinetic turbulent heating (from PNL) offsets turbulent transport[25] _ high-k ETG electron gyroradius with kinetic ions coupled to low-k ITG/TEM [26]

5 What s ahead for gyrokinetic codes? Steady state gyrokinetic gyrobohm core transport for ITER Q given Tped & nped 2-3 year horizon.have SciDAC funding in place as SAP to FACETS ITER core (at 5x smaller ρ* than DIII-D) will have local gyrobohm transport Master fusion source code can drive parallel slave copies of GYRO flux-tubes (one at each radius r j ) with local feedback adjustments on the gradients (z) converging to power balance: Ptran =Pheat: (α feedback strength).at each iteration z j = -d ln T/dr: Δz j /z j = - α Δ(P tran_j -P heat_j )/ P heat_j each radius r j Shoot up gradient from Tped: Tj-1 = Tj+1 - Tj zj see examples in Ref. [11,12] A global code for ITER-core would be very expensive BUT not really needed!!!! Full-f code required for edge (and maybe ITB transitions) where there is no time scale sepation between micro-turbulence and transport, ELM crash & reheat time scales. 5 year horizon Just as delta-f global code must converge to delta-f flux-tube code at vanishing ρ*, so a full-f edge code must approximate a global delta-f code in the core at finite ρ* Will be very challenging computationally. Conservative form. Full-f Poisson Eq.

6 Newest GYRO simulations of coupled ITG/TEM-ETG with ExB shear Expensive large-box and high-resolution GYRO spanning low-k ITG/TEM and high-k ETG turbulence done to determine how much extra electron energy transport can be picked up extending usual simulations to higher-k. First simulation to treat ETG transport with gyrokinetic electrons and ions, as well as ExB shear. ρ si /ρ se = µ = 30 64ρ si x 64ρ si 192 modes to k y ρ se = 0.7 GA-std case χ i /χ gbi χ e /χ gbi D/χ gbi ITG/TEM k y ρ si < ETG_1 k y ρ si > ETG adds 10% to electron energy transport from 1< k y ρ si < 21 but most k y ρ si < 5 GA-std case: r/a=0.1, R/a=3, q=2, s=1, a/lt=3, a/ln=1

7 Small box high-k ETG-ki simulations can get total ETG transport χ e (ETG_1) correct, but large-box & high-resolution simulations required for physical high-k spectrum Large (physical) box Small (cheap) box Large (physical) box transport decreases monotonically in ky from a single peak in low-k.

8 Eddies vs streamers as they appear in x-space and k-space ETG streamers can only x:y 5:1 streamers exist in less physical small boxes which preclude large isotropic ion scale ITG/TEM eddies. High-k ETG δn/n more physical large box less physical small box kx:ky 1:5 spectrum is very isotropic and driven largely by low-k modes.

9 ITG driven ETG transport in an ETG stable plasma!!! Turning down electron temperature gradient a/lte = 3-->1, ETG is below threshold of 1.3. ETG_1 is nonlinearly driven by ITG but still about 6% of total electron energy transport. χ(etg_1) is only slightly reduced but the ETG_1 transport is reduced by 3.5-fold. Per-mode quasilinear transport models must be modified with mode coupling to handle sub-critical ETG transport.

10 ExB shear preferentially stabilizes low-k ITG/TEM making high-k ETG likely dominant electron transport inside barriers For the GA-std case ETG is about 10% of total electron transport which in turn is about 30% of total transport. GA-std case is typical of DIII-D midradius core where ITG/TEM low-k electron (and ion) transport is often 50% ExB shear stabilized making ETG 20% of total electron transport in this case.

11 GYRO work started on DIII-D L- and H-mode edge simulations with K. Hallatscheck Looking at DIII-D shot edge slices r/a=[0.9,0.99] r_norm/a = 0.95 with full physics cyclic flat profile (with ExB shear) at 0.95 time slice " e#i#ave sim /"gb L H (electrostatic) " e#i#ave exp /"gb " [m 2 /sec] gb _Experimental levels likely lower with proper recycling accounted: will use GYRO prediction. _Looks consistent with L/H bifurcation: What s the mechanism? GYRO with profile variation _H-mode data may be exceeding beta limit

12 New TGLF model gets very fast and accurate linear growth rates 15 (4) gyrofluid closure moments of linear gyrokinetic equation plus 4 (1) θ basis functions per species for TGLF (GLF23). Covers trapping from seamlessly from low-k ITG/TEM to high-k ETG with very accurate gyro-averaging. Dispersion matrix 96x96 (13x slower than GLF23 with 8x8) still gets linear growth rates 100 s X faster than GKS (GS2) to ~12% accuracy (GKS-GYRO differ by 11%) Examples: STD core NCS PED GLF23 21% 32% 54% TGLF 10% 10% 14% MHD βcrit ˆ s shear

13 Linear stability analysis of data: radial profile _ TGLF vs GKS DIII-D NCS discharge at 1.3sec Profile of the normalized linear growth rate for k y =0.3 for three different physics settings (A) comprehensive physics (B) collisionless, electrostatic (C) s-alpha geometry dilution, collisionless, electrostatic Extension to the very edge sometimes requires high numerical resolution and precise magnetic equilibrium instead of the Miller shaped geometry model. TGLF code is 100 s times faster than GKS & requires no convergence fiddle

14 Linear stability analysis of data: k-spectrum _ TGLF vs GKS k-spectrum of the linear eigenmodes (left) and quasilinear weights (right) DIII-D NCS discharge 1.3sec & r/a=0/7 TGLF can also display subdominant modes; GLK initial value code cannot

15 TGLF has more accurate fit to gyrokinetic transport (GYRO) than GLF23 The GLF23 electron energy flux is systematically too high compared to GYRO. _Expect better ITER - Q projection STD case with kinetic electrons The inaccurate trapped electron model in GLF23 can give transport at low temperature gradient GLF23 was fit to this STD temperature gradient scan with adiabatic electrons. STD = ( a L ne =a L ni =1, a L Te =a L Ti = 3, T i T e =1, q = 2, s ˆ =1, " = 0, r /a = 0.5, R /a = 3.0)

16 TGLF transport model has excellent fit to GYRO nonlinear simulations Fit to 3 channel 16-mode spectrum of [χi, χe, D] from 86 GYRO simulation with total fractional deviations: " Qi =17%, " Qe =15%, " # = 28%, s"# model Initial Miller shaped geometry tests look good.

17 TGLF transport model is much more accurate than GLF23 RMS errors computed between model and GYRO for scans in shat, q, a/lt, a/ln, Ti/Te, ν ei, r/a, R/a, κ, δ, ExB shear Avg errors for entire dataset for χ i, χ e, D are [0.25,0.26,0.65] for TGLF compared to [0.49,1.54,1.10] for GLF23 v1.61 Large errors in D attributed to D being close to a null pt in many cases 2.0 TGLF GLF23 v1.61 χ i χ e D STD Case 2.0 TGLF GLF23 v1.61 STD Case 2.0 TGLF GLF23 v1.61 STD Case 1.5 Fit database other scans ! "i 1.0! "e 1.0! D Scan No. Scan No. #20=γ E -scan, #21=κ-scan, #22=δ-scan, #23=A-scan, #13,14 = s, a/lt scans w/ PED parameters σ x =[ i (X GYRO i -X TGLF i ) 2 / i (X GYRO i ) 2 ] 1/2 where X = χ or D ^ Scan No.

18 Transport Database for TGLF Testing Scan Set STD STD STD STD STD STD STD STD TEM1 TEM1 TEM2 STD STD PED PED STD TEM2 TEM1 TEM2 STD STD STD STD STD Scan type Magnetic shear MHD alpha=0, 1, 2 Magnetic shear a/ln=0.5, 1.5 Magnetic shear q=1.25 Safety factor s=1.0 a/l T a/l n =0.5, 1.0, 1.5 T i / T e scan (0.5, 1.0, 1.5, 2.0) r/a scan (0.10, 0.25, 0.50, 0.75, 1.0) R/a scan (1.75, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0) Safety factor scan (q=1.1, 1.25, 2.0, 2.5, 3.0, 4.0) Magnetic shear q=2 Safety factor scan (q=1.1, 1.5, 2.0, 2.5, 3.0, 4.0) a/l n a/l T =2.0, 3.0 a/l T r/a=0.75 Magnetic shear scan (s=1.0, 1.5, 2.0, 2.5, 3.0) a/l T scan Safety factor s=-0.5, 1.0, 1.5 Safety factor s=1.0 Magnetic shear q=2 Magnetic shear q=2 Collisionality scan (ν ei = ) ExB shear scan (γ E = , γ p =0) Elongation scan scan w/ Miller geometry, κ=1.0 (κ= ) Triangularity scan w/ Miller geometry, κ=1.0 (δ= ) Aspect ratio (R/a) scan w/ κ=1.0 (A= ) # pts =193 out of 350+ GYRO runs STD case: R/a=3.0, r/a=0.5, q=2.0, s=1.0, α=0, a/l T =3.0, a/l n =1.0, T i /T e =1.0, ν=0, β=0 TEM1 Case: STD w/ a/l n =2, a/l T =2 TEM2 Case: STD w/ a/l n =3, a/l T =1 PED case: R/a=3.0, r/a=0.75, q=4.0, s=3.0, α=5, a/l T =10.0, a/l n =3.0, T i /T e =1.0, ν=0, β=0

19 Remaining development of TGLF transport model Comprehensive linear TGLF code available now for fast experimental profile analysis In addition to [" i," e,d] easy to add turbulent heating[25] H e,i r r ="j e,i "E e,i +" j e,i $" E e,i D# # Like to add toroidal viscosity for momentum channel Requires easy addition of parallel velocity shear " p = R#[u /R]/#r to TGLF equations BUT also (to be accurate) more difficult addition of ExB shear mode stability avoiding " net ="k #$ " quench rule. k E E " E directly into toroidal Addition of high-k tail of ETG electron energy transport will require modification of quasilinear per ky-mode nonlinear saturation rule. Recent GYRO coupled ITG/TEM-ETG simulation demonstrate transport carried by stable high-k modes. Easy to modify present local gyrobohm TGLF model to include nonlocally broken gyrobohm transport[13]. Speed-up TGLF transport model and develop new power balance transport algorithum.

20 BACKGROUND.

21 BACKGROUND: theory > simulation > models > experiment > projection From the early 90s, development of toroidal gyrofluid (or Gyro-Landau-Fliud) equations [1,2] and nonlinear simulations of turbulent transport in the tokamak core [3] lead to three key advances of the mid 90s: _Small scale ExB shear from n=0 radial modes (or zonal flows) is the dominant nonlinear saturation mechanism[3] _Equilibrium scale ExB shear can quench transport when shear rates are comparable[2,3,4] to maximum linear growth rates _Core transport is very stiff (very large power needed to lift away from critical temperature gradient hence fusion performance strongly dependent on Tped Dispersion relations from reduced GLF equations, combined with quasilinear recipes and mixing length rules, were used to construct physically comprehensive, but purely theoretical, transport code models like the 97 GLF23 model [5] with an excellent fit to a wide range of gyrofluid (or better gyrokinetic) simulations. Only then are the transport models tested against experiment.

22 Data test for GLF23 & ITER performance Q projection[6] 8.7% is good fit for τ E =W th /P but Fusion Product = nt τ E has ~2 x 8.7% rms error. Q from stiff core model highly dependent on T ped for which there is no theoretical prediction or simulation at present. Empirical fits have 30% rms error. Q"V tot (n T ) 2 /P ped ped aux for a perfectly stiff core model. Transport models derived form gyrofluid and gyrokinetic simulations (e.g. 94- IFS/PPPL, 97-GLF23) are very stiff tend ride close to the critical temperature gradient.

23 BACKGROUND continued Gyrofluid closures have difficulty treating n=0 radial modes (zonal flows and GAM s) accurately [7] and k-space is awkward for nonlocal broken gyrobohm transport. Building on the physically comprehensive continuum gyrokinetic flux-tube code GS2, development of the global code GYRO[8] began 1999 and largely completed 2003[9,10] Main purpose of GYRO was to explore: finite- ρ*and broken gyrobohm scaling matching the Bohm DIII-D L-modes [9,10,11,12] look for nonlocal transport and develop a heuristic model of nonlocality [13] then build a parameter scan DATABASE of local transport simulations [14,15] The GYRO website: _ transport DATABASE (details >350 simulations)[16], Technical History of GYRO, all publications, posters, talks, movies, code download and documentation DATABASE being used to develop a more accurate and physically comprehensive successor to GLF23 based on Staebler s advanced TGLF gyrofluid equations[17]

24 Discovery of large profile corrugations at q-min = 2/1 triggering an ITB[19]

25 ITB formation in low power DIII-D at q-min=2 As q-min decreases in time, the plasma is clearly sensitive to the rational q-min points (4/1, 3/2, 2/1) entering the plasma An ITB clearly forms at q-min=2/1 where there is a gap in rational surface density

26 At q-min=2 the ECE sees a bump-dip-bump profile corrugation in the electron temperature gradient Since the minimum-q profiles is drifting downward at the rate dq-min/dt = -0.02/10ms past the ECE channels, the time traces can be mapped to radial traces and very accurate a/lt gradients can be determined. The q-min = 2/1 splits into inward and outward moving 2/1 surfaces.inward 2/1(-) at left

27 GYRO simulations match the bum-dip-bump at the q-min=2/1 and follow the q=2/1(-) and 2/1(+) split corrugations[19] Ion gradient corrugations are weak since electron transport is most closely bound to rational surfaces Simulation use q-min=1.98 data exp. profiles with a/lti +10%. The q(r ) profiles are remapped to qmin=2.06, 2.01, 2.00(-) etc The q-min=2/1 corrugation is larges where the gap in singular surface density is largest. The see a bump precursor before q-min=2/1 may be apparent in q-min=2.01 simulation

28 There is no gap in transport associate with the gap in singular surface density at q-min=2/1 or s=0[19] Instead there is gap in ion transport just interior to the q-min = 2/1(-) surface associated with a large ExB shear layer (next slide). This gap in ion transport means interior heat can not escape so ITB forms, hence the ExB shear layer corrugation is the ITB trigger We have no explanation or understanding as to why the transport plunges by 3-fold across the whole simulation going from q-min=2.02 to q-min=2.01(-) gap

29 The ExB shear layer interior to q-min=2.00(-) is the ITB trigger[19] Zonal flows fluctuate in time (black lines) but the time averaged corrugations persist an follow the low order surfaces as they move with the sinking q-min. There is actually little flow in a zonal flow: V y =V ExB y +Vy * ~ 0 where Vy ExB = "c(#$/#r)/b and V* y = "(c/n0 )(#P i /#r)/(eb) i.e. force balance implied P i = n 0 T i + n T i0 = "en 0 # Zonal flows have a weak density component only for adiabatic electrons. Profile corrugations in the density gradient s should be observable. Large corrugations in the current density gradient are expected at q-min=2/1 which should effect tearing mode stability. (Hinton, Waltz, & Candy: Phys. Plasma 2004)

30 Conclusions about q-min profile corrugations[19] The ECE measured bump-dip-bump electron temperature gradient in the DIII-D off axis q-min discharges confirms the physical reality of the long observed low order singular surface profile corrugations found in GYRO simulations. There is no gap in transport at the q-min =2/1 or s=0 point where the density of singular surfaces as a gap However the q-min = 2/1 corrugation has a huge ExB shear layer just interior to the q-min = 2/1 which provides a gap in ion channel transport triggering the ITB which follows. Zonal flows appear to have rather small flow with ExB canceling the diamagnetic flow.

31 Some general characteristics of profile corrugations in monotonic q-profiles[19] Standard GYRO partial torus (e.g. Δn=6) radial slice simulations give same transport diffusivity (within a few percent) as full torus (i.e. Δn=1) simulations. (e.g. 16 to 96 mode rho-star = DIII-D L-mode cases) Corrugations occur at all lowest (N=1) next lowest (N=2) singular surfaces (q=m/(nδn where Δm=1 ) where there is a dip in singular surface density (red) and m/n mode density (black). For physical Δn=1 (96 mode case) there is only one next lowest surface q=3/2 within the slice 0.45<r/a<0.71)

32 Bohm scaled L-modes, gyrobohm scaled H-mode, and the problem of perfect profile similarity in ρ*-scaling pairs [18]

33 GYRO DIII-D experimental analysis of ρ*-scaling pairs[18]: Mechanisms for breaking gyrobohm scaling to Bohm scaling rho_star : " * # " s /a<1%, " s # c s /$ gyrobohm: " gb =[c s /a]# s 2 =[cs # s ]# * = " B # * Bohm: " #[c B s $ s ]= " /$ gb * Tokamak transport gyrobohm sized and Bohm scaling should theoretically extrapolate to gyrobohm scaling Local profile shearing (LPS)[25]: similar to local ExB shear stabilization " = ˆ "(0)" minus = stabilization gb [1#$ /$ (crit)] * * Nonlocal turbulence spreading (NTS) " = ˆ "(0)" can give Bohm in unstable draining region like LPS gb [1#$ /$ (crit)] * * " = ˆ " (0)" gb [1+# * /# * (crit)] can give super-gyrobohm in stable spreading region Empirical Mixed Bohm-gyroBohm Model separate mechanisms. Extrapolates to ITER Bohm scaling at very small ρ* " = ˆ "(0)" B [# (crit)+# ] * *

34 GYRO DIII-D experimental analysis of ρ*-scaling pairs [18] : L-mode pair has Bohm scaling with good profile similarity A ρ*-scaling experiment is a power scaling to match a dimensionally similar[26 ] temperature profile. For a ρ* = pair, the difference between Bohm and gyrobohm is just outside temperature error bars of 9%. Accuracy of profile similarity crucial particularly with stiff core. Perfect profile similarity can be obtained by projecting data: B α 1/(ρ*) 3/2, T 0 (r) α 1/ρ*, n 0 (r) α 1/ρ*, v φ0 (r) α 1/(ρ*) 1/2, φ 0 (r) α 1/ρ* Super-gB Bohm 1x => ρ*= x, 2x => ρ*= x => ρ*= x => ρ*= Evidence of super-gyrobohm in stable (or less unstable) inner core suggests NTS is dominant here.

35 GYRO DIII-D experimental analysis of ρ*-scaling pairs [18] : L-mode ρ*- projected data and sensitivity to grad-ti and ExB shear DIII-D core is very stiff and strong ExB shear (high toroidal rotation) accounts for much of the good confinement. GYRO simulations match experiments with 15% reduction of grad-t or 15% increase in ExB shear: Reduces 2-fold larger transport to well within experimental error bars.

36 GYRO DIII-D experimental analysis of ρ*-scaling pairs [18] : H-mode larger ρ*- pair measured gyrobohm with poor similarity Puzzle: Going from Bohm L-mode ρ*= to H-mode at larger ρ*= EXPECT more Bohm-like NOT gyrobohm? GYRO simulations match experiments (with 15% reduction ) with actual H-mode data and show gyrobohm scaling(1x vs 2x);.but projected data with perfect similarity show Bohm (1x vs 1x_P_2x). 1x => ρ*=0.004 ; 2x => ρ*=0.006

37 Status and plans for new TGLF transport mode [17] GLF23 comprehensive physics for STD-core separate trapped/passing electron, passing ions, impurities.itg/tem + ETG collisions finite-β (not used because s-α circular geo has low β limit) pinched plasma, energy, and momentum flow from quasi-linear and mixing rule parallel shear KH drive ExB shear stabilization rule built into mixing rule local mixing rule with multiple e and i branches separate ITG-ae & ETG-ai fit to a few ITG-ae & isoetg gyrofluid (later ITG-ae & ETG-ai gyrokinetic simulations) TGLF much more accurate, covers STD-core and high shear & gradient PED, NCS seamless trapped/passing low-k ITG/TEM to high-k ETG and much better collisions PLUS real (Miller shaped [28] )geometry covering all aspect ratios and finite-β all turbulent transport channels plus turbulent heating (later) ExB shear stabilization from linear quasi-mode stability rather than rule local mixing rule (later) generalized to treat nonlocality [19] fit to >350 DATABASE kinetic-e ITG/TEM-ETG simulations

38 GA work on gyrofluid models & gyrokinetic simulations [1] [Waltz 1992] R.E. Waltz, R.R. Dominguez, G.W. Hammett, "Gyro-Landau Fluid Models for Toroidal Geometry," Phys. Fluids B4, 3138 [2][Waltz 1995] R.E. Waltz, G.D.,Kerbel, J, Milovich, J.,and G. W. Hammett, "Advances in the Simulation of Toroidal Gyro-Landau Fluid Model Turbulence,'' Phys. of Plasmas 2, 2408 [3][Waltz 1994] R. E. Waltz, G.D. Kerbel,.,and J. Milovich, "Toroidal Gyro-Landau Fluid Model Turbulence Simulations in a Nonlinear Ballooning Mode Representation with Radial Modes," Phys. of Plasmas 1, 2229 [4][Waltz 1998] Waltz, R.E.. Dewar, R.L. and Garbet, X., "Theory and Simulation of Rotational Shear Stabilization of Turbulence", Phys. Plasmas 5, 1784 [5][Waltz 1997] R. E Waltz, G. M., Staebler, G.W. Hammett,, M. Kotschenreuther, M., and J. A. Konings, "A Gyro-Landau-Fluid Transport Model", Phys. Plasmas [6][Kinsey 2003] J. E. Kinsey, G.M. Staebler, R. E. Waltz, "Burning Plasma Confinement Projections and Renormalization of the GLF23 Drift-Wave Transport Model", Fus. Sci. & Tech. 44, 763

39 GA work on gyrofluid models & gyrokinetic simulations [7] [Rosenbluth 1998] M. N. Rosenbluth and F.L Hinton. "Poloidal flow driven by iontemperature gradient turbulence in tokamaks. Phys. Rev. Lett [8][Candy 2003a] J. Candy and R.E. Waltz, An Eurlerian Gyrokinetic-Maxwell Solver, J. Comp. Phys. 186, 545 [9] [Waltz 2002] R.E. Waltz, J. Candy, and M.N. Rosenbluth, Gyrokinetic turbulence simulation of profile shear stabilization and broken GyroBohm scaling, Physics of Plasmas 9, 1938 [10][Candy 2003b] [J. Candy and R.E. Waltz, Anomalous Transport Scaling in DIII-D Tokamak Matched by Supercomputer Simulation. Phys. Rev. Lett [11][Waltz 2005c] R.E. Waltz, "Rho-star scaling and physically realistic simulations gyrokinetic simulations of DIII-D", Fus. Sci. and Tech. 48,1051. [12] [Waltz 2005b] R.E. Waltz, J. Candy, F.L. Hinton, C. Estrada-Mila, and J.E. Kinsey, "Advances in comprehensive gyrokinetic simulation of transport in tokamaks", Nucl. Fusion 45, 741

40 GA work on gyrofluid models & gyrokinetic simulations [13][Waltz 2005a] R.E. Waltz and J. Candy, Heuristic Theory of Nonlocally Broken Gyro- Bohm Scaling, Phys. Plasmas 12, [14][Kinsey 2005] J.E. Kinsey, R.E. Waltz, and J. Candy, "Nonlinear Gyrokinetic Turbulence Simulations of ExB Shear Quenching of Transport", Phys. Plasmas 12, [15][Kinsey 2006] J.E. Kinsey, R.E. Waltz, and J. Candy, "Effect of Shear and Safety Factor on Turbulent Transport in Nonlinear Gyrokinetic Simulations", Phys. Plasmas 13, [16] [Kinsey 2006] J.E. Kinsey, R.E. Waltz, and J. Candy, Parametric dependence of transport using gyrokinetic simulations, BAPS 51 April 2006 p111invited talk I16 3 Sherwood (GYRO website) [17] [Staebler 2005] G.M. Staebler, J.E. Kinsey, and R.E. Waltz, "Gyro-Landau-Fluid equations for trapped and Passing Particles", Phys. Plasmas 12, [18] [Waltz 2006a] R. E. Waltz, J. Candy, and C.C. Petty, " Projected profile similarity in gyrokinetic simulations of Bohm and gyrobohm scaled DIII-D L- and H-modes " General Atomics Report GA-A25345 submitted to Phys. Plasmas. (GYRO website)

41 GA work on gyrofluid models & gyrokinetic simulations [19] [Waltz 2006b] R. E. Waltz, M.E. Austin, K.H. Burrell, and J. Candy, " Gyrokinetic simulations of off-axis minimum-q profile corrugations" General Atomics Report GA-A25309 accepted Phys. Plasmas. (GYRO website) [20a] [Estra-Mila 2005] C. Estrada-Mila, J. Candy, R.E. Waltz, Gyrokinetic simulation of ion and impurity, Phys.Plasmas 12, [20b] [Estra-Mila 2006] C. Estrada-Mila, J. Candy, R.E. Waltz, Density peaking and turbulent pinch in DIII-D discharge, Phys.Plasmas 13, [21] [Estra-Mila 2006] C. Estrada-Mila, J. Candy, R.E. Waltz, Turbulent transport of alpha particles in reactor plasmas, General Atomics Report GA-A25314, Fusion (GYRO website) [22] [Hinton 2004] F.L. Hinton, R.E. Waltz, and J. Candy, Effect of Electromagnetic Turbulence in the Neoclassical Ohm s Law, Phys. Plasmas 11 (2004) pp [23] [Candy 2006a] J. Candy and R.E. Waltz, Velocity-space resolution, entropy production, and upwind dissipation in Eulerian gyrokinetic simulations, Phys. Plasmas 13 (2006)

42 GA work on gyrofluid models & gyrokinetic simulations [24] [Candy 2006b] J. Candy, R.E. Waltz, Y. Chen, and S.E. Parker, Relevance of the parallel nonlinearity in gyrokinetic simulations of tokamak plasmas, Phys. Plasmas 13 (2006) [25] [Hinton 2006] F.L.Hinton and R.E. Waltz, Gyrokinetic turbulent heating, General Atomics Report GA-A25111, accepted Phys. Plasma (Sept 2006) [26] [Waltz 2006c] R.E. Waltz, J. Candy, and M. Fahey, Coupled ITG/TEM-ETG gyrokinetic simulations: General Atomics Report GA-A25609 submitted to Phys. Plasma APS edition [27][Waltz 1990] [Waltz 1990] R. E. Waltz, J. C. DeBoo, and M.N. Rosenbluth, "Magnetic Field Scaling of Dimensionally Similar Tokamak Discharges," Phys. Rev. Lett. 65, 2390 [28] [Waltz 1999] R. E. Waltz,R.E. and R. M. Miller,.,"Ion Temperature Gradient Turbulence and Plasma Flux Surface Shape", Phys.Plasmas 6, 4265.

43 EXTRAs

44 TGLF available for linear stability analysis of experimental profiles To illustrate the accuracy of TGLF with comprehensive physics (shaped geometry, electron-ion collisions, electromagnetic, dynamic electron, ion and impurity) an analysis of DIII-D discharge at 1.3 sec. made. L. Lao et al. Phys. Plasmas 3 (1996) 1951.