the eped pedestal model and proposed tests on alcator c ... · pdf filepb snyder/c-mod...

The EPED Pedestal Model and Proposed Tests on Alcator C-Mod and NSTX

Coworkers: M. Beurskens, R.J. Groebner, J.W. Hughes, R. Maingi, T.H. Osborne, H.R. Wilson, J. Canik, Y. Kamada, A. Kirk, C. Konz, A.W. Leonard, N. Oyama, H.Urano, X. Xu and the ITPA Pedestal group

C-Mod/NSTX Pedestal Workshop Princeton, NJ 7 September 2010

Philip B. Snyder

General Atomics, Theory and Computational Science

PB Snyder/C-Mod NSTX/Sept2010

EPED Model Combines Peeling-Ballooning and KBM Physics to Predict Pedestal Height and Width

A.  The Peeling-Ballooning Model –  “Global” constraint on pedestal height vs width –  Successfully tested across wide range of cases

B.  Kinetic Ballooning Mode Onset –  Local constraint from ballooning/GK theory –  Integrate to get 2nd relation on width vs height

C.  Combine 1&2 to Develop Predictive Model (EPED) –  2 “equations” for 2 unknowns: pedestal height and width –  Simplified model (EPED1), and present model (EPED1.6)

•  EPED1.6 calculates both P-B and KBM constraints directly for each case •  First principles, no use of measurements in any part of model, but still simple & predictive

D.  Validate Model Against Experiment –  Dedicated experiments on C-Mod and DIII-D, tests on JET and JT-60U –  Plans for further tests on C-Mod and tests on NSTX

Develop a model based on two fundamental physics constraints, which are directly calculable, but simple enough to be predictive and easily testable


Peeling-Ballooning Model and Validation


The Peeling-Ballooning Model Explains ELM Onset and Pedestal Height Constraint

•  Pedestal is constrained, and (“Type I”) ELMs triggered by intermediate wavelength (n~3-30) MHD instabilities –  Driven by sharp pressure gradient and bootstrap current in the edge barrier (“pedestal”) –  Complex dependencies on ν*, shape etc. due to bootstrap current and “2nd stability” –  P-B limit increases with pedestal width (Δψ), but not linearly (roughly βNped~Δψ3/4)

The P-B constraint is fundamentally non-local (effectively global on the scale of the barrier)

•  ELITE code, based on extension of ballooning theory to higher order, allows efficient and accurate computation of the intermediate n peeling-ballooning stability boundary

H.R. Wilson, P.B. Snyder et al PoP 9 1277 (2002). P.B. Snyder, H.R. Wilson et al PoP 9 2037 (2002). P.B. Snyder, K.H. Burrell, H.R. Wilson et al Nucl Fusion 47 961 (2007).

ELITE

mode width


Peeling-Ballooning Model Extensively Validated Against Observation

•  High resolution measurements allow accurate reconstructions and stringent tests of P-B pedestal constraint & ELM onset condition

•  Onset of each ELM consistently found to correlate to crossing the P-B stability boundary [Multiple machines, >100 cases studied]

–  Changing pedestal stability (shape, q, ν* etc) yields predictable change in the pedestal height

–  Statistical study of 39 recent DIII-D discharges finds good agreement: 1.05 ±0.19 •  Consistent with conclusion the P-B stability limits pedestal height and triggers ELMs

•  P-B stability studies on NSTX challenging, often find ELMs appear to be triggered by low-n kink/peeling modes [Maingi09, Canik09…]

–  Low-A, strong shaping -> strongly stabilizing to ballooning and P-B modes

DIII-D

NSTX


ELMs on Alcator C-Mod Consistent with Peeling-Ballooning Model

•  ELM-free and EDA shots are peeling-ballooning stable •  Peeling-Ballooning modes consistently unstable just before ELMs •  Limiting modes tend to be higher mode number (n~10-30) •  Appearance of ELMs likely due to physics of both the QC mode and

P-B modes

D.A. Mossessian, P.B. Snyder, A. Hubbard, J.W. Hughes et al, PoP2003


Stability Studies Using Model Equilibria Useful for Predictions in Present and Future Devices

•  For predictions it is useful to conduct pedestal stability analysis on series of model equilibria –  Simplified shape and profiles, with tanh pedestal and Sauter bootstrap current –  Predict pedestal height as a function of (Δψ ,Bt, Ip, R, a, κ, δ, ne,ped, βN, global) –  Calculations using pedestal width (Δψ) as an input find good agreement with observation (model equilibria

capturing important stability physics) [Snyder04,09] •  Δψ defined as the average of the electron density and temperature widths, fit to a tanh

•  Can accurately quantify stability constraint [height=f(width)], but need second constraint for fully predictive model of pedestal height and width

–  Many mechanisms cause transport in the pedestal (neo, turb etc), but we’re looking the for one which stops the profiles from evolving (keeping in mind ExB shear generally increases with pprime)

[focus on low collisionality, high power regime where QCM and TIII ELMs don’t come in]

Δψ

Tped


Kinetic Ballooning Mode Onset


Propose Pedestal Constrained by KBM Onset Near Ideal Ballooning αcrit

•  EM GF and GK simulations: onset of strongly driven kinetic ballooning mode (KBM) near ballooning αcrit [Snyder99, Scott01, Jenko01, Candy05…] –  Abrupt, strong onset -> profile stiffness –  kinetic effects drive onset slightly below ideal

ballooning (n->infinity) boundary αcrit –  ExB shear can impact onset somewhat but

not suppress –  ideal αcrit useful marker for KBM

•  onset near nominal αcrit even with 2nd stability €

KBM has sharp onset, drives large fluxes. Much shorter correlation time than ITG

Snyder & Hammettʼ99,ʼ02

Typical ExB shearing rate in edge barrier


Calculate KBM Constraint in Terms of Measurable Parameters “ballooning critical pedestal”

“Ballooning critical pedestal” calculations to quantify KBM model

•  Model equilibria as in peeling-ballooning calculation (ITER-like, DIII-D cases shown)

–  Increase pedestal height at each pedestal width until pedestal reaches “ballooning critical” point

•  “ballooning critical” when central half of edge barrier unstable (or 2nd stable inside) [baloo code, R.L. Miller]

–  Effectively integrates the local constraint, to allow it to be written in terms of measurable “global” pedestal parameters (Δ, βp,ped, ν, ε...)

•  Find expected dominant dependence, weak variation with other parameters

•  DIII-D, ITER-like cases:

•  Lump weak dependencies into G function, where we calculate <G>~0.07-0.1 for standard aspect ratio tokamaks (0.084+-0.10 for ensemble of 16 cases)

€

€

ΔψN= βp,ped

1/ 2 G(ν*,ε...)

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ΔψN~ βp,ped

1/ 2 No observations on this plot, points are calculations and lines are fits to calculations

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ΔψN≈ (0.09 ± 0.02)β p,ped

1/ 2


Implementing and Testing the EPED Model


Mechanics of the EPED Predictive Model •  Input: Bt, Ip, R, a, κ, δ, nped, βglobal •  Output: Pedestal height and

width

A.  P-B stability calculated via a series of model equilibria with increasing pedestal height

–  ELITE, n=5-30; EPED1: EPED1.6: non-local diamag model

B.  KBM Onset: EPED1: –  EPED1.6: directly calculated with

ballooning critical pedestal technique

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ΔψN= 0.076β p,ped

1/ 2

•  Different width dependence of P-B stability (roughly pped~Δψ3/4) and KBM onset (pped~Δψ2) ensure unique solution, which is the EPED prediction (black circle)

-can then be systematically compared to existing data or future experiments P-B stability and KBM constraints are tightly coupled: If either physics model (A or B) is

incorrect, predictions for both height and width will be systematically incorrect Effect of KBM constraint is counter-intuitive: Making KBM stability worse increases pedestal

height and width

€

γ >ω*pi /2

P.B. Snyder et al PoP 16 056118 (2009)


Experimental Tests of the Simplified EPED Model (EPED1)

Simplified model has a longer history and has been used predictively on DIII-D and C-Mod and compared to JET & JT60-U


EPED Predictions in Good Agreement with Dedicated DIII-D Experiment (EPED1 Model)

•  Experiment planned to yield large pedestal variation via scans in Ip, Bt and δ (~factor of 3 variations, 17 discharges) [Groebner08]

•  EPED1 predictions made and presented before the experiment –  Good agreement, reproduce observed trends

•  Using achieved inputs, find very good agreement in predicted/measured height 1.03 ±0.13 and width 0.93 ±0.15 –  Height varied more than a factor of 10, width varied by factor of ~3


Tests of EPED on JET and JT-60U (EPED1 Model)

•  Initial test on 11 JET shots [Beurskens09], 1.05 ±0.14 •  Trends with time on JT-60U [Urano08] accurately reproduced

–  Changes in time of pedestal explained by βglobal and nped variation

•  Predicted/Measured pedestal height= 1.02 ±0.14 (21 DIII-D, 16 JT-60U,11 JET)

•  Automation of EPED1 runs allows comparisons to large data sets


Experimental Tests of the Full EPED Model (EPED1.6)

Full model is first principles, in that both P-B and KBM constraints calculated directly, with no reference to observation. A more sophisticated model of diamagnetic stabilization of P-B is included, allowing better comparison with C-Mod.


Dedicated Experiment on C-Mod to Test EPED Model in Plasma Current Scan

•  Previous C-Mod studies highlighted importance of ω* stabilization –  EPED1 uses simple model: for instability [Roberts, Tang]

–  ω*pi varies rapidly in edge barrier. EPED1 used effective (half max) value, EPED1.6 adds effect of rollover at high-n via bi-linear fit using BOUT++ results

•  Ip scan expt on C-Mod, 9/09 [Hughes, C-Mod team, Snyder, Groebner] –  Two values of current (Ip=940 & 680 kA, Bt=5.3T) –  C-Mod analysis shown here is preliminary, average over arbitrary ELM timing

•  Good agreement in predicted/measured height: 1.02 ± 0.20 –  6 C-Mod (1.03±0.19), 7 JET, 7 DIII-D discharges, factor of >20 in ped pressure

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−γMHD2 =ω(ω −ω*pi)→γMHD >ω*pi /2


Proposed Tests of EPED Model on C-Mod and NSTX


Further Tests of EPED Model on C-Mod

•  Challenge is to extend the parameter range with ELMs and best quality pedestal data –  C-Mod in unique parameter space, very high field and density. Pedestal pressures

approaching ITER regime possible –  High power needed to reach ELMing or QH mode at high field, density

•  Ip scan expt on C-Mod, 9/09 [Hughes, C-Mod team, Snyder, Groebner] –  Two values of current (Ip=940 & 680 kA, Bt=5.3T) –  Extend to higher and lower values of current, ion data if possible

–  Secondary shape and density variation to extent feasible: hypothesis, bad shapes for ELMs –  Comparison expt with DIII-D: match shape, A, q, etc


Tests of EPED Model on NSTX

•  Many challenges/differences with NSTX –  KBM more stable, kinetic effects more important, need GK EPED2, NEO bootstrap? –  P-B also more stable, ELMs appear to usually be low-n kink/peeling

•  Proposed approach: Start by shining a streetlight down on a patch of parameter space we think we can understand, move out from there –  Proposed expt (see also A. Sontag): Operate NSTX in stability regime similar to standard

aspect ratio tokamaks, and extend out from there –  Triangularity scan to lowest feasible value, moderate to low elongation, with Ip scan at

lowest attainable triangularity values (max Bt) •  Maintain high power, relatively low collisionality, optimize for pedestal diagnostics •  Secondary density scan, go as low as possible

NSTX

NSTX P-B analysis, without (left) and with (right) lithium (Maingi09)


Summary •  P-B stability constrains pedestal height, explains range of observations

–  “Global” but calculable constraint on pedestal height as function of width –  Explains Type I ELM onset, linear & early nonlinear mode structure, QH…

•  KBM onset near αcrit provides 2nd testable constraint on pedestal •  Combine P-B calculations on model equilibria with a KBM onset model

to yield predictive pedestal model (EPED) –  EPED1: simple KBM model –  EPED1.6: KBM constraint directly calculated on model equilibria (first principles)

•  Input: Bt, Ip, R, a, κ, δ, nped, βglobal Output: Pedestal height and width •  If any part is wrong, both height and width predictions wrong (test vs. height) •  KBM model acts as amplifier of P-B stability physics: most complexity in P-B stability

–  EPED1 successfully tested: 1.02 ±0.14 (21 DIII-D, 16 JT-60U,11 JET) –  Successful initial test of EPED1.6: 1.02 ±0.20 (7 DIII-D, 7 JET, 6 C-Mod) –  ITER pedestal height and width predicted βNped~0.6, Δψ~0.04

•  Proposed experiments to test model on C-Mod and NSTX –  Ip scan, weak shaping on NSTX, secondary density scan, C-Mod DIII-D comp

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ΔψN= 0.076β p,ped

1/ 2

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ΔψN= βp,ped

1/ 2 G(ν*,ε...)

EPED1 refs: P.B. Snyder et al PoP 16 056118 (2009); P.B. Snyder et al Nucl Fusion 49 085035 (2009).


Future Work •  Test and Improve upon EPED model

–  Further tests of EPED on multiple tokamaks (including dedicated expts) •  Systematic tests on C-Mod, DIII-D, NSTX for 2011 US Milestone

–  Extend physics of KBM model, and peeling-ballooning model (EPED2) •  Improved treatment of diamagnetic effects in P-B •  KBM from linear (eventually nonlinear) GK simulations, including weak

dependencies of G (aspect ratio, collisionality…)

•  Couple EPED to core transport (TGLF, MM etc) for global profile prediction: global optimization, control


Extra Slides


KBM relationship between width and height is consistent with observations

•  Scaling of first found by Osborne99 •  DIII-D, C-Mod, MAST, AUG find Δ~βp,ped

1/2 dependence in T1 discharges –  Accounting for this dependence, weak dependence on other parameters (q, ν*, ρ*i, ρ*θ, β) –  KBM calc: , <G>~0.07-0.1; data: <G>~0.076 (DIII-D), <G>~0.084 (C-Mod)

•  Isotope variation expts on JT-60U [Urano08], DIII-D [Groebner08] find no dependence of width on mass (consistent with KBM, inconsistent with gyro or banana radius models)

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Δ /a∝βp,ped0.4

C-Mod, 18 discharges

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ΔψN= βp,ped

1/ 2 G(ν*,ε...)


Initial tests of KBM constraint in slowly evolving DIII-D discharges [Groebner10]

Peeling-ballooning modes limit pedestal height, boundary crossed ~ELM

Kinetic ballooning modes constrain “local” pressure gradient prior to ELM

•  Easy test: Integrate up critical KBM constraint and compare to observed relation between width and height (previous pages)

•  Harder test: Look at dynamic evolution and contact with “local” KBM critical gradient (shown) – challenging to diagnostic resolution

•  Harder yet: Look at dynamic evolution to find contact with KBM critical gradient and study resulting turbulence/bursts/~coherent mode – challenging to sim/diagnostics


Pedestal Prediction for ITER

•  βN,ped useful metric for predictions

•  For ITER baseline, EPED predicts a pedestal height of βN,ped~0.6 and a width Δψ~0.04 (~4.4cm)

•  At ITER reference density, and typical density peaking, one expects nped~7x1019 m-3 , at this density, βN,ped=0.6 corresponds to Tped=4.1 keV

–  Note: Predictions are for pedestal top (ψ~0.96, ρ~0.95). Core transport studies often use a BC further in (eg Kinsey ρ=0.86). Using model profiles, our prediction corresponds to roughly βN,ρ=0.86~0.9, Tρ=0.86~5.5 keV, ne,ρ=0.86~8x1019 m-3


Magnetic shear dependence of KBM onset implies approximately: Δψ~βp,ped

1/2

•  Strong KBM onset implies: –  Integrating across edge barrier:

(Δψ is width in normalized poloidal flux, overbar represents average across barrier)

–  Strong dependence of Δψ on βp,ped, sub-linear due to s dependence of αcrit

–  Pedestal is predominantly in a low (local) magnetic shear region

–  Higher Jbs~βp reduces shear (s~1/Jbs) and increases αcrit, expect approximately

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α ∝α crit ∝β p,ped Δψ

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Δψ ∝βp,ped α crit

€

€

αcrit ~ 1/s1/ 2

Stable Unstable

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α crit ∝βp,ped1/ 2 ⇒Δψ ∝βp,ped

1/ 2

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α ∝αcrit

This is the leading order behavior, need to calculate quantitative details


ELITE Code Extensively Verified

•  Extensive successful benchmarks have been carried out between codes (GATO, MARS, MARG2D, MISHKA, CASTOR, M3D, M3D-C1, NIMROD, BOUT++) –  Good agreement in both limiter and near-separatrix geometry –  Good agreement both with and without flow shear (MARS and CASTOR)

Cutoff=99.8%

the eped pedestal model and proposed tests on alcator c ... · pdf filepb snyder/c-mod...

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