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Pedestal stability comparison and ITER pedestal prediction

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2009 Nucl. Fusion 49 085035

(http://iopscience.iop.org/0029-5515/49/8/085035)

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IOP PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY NUCLEAR FUSION

Nucl. Fusion 49 (2009) 085035 (8pp) doi:10.1088/0029-5515/49/8/085035

Pedestal stability comparison and ITERpedestal predictionP.B. Snyder1,a, N. Aiba2, M. Beurskens3, R.J. Groebner1,L.D. Horton4, A.E. Hubbard5, J.W. Hughes5, G.T.A. Huysmans6,Y. Kamada2, A. Kirk3, C. Konz4, A.W. Leonard1, J. Lonnroth3,C.F. Maggi4, R. Maingi7, T.H. Osborne1, N. Oyama2, A. Pankin8,S. Saarelma3, G. Saibene9, J.L. Terry5, H. Urano2 andH.R. Wilson10

1 General Atomics, PO Box 85608, San Diego, CA 92186-5608, USA2 Fusion Research and Development Directorate, JAEA, Naka, Ibaraki, Japan3 EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, UK4 Max-Planck-Institut fur Plasmaphysik, EURATOM Association, Garching, Germany5 MIT Plasma Science and Fusion Center, Cambridge, MA, USA6 Association Euratom/CEA, Cadarache, St Paul-lez-Durance, France7 Oak Ridge National Laboratory, Oak Ridge, TN, USA8 Department of Physics, Lehigh University, Bethlehem, PA, USA9 EFDA Close Support Unit-Garching, Garching, Germany10Department of Physics, University of York, York, UK

E-mail: snyder@fusion.gat.com

Received 31 December 2008, accepted for publication 18 June 2009Published 28 July 2009Online at stacks.iop.org/NF/49/085035

AbstractThe pressure at the top of the edge transport barrier (or ‘pedestal height’) strongly impacts fusion performance,while large edge localized modes (ELMs), driven by the free energy in the pedestal region, can constrain materiallifetimes. Accurately predicting the pedestal height and ELM behavior in ITER is an essential element of predictionand optimization of fusion performance. Investigation of intermediate wavelength MHD modes (or ‘peeling–ballooning’ modes) has led to an improved understanding of important constraints on the pedestal height andthe mechanism for ELMs. The combination of high-resolution pedestal diagnostics, including substantial recentimprovements, and a suite of highly efficient stability codes, has made edge stability analysis routine on several majortokamaks, contributing both to understanding, and to experimental planning and performance optimization. Here wepresent extensive comparisons of observations to predicted edge stability boundaries on several tokamaks, both forthe standard (Type I) ELM regime, and for small ELM and ELM-free regimes. We further discuss a new predictivemodel for the pedestal height and width (EPED1), developed by self-consistently combining a simple width modelwith peeling–ballooning stability calculations. This model is tested against experimental measurements, and usedin initial predictions of the pedestal height for ITER.

PACS numbers: 52.55.Fa, 52.55.Tn, 52.65.Kj, 52.65.Tt, 28.52.Av

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Accurate prediction of both the pedestal height (pressure atthe inside of the shaded region in figure 1(a)) and edgelocalized mode (ELM) behaviour is essential for predictionand optimization of ITER performance. Pedestal stabilitystudies have developed into a useful and standard tool

a Author to whom any correspondence should be addressed.

for understanding both pedestal and ELM characteristics intokamaks. Here we discuss the basics of peeling–ballooningtheory and numerics (section 2), calculations of edge stabilityin experiment and their implications (section 3) and pedestalwidth physics (section 4). Finally, we combine stabilitycalculations and a simple pedestal width model to develop apredictive model of the pedestal height and width (EPED1),test it against experiment and present initial predictions for theITER pedestal height (section 5).

0029-5515/09/085035+08$30.00 1 © 2009 IAEA, Vienna Printed in the UK

Nucl. Fusion 49 (2009) 085035 P.B. Snyder et al

(a) (b) (c)

Figure 1. (a) Typical profiles in the edge barrier or ‘pedestal’ (shaded) region. (b) Schematic diagram of pedestal stability, including impactof shaping and collisionality. (c) Typical peeling–ballooning mode structure in 3D (n = 18).

(a) (b) (c)

Figure 2. Benchmark of MHD codes in (a) limiter and (b) near X-point geometry, with n = 11 mode structure inset. (c) Effect of cutoff onstability bound for a QH mode discharge, showing that impact of cutoff location is small provided the cutoff is sufficiently far out (here>∼ 99.3%).

2. Peeling–ballooning theory and code development

The peeling–ballooning model of ELMs posits that freeenergy from the strong pressure gradient and the resultinglarge bootstrap current (figure 1(a)) in the edge barrier, or‘pedestal’, region drive intermediate wavelength (typicallyn ∼ 3–30) modes, which constrain the pedestal height andtrigger ELMs. A schematic diagram of peeling–ballooningstability boundaries is given in figure 1(b) and a typical 3Dstructure of such a mode is shown in figure 1(c) [1]. Asshown in figure 1(b), the stability boundary varies stronglywith the plasma shape, and the trajectory with which thepedestal evolves towards its stability limit is strongly impactedby collisionality (ν∗). At high ν∗, the bootstrap currentis suppressed and relatively high n ∼ 10–30 ballooningmodes are limiting, while at low ν∗, relatively low n ∼ 3–6current-driven kink/peeling modes are limiting. Many highperformance H-mode discharges operate at moderate ν∗, wherethe pedestal height and overall performance are maximized,and intermediate n ∼ 5–20 peeling–ballooning modes providethe constraint on the pedestal height.

The hypothesis that ELMs are driven by magnetohydrody-namic (MHD) modes has existed essentially since the discov-ery of H-mode. The importance of peeling–ballooning cou-pling was noted in high-n studies [2]. Extension of the theoryto higher order, and its numerical implementation in the ELITEcode, allowed quantitative treatment of intermediate-n modes,and successful comparisons with experiment [3, 4]. In par-allel, a number of efficient MHD codes, including MISHKA[5, 6], MARG2D [7, 8], KINX [9], ILSA [10], DCON and

GATO [11], were employed for the study of intermediate-nedge stability in tokamaks. A pair of recent reviews [6, 12]summarize many results. The availability of multiple codesemploying a variety of numerical methods has allowed exten-sive code verification (e.g. [3, 4, 8, 13]). An example is given infigure 2(a). Here GATO (n = 1–19), MARG2D (n = 5–160)and ELITE (n = 4–1000) find good quantitative agreement ingrowth rate for a D-shaped equilibrium with pedestal profiles,but no X-point. The ELITE results for n = 800–1000 are usedto extrapolate the n = ∞ growth rate (using a least squares fitto the high-n analytic ballooning expansion �2 = c1 + c2/n,where c1 is the square of the n = ∞ growth rate), showinggood agreement with BAL-MSC. This result also shows thatinfinite-n theory, while quite useful for insight, cannot be usedquantitatively in the pedestal, as it is generally only valid forn’s higher than the diamagnetic (n ∼ 25–60) or finite Larmorradius (n ∼ 40–300) cutoff.

Obtaining precise growth rates for experimental equilib-ria, which often contain X-points, can be more challenging.Relatively high RZ grid resolution is required for high preci-sion, particularly in the vicinity of the X-point. Figure 2(b)shows a comparison of MARG2D and ELITE for a JT-60U-like equilibrium cutoff at the 99.6% flux surface. Good agree-ment is obtained here, and at a 99.8% cutoff, via use of avery high-resolution (1025 × 1025) equilibrium, with only avery small difference in growth rate between the 99.6% and99.8% cutoffs (<5% difference in peak growth rate for bothELITE and MARG2D). ILSA finds somewhat lower growthrates for this case [10]. Fortunately, for experimental compar-isons, such very high precision is generally not needed. One

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Nucl. Fusion 49 (2009) 085035 P.B. Snyder et al

Figure 3. Pedestal stability diagrams for experiments on (a) AUG (b) DIII-D (c) JET and (d) JT-60U, showing Type I ELMs occurring nearthe stability bound (AUG 20116 (crosshair), DIII-D 126443 (black crosshair), JET 70355 (star), JT-60U 43075 (asterisk)), while other typesof discharges (RMP ELM-free DIII-D 126442 (white crosshairs) and radiative Type III JET 70281 (open diamond) and 70289 (filleddiamond)) can lie in the stable region.

needs only to determine the stability boundary (figure 1(b)or figures 3(a)–(d)) to substantially better accuracy than themeasurement error bars for the pressure gradient (typically∼10–25%), which can typically be achieved at RZ resolutionsof ∼129 × 129. Similarly, while the effect of the approach tothe X-point is a subject of significant theoretical and numeri-cal interest [6, 9, 10, 15], for realistic experimental profiles, theimpact of cutoff changes on the stability boundary is found, inthe several cases studied, to be small compared with measure-ment uncertainty, provided the cutoff is sufficiently far out toretain the sharp gradient region. Typically ψ >∼ 99.4% issufficient, as shown in figure 2(c). Most results discussed hereuse a cutoff in the range 99.5–99.7%.

3. Pedestal stability calculation in experiment andits implications

The implementation of high-resolution diagnostic systems hasallowed accurate measurement of profiles in the edge barrierregion necessary for rigorous testing of the peeling–ballooningmodel of ELMs. Such tests, involving high-resolutionequilibrium reconstruction and MHD stability calculation overa wide range of mode numbers (typically n ∼ 3–30), havenow been successfully conducted on all of the world’s majortokamaks in well over 100 different discharges [1, 4, 6, 9, 11–14, 16–23, 25, 28, 41]. Numerous comparison experiments

between machines, as well as dedicated experiments onparticular devices, have elucidated the role of shape [4, 8, 9, 18,20], aspect ratio [14, 18, 19], beta [13, 20], collisionality [4, 12]and rotation [13, 14, 19, 22] on edge stability.

A direct method to study experimental pedestal stability isto construct a time series of equilibria, or a single equilibriumnear the maximum observed pedestal height (shortly beforean ELM), and then perturb the values of pressure gradientand current around the observed reference point, calculatingn ∼ 3–30 stability at each point to construct a stabilityboundary, as in figures 3(a)–(d). These pedestal stabilitydiagrams are typically cast in terms of two local parameters,such as the maximum normalized pressure gradient (α) in theedge barrier (figures 3(a)–(d)), and a characteristic parallelcurrent (figures 3(a)–(c)) or magnetic shear (figure 3(d)) in thepedestal. It is important to keep in mind, however, that pedestalstability is not local. In general the shape of the profilesthroughout the pedestal region and into the core impactspedestal stability. Because a broadened pedestal increasesoverall drive for broader modes, the maximum stable gradientis not a constant but rather decreases with pedestal width,roughly pped ∼ �3/4, where � is the pedestal width in thenormalized poloidal flux (ψN).

For standard Type I ELMing discharges, such as ASDEXUpgrade (AUG) 20116 in figure 3(a) [24], DIII-D 126443 infigure 3(b) [25], JET 70355 in figure 3(c) [16, 17] and JT-60U43075 in figure 3(d) [19, 22], it is consistently found that the

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Nucl. Fusion 49 (2009) 085035 P.B. Snyder et al

pedestal approaches the stability boundary and crosses it atapproximately the time an ELM is observed (at which timethe pedestal height is reduced, and the cycle repeats). Infigures 3(a) and (d) the most unstable toroidal mode numberis indicated by numerals on the stability diagram, showing thelimiting mode is in the expected range of peeling–ballooningmode numbers (n ∼ 10 in figure 3(a), and n ∼ 4 infigure 3(d)), and also indicating the expected pattern of limitingmode numbers discussed above. We note that the diagrams infigure 3 have been constructed with multiple equilibrium andstability codes, and that in the case of figure 3(a), the stabilitycalculations have been conducted both with ILSA/MISHKAand with ELITE, finding good agreement in the location of theoperational point relative to the stability bound.

We define a relative distance to the stability boundaryD as the ratio of the distance from the origin of a j − α

diagram (e.g. figures 3(a)–(c)) to the operational point, overthe distance from the origin to the stability bound along aline passing through the operational point. For 39 recentDIII-D Type I ELM discharges, for which stability has beenstudied shortly before an ELM, we find a ratio D = 1.05 ±0.19, consistent with the conclusion that peeling–ballooningstability constrains the pedestal and triggers observed ELMs.

Because of the potential impact of large ELMs on plasmafacing materials on ITER, a number of techniques have beendeveloped to avoid or mitigate large Type I ELMs. Theseinclude both passive (e.g. quiescent H-mode, Type II, Type III,grassy ELMs, EDA) and active (e.g. resonant magneticperturbations (RMP), pellets, impurity injection) ELM controltechniques. Stability studies on these discharges provideimportant insight into the mechanisms for ELM mitigation.

The use of imposed RMP to suppress ELMs has beendemonstrated on DIII-D [25], and is planned for ITER. Edgestability studies on ELM suppressed discharges (using 2Dequilibria) have consistently found them to lie in the stableregion. A typical example is shown in figure 3(b) (whitecrosshairs). Both the low and high power phases of discharge126442, where ELMs are suppressed with 6.3 kA of current inn = 3 coils, lie well inside the stable region. An otherwiseidentical discharge with zero RMP current, 126443 (blackcrosshair), evolves to the stability limit and exhibits Type IELMs. Stability analysis of 15 DIII-D RMP ELM-free timeslices finds a stability ratio D = 0.83 ± 0.10, consistent withoperation significantly below the stability boundary. Hence noeffect of the RMP perturbation on stability itself is required toexplain ELM suppression. Rather, enhanced particle transportassociated with the RMP appears to hold edge gradients belowthe stability limit. Recently, ELMs have been mitigated, butnot fully suppressed, via n = 1 perturbations from the errorcorrection coils on JET [26]. Analysis of these dischargesfinds a reduction in pedestal density, but pedestal temperature issufficiently high that the pedestal is near the low-n kink/peelingstability boundary [17]. It is suggested that these current-driven modes are responsible for the remaining small ELMs inthese discharges.

In a recent set of advanced tokamak JET experiments [16],it was found that increasing the radiation fraction by eitherimpurity injection or deuterium gas puffing can significantlyreduce ELM size. Stability analysis [16, 17], taking advantageof high-resolution profile measurements from a new Thomson

scattering system, finds that for low radiation fractions whereType I ELMs occur, the stability boundary is reached asexpected before an ELM (figure 3(c)). However, at higherradiative fraction the pedestal height is reduced and the ELMsbecome Type III. The stability analysis finds these highlyradiative discharges 70281 and 70289 to be stable. Moregenerally, it has been consistently found that high-densityType III ELM discharges lie below the ideal stability boundary,and these small ELMs are presumably driven by resistive orkinetic instabilities. Similarly, on Alcator C-Mod, ELM-freeEDA mode operation is found to occur below the stabilityboundary, while the appearance of small or large ELMs isassociated with crossing the stability boundary [23].

It has long been suggested [2, 4] that higher-n ballooningmodes (figure 1(b)), which are relatively narrow and do notneed to dissipate current to stabilize themselves, should triggerrelatively small ELMs. This is consistent with the widelyobserved decrease in ELM size at high collisionality [27]. Boththe onset of small Type I ELMs in DIII-D [4] and grassy ELMsin JT-60U, which occur at low collisionality [21], have beenassociated with relatively narrow modes. A recent study ofType II ELM discharges on JET [17], finds that the combinationof deep second stability access near double null, and relativelyhigh collisionality, causes the discharges to be limited byrelatively high-n modes with narrow mode structures.

Quiescent H-mode (QH) discharges, with no ELMsand a relatively steady, rotating low-n mode in the edge,were first discovered at very low density with neutral beaminjection in the counter-current direction. Stability analysis(figure 2(c)), finds that QH modes operate near the low-n kink/peeling stability boundary in the edge [13, 28]. Byoptimizing discharge shape to increase the density at whichthe transition from kink/peeling limited to peeling–ballooninglimited occurs, more than a doubling of allowed density in QHmode discharges was achieved [13], up to Greenwald fractionsof 0.6. Furthermore, it was suggested that strong rotationshear facilitates the saturation of the low-n kink/peeling mode,resulting in the edge harmonic oscillation (EHO), and it waspredicted that co-rotating QH modes should be possible ifsufficient flow shear could be achieved. In 2008, co-rotatingQH mode discharges with very strong edge rotation shearwere produced on DIII-D [28]. This stability model allowsprediction of required pedestal density for QH operation inITER, but further work is needed to fully quantify requirementson rotation shear.

While stability studies have elucidated several aspectsof operation in small and no-ELM regimes, a number ofimportant, ITER-relevant issues remain unresolved. Theseinclude quantitative understanding of transport, sources andsinks in RMP discharges, quantitative predictions of ELMsize, quantification of the conditions for robust EHO saturationand successful QH-mode operation, requirements for andcharacterization of pellet-induced ELMs and theoreticalunderstanding of conditions for operation in Type III and EDAregimes.

4. Pedestal width observations and models

Pedestal stability calculations provide a constraint on thepedestal height as a function of pedestal width. Studies

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Nucl. Fusion 49 (2009) 085035 P.B. Snyder et al

Figure 4. The pedestal width in normalized poloidal flux is plotted as a function of pedestal poloidal beta for (a) DIII-D and (b) AlcatorC-Mod. Similar scaling of the pedestal width, approximately with β

1/2p,ped, is found for discharges with ELMs in both devices.

have demonstrated that it is possible to predict these stabilityboundaries with reasonable accuracy by using model equilibriacharacterized by a relatively small set of parameters [18].These calculations can be used to predict the pedestal height,provided the pedestal width (more specifically, an independentrelationship between width and height) is known beforehand.

A number of models have been developed and experimentsperformed to study physics of the pedestal width, see forexample [29] and references therein. Many of these widthmodels are based in part on ion orbit physics or E×B shear (e.g.[30–34]), and many lead to a poloidal or toroidal gyroradiusscaling of the width. In some cases, width models have beencombined with local expressions which approximate an MHDstability constraint to allow predictions of pedestal height (e.g.[29, 35–36]).

However, until recently, studies of pedestal width havebeen somewhat ambiguous, due in part to two key issues:(1) the narrow pedestal width is difficult to measure accurately,and small changes in the width are particularly difficult toassess and (2) stability physics imposes a strong and complexconstraint on the pedestal width, schematically pped ∼ �3/4S,where S has a complex dependence on shape, collisionality,current, field, etc, and � is defined in the normalizedpoloidal flux space [18]. This constraint implies that thewidth will correlate strongly with most parameters related topedestal height, and that complex dependences on densityand temperature can be due to the stability constraint (viacollisional effects on bootstrap current), rather than the physicsinherent to the width itself. Three important methods havebeen used to clarify pedestal width physics: (1) carefullydesigned experiments using high-resolution edge diagnostics,(2) accounting for the stability constraint when analyzingresults of width experiments and (3) combining pedestalwidth models with full calculations of edge stability to yieldpredictive models which can be tested directly against pedestalheight, which can be measured more accurately (section 5).

Early studies on DIII-D found pedestal width correlationswith pedestal beta poloidal (βp,ped) and with poloidal andtoroidal gyroradius (ρθ i, ρi), with time dependent analysisfavouring pedestal beta poloidal scaling [37]. An experimenton JT-60U varied species mass (H and D) and found noappreciable difference in pedestal width for similar discharges

in H and D, implying weak or no ρi or ρθ i dependence,and consistency with a β

1/2p,ped scaling [38]. Recent hydrogen

experiments on DIII-D appear to confirm this result [39].Similarly, dimensionless scans on JET and DIII-D find weakor no dependence on ρ/a. Measurements of Te width on AUGalso appear consistent with a β

1/2p,ped scaling, though weak or no

dependence is seen in the density width [24].A comparison experiment between AUG and DIII-D

investigated global β (or power) dependence of the pedestal[20, 24]. Investigation of the pedestal height increase withglobal β [20] found that much of the effect could be explainedby the expected impact of Shafranov shift on edge stability[13]. But this effect alone, at constant pedestal width,was not sufficient to explain the observed pedestal height.However, combining a pedestal width which increases withβ

1/2p,ped with the stability calculations, it is possible to explain the

observations in both beta and shape scaling experiments [20].Detailed examination of a large DIII-D database of

pedestal width measured in the last 20% of the Type I ELMcycle reveals a β

1/2p,ped width scaling. A database of pedestal

widths from 4122 time slices from 120 discharges is binnedby value of β

1/2p,ped and plotted in figure 4(a). While there is

significant scatter, there is very good agreement in the binaverages with � = 0.076β

1/2p,ped, with correlation coefficient of

0.68 and standard deviation of 0.20. Here � is the average ofthe ne and Te widths in the normalized poloidal flux (ψN). And,notably, once this strong correlation of the width with β

1/2p,ped

is taken into account, dependence of the normalized pedestalwidth on other dimensionless parameters (q, ρi, ρθ i, ν∗, β) isweak [40]. Specifically, the absolute value of the correlation of�/β

1/2p,ped with each of these dimensionless parameters is <0.2.

The independence of �/β1/2p,ped on density over a wide range

further suggests that β1/2p,ped, rather than poloidal or toroidal

gyroradius, is the correct dependence of the width.Similarly, measurements of the pedestal width in

discharges with ELMs on Alcator C-Mod find a strongcorrelation with βp,ped, as shown in figure 4(b). Both theexponent and coefficient of the variation with βp,ped aresimilar to that found for DIII-D. In addition, once the ∼β

1/2p,ped

dependence is accounted for, weak or no dependence oncollisionality or gyroradius is found.

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Nucl. Fusion 49 (2009) 085035 P.B. Snyder et al

Figure 5. An illustration of the application of the EPED1 pedestal model. The solid line, calculated by ELITE using a series of modelequilibria, is the peeling–ballooning stability constraint. The dashed line corresponds to the relation � = 0.076β

1/2p,ped, and their intersection

(solid circle) gives the EPED1 predicted pedestal height and width. The model is illustrated for input parameters corresponding to DIII-Ddischarges (a) 132003 and (b) 132017, with the measured values of pedestal height and width shown by open squares with error bars.

In summary, there is strong observational evidence onmultiple machines supporting exploration of a scaling of thepedestal width similar to β

1/2p,ped.

5. Tests of a predictive pedestal model andpredictions for ITER

Combining insight from theory and observation, a predictivemodel of the pedestal height (EPED1) has been developed.The EPED1 model consists of two hypotheses which togetherallow a well-defined predictive model: (A) that the pedestalheight in high performance H-modes is constrained byintermediate-n edge stability, which can be characterized viaa series of 2D model equilibria and a γ > ω∗pi/2 criterionand (B) the pedestal width can be characterized as � =0.076β

1/2p,ped, where � is the average of the ne and Te widths in

normalized poloidal flux (ψN) [40]. The form of the constraint� = c1β

1/2p,ped (notably including a prediction of the lack of

strong dependence on parameters other than βp,ped) and theapproximate value c1 ∼ 0.1 follow from an argument based onkinetic ballooning mode onset, and its variation with magneticshear [40]. The precise value of c1 ∼ 0.076 is taken froma database (figure 4(a)), but because of the expected lackof dependence on other parameters, this value is taken as aconstant for all standard aspect ratio tokamaks.

The EPED1 model employs stability calculations withELITE (n = 5–30) on series of full 2D model equilibria(as in [18, 40]), in which the pedestal temperature is raiseduntil the stability boundary is found. By keeping the pedestalwidth consistent with � = 0.076β

1/2p,ped we locate the point

satisfying both hypotheses (A) and (B). Essentially, thestability calculations give the predicted height as a functionof predicted width, and the width model gives the predictedwidth as a function of predicted height. Because the widthdependence of the stability constraint (roughly pped ∼ �3/4)

and the width model (pped ∼ �2) are quite different, there isgenerally a single non-trivial point in the (pped, �) parameterspace where the two constraints intersect: this is the EPED1height and width prediction. The inputs to the model are a setof operational parameters (BT, Ip, R, a, κ , δ) and a pair ofplasma physics parameters (global β, ne,ped), and the outputs

are pedestal height and width. Note that if any of the inputsare not known precisely beforehand, predictions can be madefor a range of input values.

An illustration of the application of the EPED1 modelis given in figure 5. Given a set of input parameters, oneconstructs series of model equilibria [18, 40] at increasingpedestal height, and evaluates their stability with ELITEto calculate the solid lines, representing hypothesis (A),in figures 5(a) and (b). The dashed lines in figure 5,corresponding to hypothesis (B) (� = 0.076β

1/2p,ped), are trivial

to calculate. The intersection of the two lines, given by theblack circle in figures 5(a) and (b), is the EPED1 prediction ofthe pedestal height and width. The cases shown correspondto input parameter sets of BT = 1.62 T, Ip = 1.16 MA,R = 1.7 m, a = 0.58 m, κ = 1.8, δ = 0.2, βN,global = 2.1,ne,ped = 5 × 1019 m−3 in figure 5(a) and BT = 2.1 T,Ip = 0.85 MA, R = 1.7 m, a = 0.59 m, κ = 1.8, δ = 0.55,βN,global = 2.3, ne,ped = 4 × 1019 m−3 in figure 5(b). Giventhese inputs, both curves and the EPED1 prediction can bemade either before or after an experiment is conducted. Theseparticular inputs correspond to DIII-D discharges 132003and 132017, and the measured heights and widths for thesedischarges are shown by open squares with error bars infigure 5, showing good agreement with the predictions withinmeasurement uncertainty. The higher triangularity and largerratio of BT/Ip lead to substantially larger pedestal height andwidth for 132017, despite similar values of Ip × BT.

The EPED1 model can be rigorously tested againstobservations of both pedestal height and width across multipledevices, and used to make predictions for future experimentsand devices. Note that hypotheses (A) and (B) are stronglycoupled in the model. If either the stability constraint or thewidth model is incorrect, both the predicted height and widthshould be systematically incorrect (that is, moving either curveup or down in figure 5 would in general change both width andheight predictions). For example, if the constant c1 = 0.076is too high (low), both height and width predictions would besystematically high (low). If there is a missing dependencein the width model (e.g. on q, ρi, ρθ i, ν∗, β, a/R) then theaccuracy of EPED1 predictions would vary systematically asa function of that parameter.

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Nucl. Fusion 49 (2009) 085035 P.B. Snyder et al

Figure 6. Test of the EPED1 model against observed pedestal height on a series of discharges from DIII-D, JET and JT-60U. Results areshown in units of (a) pedestal pressure in kPa and (b) normalized pedestal beta. Perfect agreement is represented by the solid diagonal line.In (b), the predicted pedestal height for ITER in its baseline operating scenario is shown by a filled square.

EPED1 has been successfully tested on systematicparameter scans on DIII-D, and against a random selectionof 20 discharges from the DIII-D pedestal database [40]. Adedicated DIII-D experiment was planned to test the modelagainst optimized pedestal measurements, with a very largevariation in the pedestal parameters produced via triangularity(δ), Ip at fixed BT and Ip at fixed Ip/BT variations of afactor of ∼3 [39, 40]. EPED1 predictions, presented beforethe experiment, agree well with observations, and producethe correct trends in each individual scan. Using the inputvalues achieved in the experiment, predicted/measured ratiosof 1.03 ± 0.13 in height and 0.93 ± 0.15 in width are foundover 17 discharges in which pedestal height varied by a factorof 10 and width by a factor of 3. In addition, a series of ITERdemonstration experiments has been conducted, in which DIII-D is used as a 1/3.7 scale model of ITER [41]. Theseprovide a useful test of EPED1 and its ability to make accuratepredictions for ITER. EPED1 predictions have been comparedwith observations for two baseline (Scenario 2, q95 ∼ 3.1)

discharges and two reduced current (q95 ∼ 4.5) discharges.The ratio of predicted/observed height in these 4 cases is1.00 ± 0.14. A comparison of predicted to observed pedestalheight for 17 discharges from the dedicated experiment and 4ITER similar discharges is shown by open circles in figure 6.Overall, for the 41 DIII-D discharges studied thus far, the ratioof predicted to observed height is 1.00 ± 0.18 and width is0.96 ± 0.19. A fairly wide range in q95 (2.7 < q95 < 13.4),pedestal collisionality (0.02 < ν∗

ped < 3.4), global β (0.6% <

β < 3.9%) pedestal Greenwald fraction (0.28 < fGW < 0.82)normalized toroidal gyroradius (0.0014 < ρi < 0.005) andnormalized poloidal gyroradius (0.006 < ρθ i/a < 0.061),and a narrow range of aspect ratio (0.31 < a/R < 0.36) iscovered by these tests.

EPED1 has been employed on four JET AT dischargesdescribed in [16]. Reasonable agreement between EPED1 andobservations is found, as shown by filled diamonds in figure 6.The small differences in predicted pedestal height for thesefour discharges are attributable to small changes in density andglobal beta (i.e. Shafranov shift effect on peeling–ballooningstability). All four of these discharges have the same inputvalues of BT = 3.1 T, Ip = 1.9 MA, R = 2.9 m, a = 0.91 m,κ = 1.74, δ = 0.42.

A set of JT-60U Type I ELM discharges described in [38]has also been studied, consisting of a total of 16 time slices

from 4 discharges (44172, 44174, 44175, 44197). It has beenobserved that in these discharges, the pedestal height can varysignificantly in time, even though the machine parameters (BT,Ip, R, a, κ , δ) remain nearly fixed. There is a complicationin comparing EPED1 to measurements on these discharges.On JT-60U, a high-resolution charge exchange system is usedto measure the ion temperature profile, and this profile isused to define the width of the pedestal, with the heightdefined to be the pressure at the radial location of the Ti

pedestal top [21, 38]. However, at the densities characteristicof these discharges, the width of the ion temperature pedestalis expected to be larger than the width � (average of Te

and ne widths) defined for EPED1. A study of DIII-D datasuggests a ratio of approximately 2. Hence, we evaluate theEPED1 pedestal pressure at a radial location correspondingto 2� for comparison with JT-60U measurements at the Ti

pedestal top. This approximation allows a direct comparisonwith data, and does not impact predicted trends. With thisapproximation, good agreement is found between EPED1 andJT-60U measurements, as shown by asterisks in figure 6.Notably, the variation in time of the pedestal height is wellreproduced by the EPED1 model, and is due to a combinationof evolving values of the global beta (i.e. Shafranov shifteffect on peeling–ballooning stability) and pedestal density(which changes collisionality, impacting peeling–ballooningstability).

Testing of the model against a broad range of devicesis continuing, and possible extensions are being considered.Nonetheless, given initial success, it is useful to apply themodel to ITER, to provide estimated pedestal values for useas inputs to other modelling and design activity. Using theScenario 2 parameters, and a small range of pedestal densities(7 × 1019 m−3 < ne,ped < 1020 m−3), EPED1 predicts apedestal height in ITER of βN,ped = 0.6–0.7 and a width� ∼ 0.04. Small variations around the Scenario 2 parametersallow optimization to values up to βN,ped ∼ 0.8–0.9. Atthe ITER reference density of 1020m−3, and typical profilepeakedness, yielding a pedestal density 7 × 1019 m−3, a valueof βN,ped = 0.65 corresponds to Tped ∼ 4.6 keV. The ITERprediction is shown by a square in figure 6(b). Note that, interms of normalized beta, the ITER pedestal prediction liesin the midst of the observed pedestals, as expected becausethe ITER values of plasma shape, aspect ratio and q are fairly

7

Nucl. Fusion 49 (2009) 085035 P.B. Snyder et al

typical of high performance discharges. Note that the givenvalues are predictions for the pedestal top, at a radial locationof around 0.96 in normalized poloidal flux or 0.95 in squareroot of normalized toroidal flux. Many transport simulations ofITER use a boundary condition significantly further in (∼0.85)where values of pressure will be significantly higher than at theactual pedestal top. We note that this prediction of the ITERpedestal height is in a similar range to that of Sugihara et al [36](Tped = 5.3 ± 1.3 keV at pedestal density of 7 × 1019 m−3)

despite significant differences in the models.

6. Summary

The combination of high-resolution diagnostics and anefficient suite of codes has allowed routine analysis of pedestalstability on many tokamaks. It is reasonably well establishedthat Type I ELM onset is associated with the intermediate-nMHD stability boundary, while RMP ELM-free, high densityType III ELM and EDA discharges operate below the idealboundary. Models have been proposed for other types ofsmall and no-ELM discharges including Type II, grassy andQH mode.

Recent experiments, and analysis accounting for pedestalstability, suggest a pedestal width scaling similar to β

1/2p,ped.

Such a scaling is similar to that expected theoretically ifgradients in the pedestal are constrained by the onset of thekinetic ballooning mode. By combining a simple width modelwith pedestal stability calculations, a predictive model forthe pedestal height and width (EPED1) has been constructed.This model has proved accurate in an initial set of tests onDIII-D, JET and JT-60U, including a dedicated experiment,with predictions made beforehand, and ITER-demonstrationdischarges on DIII-D. Initial predictions for ITER have beenmade using this model. Work is ongoing to further test andrefine models, build theoretical understanding of the physicssetting the pedestal width and better understand small ELMregimes, including pellet induced ELMs.

This paper has focused primarily on standard aspect ratiotokamaks, specifically Alcator C-Mod, AUG, DIII-D, JET, JT-60U and the planned ITER device. Further exploration ofpedestal width models and observations on low aspect ratiotokamaks is an important avenue for future work.

Acknowledgments

This work was supported in part by the US Department ofEnergy under DE-FG03-95ER54309, DE-AC05-00OR22725,DE-FG02-92ER54141 and DE-FC02-99ER54512. The workinvolves contributions from many members of the ITPAPedestal and Edge Physics group, and the larger pedestalphysics community. This report was prepared as an accountof work by or for the ITER Organization. The Membersof the Organization are the People’s Republic of China, theEuropean Atomic Energy Community, the Republic of India,Japan, the Republic of Korea, the Russian Federation and the

United States of America. The views and opinions expressedherein do not necessarily reflect those of the Members or anyagency thereof.

References

[1] Snyder P.B. et al 2005 Phys. Plasmas 12 056115[2] Connor J.W. et al 1998 Phys. Plasmas 5 2687[3] Wilson H.R. et al 2002 Phys. Plasmas 9 1277[4] Snyder P.B. et al 2002 Phys. Plasmas 9 2037[5] Mikhailovskii A.B. et al 1997 Plasma Phys. Rep. 23 844[6] Huysmans G.T.A. 2005 Plasma Phys. Control. Fusion

47 B165[7] Aiba N. et al 2006 Comput. Phys. Commun. 175 269[8] Aiba N. et al 2007 Plasma Fusion Res. 2 010[9] Medvedev S. Yu. et al 2006 Plasma Phys. Control. Fusion

48 927[10] Konz C. et al 2008 Cross-machine and cross-code

comparisons in linear MHD stability analysis for tokamaksProc. 35th EPS Conf. on Plasma Physics (Hersonissos,Crete, Greece, 9–13 June 2008) (European PhysicalSociety) (ECA) vol 32D P2.071http://epsppd.epfl.ch/hersonissos/pdf/P2 071.pdf

[11] Turnbull A.D. et al 2003 Plasma Phys. Control. Fusion 45 204[12] Wilson H.R. et al 2006 Plasma Phys. Control. Fusion 48 A71[13] Snyder P.B. et al 2007 Nucl. Fusion 47 961[14] Saarelma S. et al 2007 Plasma Phys. Control. Fusion 49 31[15] Huysmans G.T.A. and Czarny O. 2007 Nucl. Fusion 47 659[16] Beurskens M. et al 2008 Nucl. Fusion 48 095004[17] Saarelma S. et al 2009 Plasma Phys. Control. Fusion

51 035001[18] Snyder P.B. et al 2004 Plasma Phys. Control. Fusion 46 A131[19] Saibene G. et al 2007 Nucl. Fusion 47 969[20] Leonard A.W. et al 2008 Phys. Plasmas 15 056114[21] Oyama N. et al 2005 Nucl. Fusion 45 871[22] Lonnroth, J-S, et al 2007 Plasma Phys. Control. Fusion

49 273[23] Mossessian D.A. 2003 Phys. Plasmas 10 1720[24] Maggi C.F. et al 2007 Nucl. Fusion 47 535[25] Evans T.E. et al 2006 Nature Phys. 2 419[26] Liang Y. et al 2007 Phys. Rev. Lett. 98 265004[27] Loarte A. et al 2004 Phys. Plasmas 11 2668[28] Burrell K.H. et al 2009 Edge pedestal control in quiescent

H-mode discharges in DIII-D using CO plus counter neutralbeam injection Nucl. Fusion at press

[29] Onjun T. et al 2002 Phys. Plasmas 9 5018[30] Shaing K.C. 1992 Phys. Fluids B 4 290[31] Itoh S.-I. et al 1994 Phys. Rev. Lett. 72 1200[32] Kotschenreuther M. et al 1996 16th IAEA Fusion Energy Conf.

(Montreal, Canada) IAEA-F1-CN-64/D1-5[33] Staebler G.M. et al 1994 Phys. Plasmas 1 909[34] Diamond P.H. et al 1995 Phys. Plasmas 2 3685[35] Hatae T. et al 2001 Nucl. Fusion 41 285[36] Sugihara M. et al 2003 Plasma Phys. Control. Fusion 45 L55[37] Groebner R.J. and Osborne T.H. 1998 Phys. Plasmas 5 1800[38] Urano H. et al 2008 Nucl. Fusion 48 045008[39] Groebner R.J. et al 2009 Progress towards a predictive model

for pedestal height in DIII-D Nucl. Fusion at press[40] Snyder P.B. et al 2009 Development and validation of a

predictive model for the pedestal height Phys. Plasmas16 056118

[41] Doyle E.J. et al 2008 Proc. 22nd Int. Conf. on Fusion Energy2008 (Geneva, Switzerland, 2008) (Vienna: IAEA)CD-ROM file EX/1-3 and http://www-naweb.iaea.org/napc/physics/FEC/FEC2008/html/index.htm

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