theory and simulation basis for …theory and simulation basis for magnetohydrodynamic stability in...

QTYUIOP

GA-A24627

THEORY AND SIMULATION BASIS FORMAGNETOHYDRODYNAMIC STABILITY

IN DIII-Dby

A.D. TURNBULL, D.P. BRENNAN, M.S. CHU, L.L. LAO,and P.B. SNYDER

DECEMBER 2004

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability orresponsibility for the accuracy, completeness, or usefulness of any information, apparatus,product, or process disclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by trade name,trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement,recommendation, or favoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of the UnitedStates Government or any agency thereof.

QTYUIOP

GA-A24627

THEORY AND SIMULATION BASIS FORMAGNETOHYDRODYNAMIC STABILITY

IN DIII-Dby

A.D. TURNBULL, D.P. BRENNAN,* M.S. CHU, L.L. LAO,and P.B. SNYDER

This is a preprint of a paper to be submitted forpublication in Fusion Science and Technology.

*Massachusetts Institute of Technology

Work supported byU.S. Department of Energyunder DE-FC02-04ER54698

GENERAL ATOMICS PROJECT 03726DECEMBER 2004

THEORY AND SIMULATION BASIS FOR MAGNETOHYDRODYNAMIC

STABILITY IN DIII-D A.D. Turnbull, et al.

GENERAL ATOMICS REPORT A24627 iii

ABSTRACT

Theory and simulation have provided one of the critical foundations for many of the

significant achievements in magnetohydrodynamic (MHD) stability in DIII-D over the

past two decades. Early signature achievements included the validation of tokamak MHD

stability limits, " and performance optimization through cross section shaping and

profiles, and the development of new operational regimes. More recent accomplishments

encompass the realization and sustainment of wall stabilization using plasma rotation and

active feedback, a new understanding of edge stability and its relation to edge localized

modes (ELMs), and recent successes in predicting resistive tearing and interchange

instabilities. The key to success has been the synergistic tie between the theory effort and

the experiment made possible by the detailed equilibrium reconstruction data available in

DIII-D and the corresponding attention to the measured details in the modeling. This

interaction fosters an emphasis on the important phenomena and leads to testable

theoretical predictions. Also important is the application of a range of analytic and

simulation techniques, coupled with a program of numerical tool development. The result

is a comprehensive integrated approach to fusion science and improving the tokamak

approach to burning plasmas.



GENERAL ATOMICS REPORT A24627 1

I. INTRODUCTION

Historically DIII-D has been a leader in magnetohydrodynamic (MHD) stability

physics, and the effort has always been characterized by a close synergistic interaction

between theory and experiment. The design for DIII-D in 1985 [1] was based solidly on

MHD stability calculations [2]. One consequence of this synergism is that numerical tool

development has also been a prominent element in the theoretical program and

calculations from stability codes have been an integral component of the experimental

program from early on. The key codes featured in the early DIII-D design work were

MBC [3] for ideal local stability, GATO [4] for ideal 2-D global stability, and CART [5]

for nonlinear resistive reduced MHD stability. This list has been greatly expanded over

the past twenty years, as is discussed below and in Appendix A.

The close interaction with the experiment implies that the theory and numerical

tools are continually tested against the experimental data. This occurs through both

predictionand verification, or a posteriori analysis. It also means that crucial numerical

and theoretical tools, useful in real systems, are developed for the observed limiting

instabilities as the need arises. A range of analytic and simulation tools has been utilized,

covering basic linear MHD theory, and nonlinear and quasi-linear extensions. This

approach has led to several successive records in " [6-9] and fusion gain [10] in DIII-D

with MHD stability calculations and theory playing a key role. (Here, " is defined as the

volume averaged pressure divided by the vacuum magnetic field pressure at the nominal

discharge center, " = 2µ0

< p > B0

2 , and the fusion gain is the ratio of fusion power to

input power, Q = Pfus/Pinp.) It has also led to the Advanced Tokamak (AT) concept

[1113], which is characterized here as a high performance discharge scenario with high

", high confinement, and long pulse or steady state obtained through equilibrium

optimization using active control.

Particular emphasis has been on numerical tool development. Realistic boundary

conditions are critical to obtaining quantitative agreement since stability is an eigenvalue

problem whose solutions are driven equally by boundary conditions and equation

structure. The codes listed here have been a crucial element in the DIII-D program. In

addition to the early codes MBC, GATO, and CART, the large scale numerical tools in

use include the local stability codes CAMINO [14] and BALOO [15], the ideal stability

code DCON [16], the edge stability code ELITE [17,18], the vacuum code VACUUM


A.D. Turnbull, et al. STABILITY IN DIII-D

2 GENERAL ATOMICS REPORT A24627

[19], the linear resistive stability codes PESTIII [20], TWISTR [21], and MARS

[22,23], the nonlinear MHD codes NIMROD [24] and NFTC [25], and the nonlinear edge

code BOUT, which can be considered as a stability code [26,27]. Further details of the

relevant features of each of these codes are provided in Appendix A.

The important limiting MHD modes in DIII-D can be characterized as: (i) n!=!0

axisymmetric modes, (ii) ideal non-axisymmetric modes, including local and global (both

internal, and external) kink modes as well as infernal modes [28,29]), (iii) quasi-MHD

modes basically ideal MHD modes but with important non-ideal extensions such as

the resistive wall mode (RWM), fast particle driven modes, and edge instabilities, and

(iv) resistive interchange and tearing modes. Theory has been crucial to progress, and in

some cases the prime driver, in understanding each of these limits.

Fast particle instabilities and edge stability are two important areas where theory

contributions coupled closely with experiments in DIII-D have been especially fruitful.

Theory and numerical calculations were critical to identifying several beam-driven

Alfvén eigenmodes in DIII-D [3032]. Much of this work is summarized in Ref. [30] and

more recently in the companion paper in Ref. [33]. In addition, several published

theoretical and numerical contributions provided important theoretical underpinning for

understanding the experimental results in DIII-D as well as other tokamaks [34,35]. In

the area of edge stability, the theory and experiments together have recently led to a new

understanding of edge localized modes (ELMs). Edge MHD stability and its relation to

ELMs has been a long term focus of MHD research at DIII-D since the link was first

investigated in Turnbull, et al. [36] and in Gohil, et al. [37]. This work is a distinct

success story for the application of MHD theory and numerical calculations to the

experiment. It is highlighted in the companion paper in this issue [38] and will not be

discussed further here. Several other recent summaries have also covered much of this

material [39-45].

The contributions to the understanding of ideal modes from theory applied to

DIII-D, including axisymmetric stability, are outlined in Sec. II and III. These will

respectively focus on the roles of predictive (a priori) and a posteriori calculations in the

DIII-D program. Section IV discusses one of the two most important of the quasi-MHD

modes in DIII-D, the RWM. The seminal contributions made in both passive stabilization

by plasma rotation and active feedback stabilization of the RWM are summarized.

Several specific contributions are described in detail; the remaining ones are discussed in

the companion article [46] in this issue. Section V reviews the contributions made to the

theory of resistive instabilities. Some of the more recent work on classical and




neoclassical tearing modes (NTMs) and resistive interchanges was specifically incited by

experiments in DIII-D and this is discussed in detail. Section VI contains some summary

comments and the conclusions.




II. PARAMETRIC AND OPTIMIZATION STUDIES

FOR IDEAL MHD MODES

There are two distinct ways of utilizing ideal MHD codes. One is, as an a priori

predictive tool using parameter scans, optimization studies, and existence proofs for

stable, optimized, high " states. Alternatively, they can be used a posteriori as a

verification and analysis tool in which the instability characteristics of reconstructed

discharges are compared to observations in quantitative detail [45]. Theory contributions

to ideal MHD stability research from DIII-D cover the whole range from predictive, to a

posteriori post-analysis stability studies and the whole range of instability types. The

ideal mode types can be categorized as global " and current limit instabilities, localized

(ballooning and Mercier interchange) modes, and internal and infernal modes.

The predictive approach generally involves parametric scans of stability limits, such

as " or current, under a set of constraints. This yields trends and provides proof of the

existence of equilibria with certain desired properties (e.g. high " ), but does not normally

provide absolute limits for experiments since the restrictions imposed are not necessarily

those that constrain experimental discharges. Absolute limits could be obtained in

principle by optimizing over all other parameters. However, most predictive stability

studies in practice are partial optimizations in the sense that the limits are optimized over

one set of parameters leaving the other parameters fixed or constrained in some way. A

classic example is the Troyon " limit [47], which optimized for " over a restricted set of

conveniently chosen profile parameters and fixed sets of cross section parameters. Key

examples from this predictive approach will be reviewed below. Section!III reviews the

contributions from the second, a posteriori, approach of utilizing detailed verification

studies.

A. Early optimization studies

Historically, ideal MHD theory has been utilized predictively, based on the

understanding that ideal limits are fundamental; if the limits are exceeded a large free

energy that is not easily mitigated is available to drive instabilities. Systematic

optimization calculations against local stability were pioneered early by Miller, et al.

[48]. These studies used an automated procedure to vary a number of cross section and

profile parameters and test ballooning stability in order to converge to a final, optimized,




high " equilibrium. The calculations identified highly elongated and indented equilibria

with " values as high as 14% and first identified the possibilities of significantly

increasing " by strong shaping. However, low n kink stability was ignored since the

appropriate computational tools were not then available.

The advent of fully 2D ideal MHD codes ERATO [49], PEST [50], and GATO [4]

around 1980 coincided with the Doublet III experiments and the planning stages for the

DIII-D tokamak. The GATO code, in particular, was written specifically to handle a

diverted plasma boundary and was utilized to map out the stable Doublet III operating

space with respect to the two key features of these codes, namely cross section shape and

internal profiles. Early calculations using GATO for low n and MBC for high n ideal

MHD stability predicted a scaling for the optimum " limit with the inverse edge safety

factor, 1/qs [51].

The parametric scans also predicted improvements with respect to plasma

elongation ! and especially with triangularity ". These and similar optimization studies

for JET-like cross sections [52] focused on optimizing " against the poloidal " , "p , and

invariably found an optimum " at intermediate values of "p set by competing

requirements of high and low n stability. In addition, the studies seriously considered the

advantages of wall stabilization as well as profile and cross section optimization. Similar

trends to the results in Refs [51] and [52] were found from calculations using the PEST

code [53]. Although there was no attempt to optimize " , those studies considered

systematic and independent variations of the geometric and profile parameters. In

particular, low aspect ratio, increased ! coupled with increased ", low qs, and broad

pressure were confirmed as being favorable for improving stability. These early

parameter scans and optimization studies became a basis for the DIII-D design. An

optimized design point for DIII-D was predicted having a " limit against both low n

ideal kink and ballooning modes and no wall stabilization, of 11% [2].

A systematic study of current and " limits, compiled for several different cross

sections, aspect ratios, and current profile parameterizations [54] corroborated many of

the trends found in Refs. [51], [52], and [53] but identified a much more complex picture

of the interaction of profile and shape optimization. The stable space was parameterized

for " = 0 against the q values at the axis q0 and plasma surface qs. Then, the effect of

increasing " on the stable parameter space was considered and the different unstable

modes systematically identified. The detailed effects of cross section and profile are

actually synergistic and cannot be easily uncoupled; a profile optimization in one cross

section can yield a different profile than for another cross section. Nevertheless, some




general qualitative trends can be identified. Figure 1 shows a schematic summary of the

results from this study for toroidal mode number n = 1 stability with respect to variations

in q0 and qs. The low " , n = 1 stability boundary is given by the solid vertical line near

q0!= 1 and the stair-stepped boundary on the right. This is drawn at constant "p . In the

regions shown with oblique (top-left to lower right) shading, the toroidal kink mode is

unstable for q0 < 1 and, on the right, the unstable modes are external kink and so-called

peeling modes. Elongation with no triangularity at low" destabilizes internal resonant

modes [54,55] for q0 just below rational values (reversed oblique shading). Axisymmetric

modes are unstable for more peaked profiles with high qs/q0, shown as the region shaded.

Increasing "p brings the ballooning and kink stability boundaries in; the right hand

pressure-driven kink and both the first and second regime ballooning boundaries move to

the left (lower q0) but the unstable region also broadens, closing off the stable space from

low qs on up, and on each side of the integer q0 bands (vertical shading). Stable regions

are left in the region q0 ! 1 at sufficiently high qs (first stable region) and in similar bands

above other integer q0 and higher qs. The latter can be considered as the global

manifestations of the local second region of ballooning stability.

6.0

5.0

4.0

4.0

3.0

3.0q0

qs

2.0

2.0

1.0

1.0

Currentdrivenkink

Ballooning

First regionstable

Second regionstable

Pressuredriven kink

BallooningBallooning

Currentdrivenkink

Currentdrivenkink

Pressuredriven kinkPressuredriven kink

AxisymmetricAxisymmetric

ToroidalKinkToroidalKink

Fig. 1. Schematic of a typical tokamak stability operational diagram in (q0,qs) parameter space showing the

unstable regions corresponding to ideal MHD modes at a fixed " p . The boundaries are for low n

current-driven kink modes (solid line), pressure-driven kink modes (long dashes), ballooning modes (short

dashes), and axisymmetric modes (dotted line). The unshaded regions are stable.




B. Profile optimization

Profile optimization studies for the Joint European Torus (JET) and the

International Tokamak Reactor (INTOR) [56] design project yielded the so-called Troyon

" Limit [47],

" = "N(I /aB) # " crit

= "N

crit(I /aB)!!!, (1)

which is equivalent to, but significantly simpler than the 1/qs scaling obtained in

Ref.![52]. In Eq. (1), "N

is the normalized " , I is the total toroidal current, a is the

plasma minor radius, and B is the vacuum toroidal magnetic field at a nominal radius Ro

centered in the plasma. The limiting "N

value "N

crit corresponding to the " limit " crit is

the Troyon coefficient. Figure 2(a) shows the early DIII-D data against this scaling. Note

that there are some exceptions where the experiments actually exceeded the predicted

" limits.

More detailed optimizations with respect to profiles, by Howl, et al. [57] and by

Lao, et al. [58] identified an important refinement of this scaling, namely that the

optimum "N

is well described by "N

crit = 4li, where

l i is the plasma internal inductance

[59]. The optimum is obtained for pressure gradient profiles peaked near the edge, and

hence for broad pressure and centrally peaked current density (high l i ). The study in

Ref. [57] also revealed that the strong scaling of "Nwith l i is only obtained for the

optimum, namely broad, pressure; for peaked pressure, the gain from increasing l i is

weak. This is shown in Fig. 3. The DIII-D data is shown against the modified scaling in

Fig. 2(b). Many of the original discrepancies in Fig. 2(a) are now resolved. The

remaining discrepancies have since been attributed to wall stabilization [9,60].

This work led to several lines of proposed high " scenarios. The high l i

discharges formed first in the Tokamak Fusion Test Reactor (TFTR) using negative

current ramps [61], and subsequently reproduced in DIII-D with both negative current

ramps [62] and positive elongation ramps (! ramps) [63], achieved record "N

values

with improved confinement and remain a serious option for AT operation. More recently,

proposals for reaching high l i with q0 << 1 using sawtooth stabilization by energizing a

fast particle population using rf waves are being considered [64,65]. A recent systematic

numerical optimization with respect to bootstrap current alignment and " for H-mode

equilibria identified a whole class of optimized states the optimized l i scenario [66]

that also shows promise as an alternative AT scenario.

The concept of negative central shear (NCS) [67-72] evolved, in part, from analysis

of discharge #69608 that reached a record " = 11.3% in DIII-D [8] and exhibited a




strongly peaked pressure profile from an internal transport barrier (ITB) with a slightly

reversed core q profile [73]. Improved confinement had also been noted previously in

JET pellet enhanced performance (PEP) mode experiments with core-reversed q profiles

[74]. It was also pointed out in several early papers that negative shear or raised central q

could be beneficial for stabilizing some modes, particularly providing easier access to

second stability for ballooning modes [75,76] and stabilizing other internal modes [77].

But NCS was generally thought to be impractical since it was expected to be highly

unstable to double tearing modes or infernal modes. However, in the light of the JET and

DIII-D experiments, it was realized that these modes may not actually be limiting.

0.00

0

2

1 2

i l/ab (MA/m/T)

4 i I/aB

3

4

6

8

10

12

0

2

4

6

8

10

12

0.5 1.0 1.5

I/aB (MA/m/T)

β T (%)

β T (%)

2.0 2.5

9.3%

3.5 I/aB

5.0 4.0

3.0 3.5

(a)

(b)βN > 43.5 < βN < 42.5 < βN < 3.5

Fig. 2. (a) DIII-D data relative to the so-called Troyon scaling limit ! < "Ncrit

(I/aB). Note the exceptions

in the data at intermediate current. [Reprinted courtesy of T.S. Taylor, et!al., Proc. 13th Int. Conf. on

Plasma Phys. and Control. Nucl. Fusion Research, Washington 1990, (International Atomic Energy

Agency, Vienna, 1991) Vol. I, p. 177.] (b) Refined scaling limit ! < 4 l i (I/aB). [Reprinted courtesy of T.S.

Taylor, et al., Proc. 14th Int. Conf. on Plasma Phys. and Control. Nucl. Fusion Research, Wurzburg,

Germany 1992, (International Atomic Energy Agency, Vienna, 1993) Vol. 1, p. 167.]




0.60

2

4

p' = (1 – ψ)

p' = (1 – 5/8 ψ)p' = (ψ1 – ψ3)

βN

6

0.8i

1.0 1.2

Ballooninglimit

60381Troyonlimit

Fig. 3. Marginal stability boundaries in (l i,"N ) space for three model profiles p!(") as indicated. The

calculations assumed a conformal wall at 1.5 times the plasma minor radius. Also shown are the ballooning

limits using the three model profiles as reference equilibria. The reconstructed equilibrium for DIII-D

discharge #60381 at 3 s is also indicated. [Reprinted courtesy of AIP , Phys. Fluids B4, 1724 (1992).]

Calculations demonstrating all the basic features of the concept [67-69] for DIII-D

were shown in workshops and conferences throughout 1992 and 1993. These features

included elevated central q everywhere above 2, a core of reversed shear with a transport

barrier near the minimum q, and high confinement across the entire plasma with high

global confinement. Two important new features were good bootstrap alignment

supplemented by modest auxiliary current drive requirements and stability from a nearby

wall. The concept was subsequently developed by Manickam and Kessel, et al., [78,79],

and Turnbull, et al. [70] in 1994.

In the NCS scenario, the profiles were optimized in an attempt to improve stability

to ideal n = 1, 2, 3, and " modes, and retain the high confinement core while also

obtaining good bootstrap alignment [70]. This was achieved by raising q everywhere

above 2 to minimize low n instability [70,80], by broadening the pressure profile, and by

invoking wall stabilization from a nearby wall. Simulations showed that a stable, high

"N

, steady state, NCS solution with high confinement and high bootstrap fraction exists

[70]. This was the first reported simulation to couple transport using transport

coefficients determined experimentally from analysis of actual discharges, stability

calculations from global stability codes, and current-drive simulations from neutral beam

(NB), fast wave (FW), and electron cyclotron current drive (ECCD).

This operational scenario is, in many ways, the quintessential AT, combining high

" and confinement with steady state through active profile control. In Refs. [67] through

[70], this was called the second stable core VH-mode (SSCVH mode) since it




incorporated a negative shear core in the second stability regime for ballooning modes, as

in discharge #69608, with a high confinement VH-mode edge; in Refs. [78] and [79] it

was called the reversed shear mode (RS mode). The core q profile was everywhere

elevated above 2 to maintain good stability and keep good bootstrap alignment. The

profiles are shown in Fig. 4 for this configuration. Auxiliary current drive from NB, FW,

and ECCD maintains the 30% of the current not provided by the bootstrap current.

Stabilization from a conducting wall placed at a moderate distance from the plasma

(about half the plasma minor radius) was necessary to obtain stability at high " . In

systematic computational studies scanning q0, the minimum q, qmin, and the pressure

peakedness defined as the ratio of the peak to the volume averaged pressure, f p = p0/" p# ,

it was found that the gain in " from optimizing over these parameters was moderate and

the " limits were relatively low with no wall [8082]. With a wall, however, the gain in

the " limit G = " wall #" no#wall can be very large.

1.5

1.0

0.5J (M

A/m

2 )

0.00.0 0.2 0.4 0.6

ρ0.8 1.0

JBS (1.04 MA)

Jtot (1.6 MA)

6.04.02.00.0

q

JNB (0.17 MA)

JECCD (0.32 MA)

Fig. 4. Target current profiles for the self-consistent NCS Advanced Tokamak scenario in DIII-D showing

the contributions from the different current sources. The radial variable ! is the normalized toroidal flux.

[Reprinted courtesy of AIP, Phys. Rev. Lett. 74, 718 (1995).]

C. Cross section shape optimization

Optimization of " crit with respect to cross section shape has been a major feature of

the DIII-D program for two decades. Early shape optimization studies, beginning with the

automated ballooning stability optimizations reported in Ref. [48], yielded configurations

with high elongation " and led to the ellipsoidal shell concept [83]. Subsequent studies

revealed the importance of triangularity # [51-55] and later of squareness $ [84] in

optimizing stability at higher elongations. The crucial importance of shape on optimum




performance was exploited in the earlier " limit records set in DIII-D [6-10]. The

optimization process used in designing the record " !=!11.3% discharge is detailed in

Ref.![8]; strong cross section shape is the key to resolving the conflicting requirements of

n = 0 axisymmetric stability, n = 1 modes, and ballooning stability. In addition, numerical

stability studies [15,85] clearly detailed the benefits to stability of low aspect ratio hinted

at in the earlier current and " limit studies [54]. Low aspect ratio tends to provide the

same stability benefits as high ! and strong " due to its similar effects on extending field

line lengths in the good curvature region and shortening connection lengths in the region

of bad curvature.

In conventional scenarios with monotonic q profiles, cross section and profile

optimization are moderately synergistic [54]. For the AT scenarios, however, the pressure

profile optimization and cross section are highly synergistic [66,81,82,86]. This is

especially true for bootstrap aligned scenarios [66,86] and the synergism is enhanced

even further by wall stabilization [82,86]. For many AT optimizations, the required target

features of " and noninductive bootstrap fraction are not attainable with insufficient

shaping, especially with low " [66,86]. Figure 5 shows the results from a parametric scan

of "N

crit for broad and peaked pressure profile equilibria against cross section shape

parameterized by the shape factor introduced in Ref. [8] as:

S = q95 I/(aB)!!!. (2)

q95 is the safety factor value at the 95% poloidal flux surface and is used to represent an

edge value in diverted equilibria equivalent to the edge q value in limited equilibria. S is

effectively the ratio of field line q to the cylindrical q, and can be thought of as

representing the effect of plasma shape and aspect ratio on the q-value. S can also be

considered as an effective aspect ratio since for a large aspect ratio circular cross section,

S # R/a. The synergistic effect is clear from Fig. 5. Peaked pressure yields little gain in

"N

crit from increased S (though there is a gain in " crit from increased I/aB). Broad

pressure, however, results in a large advantage to stronger shape. On the other hand, in a

circular cross section, there is little gain from optimizing the pressure profile alone.

The studies in Refs [80-82], [86], and [87] all point to the fact that high

performance in NCS cannot easily be achieved with either weak shaping or peaked

pressure and not at all with both. The key aspects of this conclusion have also been

qualitatively confirmed by a number of others using different assumptions and constraints

[77,88-90], suggesting that the result is quite general. The results of a parametric scan in

"N

crit with respect to pressure peaking factor fp, based on model equilibria with profiles

and cross section closely matching those in DIII-D NCS discharges is shown in Fig. 6




[87]. At high peaking factors, "N

is limited by a global n = 1 pressure driven ideal kink

mode. For broad pressure, however, the n = 1 "N

limit is much higher and "N

is limited

by a more strongly edge-localized ideal intermediate n instability with features similar to

the first large ELM, or X event that typically terminates VH-mode confinement

[40,71]. The discharges with an L-mode edge exhibited strong pressure peaking and all

disrupted at relatively low "N ~ 2. Those undergoing an H-mode transition, however,

reached much higher"N

, above "N

= 4 . Note that, at fixed qmin, neither the NCS

experiments nor the calculations find a significant dependence of the stability limits on q0

or !q = q0 - qmin.

In experiments in 1995, DIII-D achieved a record in fusion performance in an

NCS discharge by simultaneously optimizing the current profile through current ramps

and timing of neutral beams to slow the current diffusion rate, the pressure profile

through timing of the H-mode transition, and the cross section shape to maximize " crit

[10]. The numerical calculations in Refs [81], [82], and [86], showed that increases in

both " crit and the critical root-mean-square limiting " , [53]

"* = Pp" = (# p2$ /# p$ 2)" !!!, (3)

can be achieved by broadening the pressure with strong shaping. The point is that " crit

increases faster than " p2# /" p# 2decreases. This result is also summarized here in Figs 5

and 6. "* is often considered a more appropriate proxy for fusion performance since it

weights the pressure more strongly in the high temperature core.

6

6 8 10

p0/⟨p⟩ = 2.4

p0/⟨p⟩ = 4.8

S

βN

5

4

4

3

2

2

1

0

Fig. 5. Ideal n = 1 "N limit for equilibria with NCS as a function of shape factor S for representative broad

(solid curve) and peaked (dashed curve) pressure profiles. The data points at the lowest S are for circular

cross section equilibria. The increase in S to the second point is largely owing to an increase in " to 1.8,

along with a small increase in # to 0.1. The remaining increases in S were obtained by increasing

#. [Reprinted courtesy of IOP, Nucl. Fusion 38, 1467 (1998).]




6H–modeP(105 Pa)

84736

2

1

00.0 0.4 0.8

87009

H Transition

Ideal

Unstable

4

2

00 2 4 6

Po/⟨P⟩8 10

β N (%–m

–T/M

A)Resistive

L–mode

Fig. 6. Critical stable "N against the n = 1 ideal kink mode limit as a function of pressure profile

peakedness. Also shown are trajectories of the L-mode edge discharge #87009, which disrupted at low " ,

and an H-mode edge discharge #84736, which reached high " without disrupting. The resistive

interchange stability boundary is also shown. [Reprinted courtesy of AIP, Phys. Plasmas 3, 1951 (1996).]

These results provided strong motivation and guidance to the experiments [10].

Discharge #87977, with "N

~ 4 and moderate l i achieved a fusion gain in deuterium of

QDD = 0.0015. The equivalent in deuterium-tritium QDT ~ 0.32 is comparable to that

achieved in larger tokamaks with much higher magnetic fields, thus demonstrating that a

smaller, more compact Advanced Tokamak could match the performance of larger, high

field machines since Q = Pfus/PNB ~ ! * "E B2. The increase in neutron rate from the

previous record in a VH-mode discharge, #78136, was a factor 3. From a 25% increase in

input power, this produced more than double the fusion yield QDD. Figure 7 shows this

achievement in the context of the optimization of this discharge in profile and cross

section. In accordance with the calculations in Ref. [87], the high QDD H-mode NCS

discharges were generally terminated by an ELM-like X event.

D. Implications for a burning plasma experiment

The key theoretical contributions from two-decade-long optimization studies of

ideal stability in DIII-D can be summarized as follows: The single most important finding

is the result that both " and "N

depend in a strongly synergistic manner on both pressure

and current profile and on cross section shape. Figure 3 illustrates the synergistic effect

with respect to current and pressure profiles for the case with a conventional current

profile and Fig. 5 shows the synergism with respect to pressure profile and cross section

shape for the AT profile case. These results are semi-quantified in the following sense.




For the optimum pressure profile, "N

crit is 4l i

. The optimum pressure in all cases is the

broadest pressure considered and the optimum cross section is that with the highest shape

factor S [Eq. (2)] considered. Despite the weighting of an additional pressure peaking

factor in Eq. (3), even "* is optimized for broader pressure in both conventional

[53,57,91] and AT [86,89,90] scenarios with sufficiently strong shaping.

QDT = 0.32EQ

0

5

10

15

20

25

0 5 10 15 20 25Neutral Beam Power (MW)

Neut

ron

Rate

(×10

15/s

)

NCS L-modeNCS H-mode

Q dd = 0.0015 profile and

shapeoptimization

Profile andshapeoptimization

1996 NCS H–Mode1996 NCS H-mode

1994 VH–Mode1994 VH-mode

Fig. 7. Fusion performance versus neutral beam power for high performance DIII-D discharges showing

the improvement obtained in the high QDD discharges over the previous VH-mode result due to optimizing

the plasma profiles and cross-section. The discharge with highest neutron rate and the discharge with the

record QDD are identified. The latter reached "N = 4 , " = 6.7% , confinement factor H ! 4, total stored

energy W = 4.2 MJ, and energy confinement time of !E = 0.4 s.

These results have many clear implications of importance to the design of any

next step fusion Burning Plasma Experiment (BPE). For example, shape and profile

optimization are key to any BPE design process. Triangularity is an especially strong

driver for improved performance, but higher order shaping specifically the squareness,

" can yield considerable stability gains. It is important to take account of the synergistic

effects of profile and shape optimization. Generally, much larger gains from profile

optimization are possible for equilibria with stronger shaping.

For a conventional tokamak, Fig. 1 provides a useful view of the effect of changing

various parameters. As the wall is moved inward, the axisymmetric stability boundary

moves to lower q0, and both the current-driven and pressure-driven kink stability

boundaries open away from each other to increase the stable region. Shaping has a

complicated effect on the kink boundaries but, in general, # is destabilizing for all kink

modes, whereas increasing $ can destabilize current-driven kink modes but is stabilizing

for pressure-driven modes. Lower aspect ratio has the effect of moving each of the

pressure-driven boundaries to higher q0, generally widening the stable region.




"N

is the key performance parameter for an AT. To see this, ", " p, and "N are

related for a tokamak plasma by a general equilibrium relation [84,85]:

""p = C0#"N2 1+ $2h(%)[ ] !!!, (4)

where ! is the inverse aspect ratio, h is a weak function of " and C0

=12.5 if " , "p , and

"N

are in percent. This global relation can be derived under mild assumptions regarding

the profiles and holds to remarkably high accuracy. A high fusion gain AT, operating in

steady state with a large bootstrap fraction, requires both large " and large "p .

Therefore, there is a necessity to, and a very large leverage from, increasing "N

and #.

While # is limited by axisymmetric stability, "N

is limited by ideal MHD

non-axisymmetric stability. This is the essence of the stability problem in the AT

program. "N

can be maximized through cross section (including both # and ") and

profile optimization, and increased through wall stabilization.

The DIII-D program, coupled closely with the theory and computational effort, has

identified a range of AT options in addition to the conventional high performance, high

" , low q95 scenario. These include NCS [and variants such as weak negative shear

(WNS)], high "p , high l i , and optimized

l i. The high

l i [57,58,6265] and optimized

l i [66] scenarios are natural options for any high performance BPE. Again, however,

there is a large potential gain in possibilities from increased shaping, especially " and

aspect ratio, which increase the field line lengths in the locally stable curvature regions.

High l iwith q0 <<1 is an especially attractive scenario worth exploring further.

The NCS and hybrid scenarios are also good options for a BPE since they can

naturally lead to steady state, in contrast to the high l i and optimized

l i scenarios that

represent compromise solutions. The studies tend to indicate that there is little benefit in

high q0 for stability, though there may still be some benefits in confinement. The key

safety factor profile parameter for stability is qmin. However, the synergistic dependence

of " crit on shape, profile, and wall proximity should be accounted for. The gain is not just

through a linear additive dependence, but is even stronger than multiplicative.

Symbolically, " crit # S $ P $W , where S , P , and W represent optimization factors from

shaping, profiles, and wall stabilization respectively. All three are needed to obtain the

full benefit of this scenario. In particular, the broad pressure associated with an H-mode

edge seems to be essential to finding a solution with both high confinement and high " .




III. VERIFICATION AND ANALYSIS STUDIES FOR IDEAL MODES

The post-analysis (a posteriori) approach also has two distinct variations and both

have been important in DIII-D. In the first type, global data from selected sets of

discharges are overlaid on computed stability limits generated from a parameter scan.

The underlying computed equilibria are taken as representative of the discharge set but

are not necessarily reconstructed from any actual discharge data. The key distinction

from the predictive approach is the attempt to verify stability limits for a particular set of

actual discharges. Such studies are used to demonstrate the validity of predicted trends

with respect to key parameters. An example of this is shown in Fig. 6 considered in the

previous section; here, the stability limits were calculated in response to the MHD

behavior of the NCS discharges. While the equilibria used in this scan are modeled after a

set of NCS equilibria, they do not correspond to any particular discharge.

Recently, the truly post-analysis approach has become more important in the DIII-D

program. In this approach, the ideal stability is verified in detail for a particular discharge

against the MHD behavior of the actual discharge [45]. That is, in contrast to the first

post-analysis approach described above, the stability calculations are performed using the

reconstructed equilibrium for a single, given discharge, and compared directly with the

actual MHD behavior of that given discharge.

This second type of post-analysis can yield unambiguous stability predictions and

reveal important new physics not available from the other approaches; the two predictive

approaches and the post-analysis of the first type cannot provide unambiguous

descriptions of a given discharge. While the quality and believability of this analysis is

variable with respect to instability mode type, this relatively new approach has yielded

several examples of unprecedented quantitative agreement [45]. However, the analysis

can only be done for a selected few cases with a complete set of diagnostic (both

equilibrium and MHD fluctuation) data available. Hence, it is important to focus analysis

on these few representative cases. It is, nevertheless, important to keep in mind that there

are generally multiple discharges with the features displayed in each example.

This section describes some of the more instructive examples of post-analysis that

have provided insight into MHD phenomena in DIII-D. The earliest post-analysis cases

tend to be of the first kind. The second type, based on describing particular discharges, is

more common in later analyses though some significant successful attempts were made

over a decade ago. The most recent and most successful cases are discussed at length in




Ref. [45] and will be passed over here cursorily in favor of a survey of the progress made

at DIII-D over the last two decades.

A. Low b stability

The early Doublet III studies [51] pioneered the idea of comparing actual

discharges against stability limits predicted from optimization studies, although without

actual equilibrium reconstructions. These represent a classic example of the first type of

post-analysis. Significant disagreements between the optimized limits and the achieved

discharge " values were found in some cases. This was attributed at the time to wall

stabilization of those discharges. However, this was subsequently found to be

unnecessary in many such cases since further profile optimization could account for a

large variation in the predicted limits [57]; to invoke wall stabilization as an

unambiguous cause requires that profile variations be ruled out by reconstructing the

actual discharge equilibria.

Figure 8 shows early low " , DIII-D discharge data superimposed on the calculated

low " stability operational space [54]. The discharges were selected from early Ohmic or

low power, auxiliary heated, low confinement (L-mode) cases. The q profile was inferred

from the best fit to the magnetics and global data only. These were low " discharges so

there was little ambiguity from the tradeoff between the two independent profiles p´(!)

and ff´(!) that are required as sources in the Grad-Shafranov equilibrium equation

[f(!)!= rB" and the prime means the derivative with respect to the normalized poloidal

flux !]. With low auxiliary heating, the standard Ohmic current profile is also expected

to be a good approximation in most cases. Nevertheless, the agreement shown here needs

to be taken with this reservation in mind.

Almost all discharges lie in the computed stable region and most are stable

operationally. The major exception is the single discharge to the right of the unstable

boundary; this discharge was, in fact, transient, and disrupted immediately. A time

history is also shown in Fig. 8 for another discharge that disrupted, #54813. This

discharge evolved with qs decreasing during the initial current ramp to below 3 and

remained in the stable region during flattop. This was followed by a programmed current

ramp-down at 3000 ms, and the discharge subsequently suffered a collapse in l i at

3440!ms as the current profile broadened significantly and q0 increased after initiation of

a locked mode. The discharge then suffered a minor disruption on reaching the calculated

external kink limit at 3460 ms with a loss of internal stored energy and a large influx of

impurities. It recovered partially at 3495 ms but finally disrupted due to a tearing mode

50 ms later.




4.0

STABLE

8.0

7.0

6.0

5.0

4.0q0

qs

3.0

2.0

2.0

1.0

1.0

UNSTABLE

Extrapolatedlimit

Compuredlimit

Minordisruption

Partialrecovery

Completedisruption

Transient dischargedisrupted immediatelyFlat top

2120–3000 ms

#54813

Ramp down3000–3440 ms

i collapse

Fig. 8. Operational stability diagram in q0 and qs for current-driven modes in DIII-D showing discharge

parameters of DIII-D low " L-mode discharges at various times during the current flattop. Also shown is

the full trajectory of discharge #54813. The single point located well in the unstable region was a transient

discharge that disrupted immediately.

B. High b stability

Detailed verification studies were performed for each of the noteworthy high "

DIII-D discharges. In contrast to some of the earlier Doublet III studies, equilibrium

reconstructions were available and these studies are invariably consistent with

observations; the post-analyses yield either stability or marginal stability. The later

studies are largely examples of the second post analysis approach; the earlier ones tend to

be closer to the first type. The results are described in detail in Refs [6] and [92] for the "

= 6.2%, discharge, in Ref. [7] for the " = 9.3% discharge and the high "N

discharges

with "N

= 5, and in Refs [8] and [73] for the " = 11.3% case.

Discharge #55390 was a diverted H-mode discharge reaching " = 6.2%. This was a

milestone in the fusion program since it is widely believed that a fusion reactor can be

viable for " values of 5% or more, and the " value exceeded previous values in high

confinement mode (H-mode) diverted discharges by a factor 2. Several similar discharges




were obtained with " in excess of 5% and stable to disruptive MHD. However, discharge

#55390 was analyzed extensively and taken as a representative example. Ballooning

stability analyses were performed using the equilibria reconstructed from measured

pressure profiles and external magnetic measurements. The results are summarized in

Fig. 11 of Ref. [92]. They indicate clearly that these discharges tend to be ballooning

limited near the H-mode edge and in the core but significantly below the ballooning limit

elsewhere.

The n = 1 kink stability analysis was performed, as in the earlier Doublet III studies

[51], by comparing the time histories of the discharges against parameter scans of the

stability boundaries in l i and "

N computed using a set of model profiles similar to those

obtained from the reconstructions. The time histories were consistent with the

parameterized stability boundaries, however, and there is no need to invoke wall

stabilization in this case in contrast to the earlier Doublet III studies. Stability against

n = 2 and n = 3 kink modes was also checked using reconstructed equilibria generated

using additional constraints on the profiles to reduce the variability of the results. The

discharge " values were generally limited by confinement saturation or " collapse.

A detailed analysis of the " = 9.3% discharge, #66493 [7] found that the measured

pressure profiles for this and similar high " discharges were clearly ballooning limited in

both the core and the edge (! > 0.87). This was considered consistent with the ELM that

was observed soon after the time of the equilibrium reconstruction. In addition to

increased NB power, the high " was largely achieved through cross section

shaping specifically using a double-null (DN) divertor at high " and #, yielding higher

I/aB. Operationally, " was limited in subsequent attempts at further increases by the

onset, after either the first ELM or sawtooth crash or both, of an m/n = 3/2 mode and

subsequently a 2/1 mode. The latter resulted in saturation or collapse of the high " phase.

No operational evidence of the proximity of an ideal, low n, " limit was found in these

discharges.

Several record "N

discharges, up to "N

= 5, at higher q were also analyzed [7].

These were also ballooning limited with the profiles at the ballooning limit over a large

fraction of the cross section. Ideal kink stability was analyzed using GATO, taking a

conformal conducting wall at 1.5 times the minor plasma radius as an approximation for

the DIII-D vacuum vessel. Again, the kink limit is generally higher than the ballooning

limit. Sensitivity analysis [7,57], however, showed that the ideal limit depends strongly

on the profiles. The limits are consistent with experimental data, and all the high "N

discharges analyzed were near these limits. Some wall stabilization may contribute since




the calculations used a conformal wall at 1.5 times the minor radius and some discharges

were found slightly unstable with no wall. But the demonstrated sensitivity of the

stability limits to the profiles did not permit a conclusion on wall stabilization.

This study made clear the importance of accurate measurements of the profile

details in obtaining reasonable predictions. The parametric scan described in Refs [7] and

[57], and summarized in Fig. 3 was, in fact, based on a reconstruction of DIII-D

discharge #!60381. This discharge reached "N

= 3.5 but two identically programmed

discharges disrupted and suffered a " collapse respectively. The parametric scan used

model pressure gradient profiles, but the results were consistent with the observed

discharge behavior in the sense that the linear model profile with the finite edge gradient

(broken curve in Fig. 3) is close to the measured profile for discharge #60381. The

sensitivity of the " limit to the pressure profile found here is also consistent with the

variation in the observed behavior for this and the similar discharges.

The record " = 11.3% discharge, #69608, was achieved through careful

optimization of the cross section shape. This was expressed in terms of maximizing the

shape factor S defined in Eq. (2). To first order, S is purely dependent on ! and " and

aspect ratio R/a, but it also takes into account the toroidal field weighting of the plasma

surface. The discharge equilibrium was developed, based on extensive stability

calculations [8], to simultaneously achieve marginal stability to ballooning, n = 1, and

n = 0 modes. This was planned to be achieved by increasing S from S = 7.25 for the

previous record " = 9.3% discharge, to the maximum predicted stable value of S = 8.25

by maximizing both ! and " using the DIII-D control system. In the actual experiment,

the shape was also modified during the high # phase to further broaden the profile. The

experiment achieved the target " = 11% as planned, but the shape factor of S = 7.8 was

slightly lower than the stability-optimized target. The discharge parameters were highly

optimized in l i ; calculations showed that slightly lower

l iwould destabilize the n = 1

mode and slightly higher l i would destabilize the n = 0 mode.

A detailed equilibrium and stability verification analysis showed, however, that the

actual equilibrium profiles were significantly different from the target and this led to a

wealth of new physics [73]. The key unexpected features were a nonmonotonic q profile

and a highly peaked core pressure bounded by flat spots at an intermediate radius. This

latter feature is now called an ITB. The q profile exhibited two q = 1 surfaces, with a

minimum slightly below 1 in between, and an on-axis value above 1. A ballooning

stability analysis using the CAMINO code showed that the strongly peaked core pressure

was well within the transition regime for second stability. This is summarized in Fig. 9

for four surfaces representative of the (s,$) diagram topology. The insets for each figure




indicate the position on the pressure profile to which that diagram refers. The profile

shows the strong ITB in the core.

5

7α

11 15

(a) (c)

(d)(b)

3

3

1

S

–1–1

5

7α

11 15

3

3

1

S

–1–1

5

7α

11 15

3

3

1

–1–1

5

7α

11 15

3

3

1

S

–1–1

0.50

a0.25

0.001.1 1.8

Radius (m)2.1

P/(B

T2 /2µ 0)

0.50

c0.25

0.001.1 1.8

Radius (m)2.1

P/(B

T2 /2µ 0)

0.50 d

0.25

0.001.1 1.8

Radius (m)2.1

P/(B

T2 /2µ 0)

0.50

b0.25

0.001.1 1.8

Radius (m)2.1

P/(B

T2 /2µ 0)

Fig. 9. Regions of ballooning and Mercier instability in (s,!) space for discharge #69608 for four points on

the profile corresponding to four separate zones (a-d) The profiles are shown in the insets and the

corresponding points of the profile with representative normalized poloidal flux values of "!= 0.71, 0.41,

0.21, and 0.07, respectively, are indicated. s and ! are defined as: s = 2 " lnq( ) " lnV( ) , and

" = µ0# $ p %( ) 2& 2( ) , with " = V #( ) 2$ 2Rm( ) . The experimental (s,!) data point for the profile value is given

by the solid circle in each case. The Mercier unstable zone is darker grey and the ballooning unstable zone

is shaded lighter grey. [Reprinted courtesy of AIP, Phys. Fluids B4, 3644 (1992).]

The intriguing feature is that each point on the profile is always near the stability

boundary. The detailed structure in the safety factor and pressure profiles conspires to

keep the profile hugging the stability boundary through the first and second stability

transition regions. Sensitivity analysis of the equilibria indicates that these effects are

real. This appears to be the first case with documented measured profiles that show a

profile genuinely reaching the transition regime. The kink stability analysis shows an




unstable n = 1 mode internal to the second q = 1 surface. This is interpreted as a toroidal

kink mode [54] since there is little displacement outside q = 1, but the growth rate

depends on the wall position. This is shown in considerable detail in Figs. 19, 20, and 21

of Ref. [73]. The mode structure is consistent with soft x-ray (SXR) data showing a

saturated structure inside the outer q = 1 surface and little outside.

The most recent record " DIII-D discharge #80108 reached " = 12.6%. This

discharge was designed to take advantage of wall stabilization by operating at low l i and

maximizing the plasma volume within the vacuum vessel [9]. This was a sawtoothing

discharge with q0 < 1 and low q95 ~ 2.5. Equilibrium reconstructions utilized 8 channels

of motional Stark effect (MSE) in addition to the Thomson pressure profile data. The

n = 1 stability calculations using GATO found an internal or toroidal kink instability for

any wall position, consistent with the observed sawteeth. Hence, an unambiguous

conclusion on whether this discharge was wall stabilized was not possible. Nevertheless,

using equilibrium reconstructions with q0 forced just above one, for several times during

the discharge evolution, it was estimated that " for this discharge was more than 30%

above the no-wall " limit at the time of the peak " . This no-wall limit was also

consistent with the estimate from the 4l i

scaling. The relevance of the procedure of

forcing q0 above 1 is discussed later in Sec. III.F.

C. Axisymmetric stability limits

In general [59], and in the " = 11.3% experiment in particular, control of the n = 0

axisymmetric stability was crucial since this imposed the limit to increasing S; this was

also true to a lesser extent for the " = 12.6% discharge #80108. There is a large gain if

one can operate very close to the ideal axisymmetric stability limit. Such operation was

made possible by a detailed analysis of the n = 0 mode in preliminary experiments

designed to test the control system limits [8]. An ideal axisymmetric stability analysis of

a strongly shaped DIII-D discharge #63422 in the DIII-D vacuum vessel was performed

using the GATO code and compared to the more usual simplified rigid body models used

in control system analysis. The latter indicates elongations up to ! = 3 would be stable

with a decay index n/ncrit > -1.

However, the GATO calculations find a much more pessimistic ideal limit at

! = 2.41 and " = 0.85. These calculations used a close numerical approximation to the

DIII-D vacuum vessel for the wall; the wall was not a conformal model as was the case in

most previous comparisons. The GATO calculations find two important non-rigid

contributions to the plasma motion: a significant m = 3 component (about 33% of the




total displacement) plus small other harmonics, and a significant non-rigid distortion of

the basic m = 1 shift in which the edge moves about 20% more than the core.

Discharge #63422 underwent an axisymmetric disruption and a reconstruction of

the displaced outer boundary from magnetic probe measurements was possible during the

early phase of the vertical plasma movement. Figure 10 shows the comparison of this

with the GATO prediction of the boundary displacement [8]. The non-rigid character is

fairly clear. This result has two important implications. The experiment and analysis

together confirmed that a well-designed control system can permit operation right up to

the ideal axisymmetric limit.

Calculatedperturbation

Conductingwall (× 1.02)

Equilibriumsurfaces

Equilibriumboundary

Measuredperturbationboundary(after 2 ms)

Fig. 10. Unstable boundary displacement of discharge #63422 predicted from GATO using a conducting

wall displaced from the real vacuum vessel by a factor 1.02 and the measured boundary 2 ms later. The

unperturbed equilibrium contours are shown as solid curves and the predicted perturbed flux contours are

shown dashed. The perturbed plasma boundary reconstructed from magnetic probe measurements is given

by the thick solid data points. [Reprinted courtesy of AIP , Phys. Fluids B3, 2220 (1991).]

Also, this example of a post-analysis study of the second kind appears to be the

first successful comparison of an ideal mode structure with experimental data and, as

such, provides an important credibility check of the ideal MHD description. While the

n/ncrit criterion does work reasonably well in discharges with lower !, whereas discharge

#63422 had extreme shaping, it is nevertheless important in practice to take the

non-rigidity into account when operating close to the ideal limit. GATO calculations for

the " != 11.3% discharge showed the m=3 component of the displacement to be about




20% of the total. From experience, the ideal limit is expected at about n/ncrit ~ -0.9 and

discharge #63422 was designed to operate at this point.

D. High bp stability

Several other early verification studies of the second kind considered the interesting

MHD activity observed in DIII-D at high"p . Discharge #67700 reached "#p ~ 2, which

is the highest "#p reported in any tokamak [93] and is close to the expected equilibrium

limit [94]. The discharge suffered a minor collapse at the peak "p and the equilibrium

and stability at this time were studied extensively. This discharge had limited current

profile data available but pressure data from Thomson scattering were obtained. A

number of alternative fits to the data were constructed with varying constraints all

were reasonable and acceptable fits to the equilibrium data. The largest uncertainty was

in q0. The stability results are summarized in Fig. 11. This figure shows the growth rate

computed as a function of the wall position for each of the acceptable reconstructed

equilibria; Rwall = 1 here refers to the actual DIII-D vacuum vessel. The inset shows the

discharge "p evolution, with the "p collapse at 1750 ms.

0 1 2 3 4 5

Rwall/RDIII-D

0.000

0.002

0.004

0.006

0.008

0.010

0.012

γ 2

γ 2A

DIII-D wall

1.50

1.501.501.50

2.05

2.05

1.10

1.10

5

0

βp

1.0 1.5Time (s)

Dα (au)

2.0 2.5

Fig. 11. Predicted n = 1 stability for several equilibrium reconstructions of high "# p discharge #67700 as

a function of wall position relative to the real DIII-D vacuum vessel. The curves correspond to different

reconstructed equilibria with q0 fixed at the values indicated. The growth rates are normalized to a poloidal

Alfvén time " p#1

, which is related to the toroidal Alfvén time "A#1

by a factor q0"1

, which varies with each

curve. The squared normalized growth rates " 2 /" p2

are therefore scaled with q02. The inset shows the

discharge time trace with a " p collapse due to an n = 1 ideal kink instability just after the time of the

equilibrium reconstruction at 1750 ms.




For all the reconstructions, the plasma is predicted to be unstable with no wall at the

time of the "p collapse but stable or marginal with wall stabilization assumed from the

DIII-D vacuum vessel wall. This was the first relatively definite, positive test of wall

stabilization in a specific discharge. The predicted unstable n = 1 mode is a global

pressure driven mode and its structure is fairly insensitive to the reconstruction details,

especially q0. This is an exceptional case since it allows a definite conclusion to be

drawn; typically the stability depends more sensitively on q0.

Discharge #77676, also a high "p discharge, exhibited a new, previously

unobserved, quiescent phase with q0 > 2 in which all MHD activity vanished completely,

the density rapidly peaked inside ! ~ 0.4, and the confinement improved markedly [95].

In the quiescent phase H ~ 2 and "N

reached "N~ 2 .

Also, the axis density n0 doubled and the bootstrap current fraction Ibs increased to

about 80%; as in the high " discharge #69608, these features are now understood as an

ITB. Equilibria were reconstructed at nine separate time slices, which included 8

channels of MSE measurements of the field line pitch for the first time in this discharge

type. The high "p resulted in a large Shafranov shift so the Thomson scattering data for

electron temperature and density, taken at a fixed major radius, reached close to the core

and the MSE straddled the magnetic axis. During the discharge evolution, q0 oscillated

around 2 as the bootstrap current evolved with the pressure peaking and the Ohmic

current evolved in response.

The observed MHD activity shows a near perfect correlation with the computed

stability. Stability analyses were performed for a sequence of the reconstructed equilibria

and the results are summarized in Fig. 12(a). Here, the time evolution of the magnetic

fluctuation spectrum is shown in comparison to the n = 1 ideal kink results at each of the

analysis times. These magnetic signals also have a high coherence with SXR signals. The

fluctuations appear, saturate, and disappear as q evolves. The predicted modes have the

characteristics of infernal modes expected in situations such as this with elevated and flat

core q profiles near rational values [28,29]. In this discharge, q0 remains near 2 and the

estimated error in q0 is ~ ±0.2 . The localized structure is also consistent with the

saturation and non-disruptive character of the observed modes. Presumably, since the

computed infernal modes are radially localized and the growth rates are weak, non-ideal

effects are important and are sufficient to cause the mode saturation [29].

This was the first reported unambiguous experimental identification of infernal

modes. The computed instabilities were inversely correlated with periods during which

the discharge exhibited improved confinement. While a full sensitivity analysis with




respect to the stability was not performed the analysis simply used the best fit in each

case the results show a definite correlation between the absence of MHD activity,

ideal kink infernal mode stability, and improved confinement, suggesting a likely causal

relation. These features also correlate with ballooning second stability access in the core,

evaluated using the CAMINO code. At the peak performance time, when the ITB formed,

MHD activity disappeared, and q0 > 2, the core of the discharge is predicted to be in the

second stability transition regime. This is shown in Fig. 12(b).

0

0 1

2.95 sm = 3

0

0 1

2.15 sm = 2

0

0 1

4.01 sUnstable Infernal Mode

Unstable Infernal ModeUnstable Infernal Mode

m = 3

0 5

1.4(b)

(a)

1.2

1.0

0.8

0.6

0.4

0.2

0.00.0 0.5 1.0 1.5

α/αnose

s/s n

ose

10 15 20 25Frequency (kHz)

Time → Stab

leStab

le

(axis)0.10

0.750.65

0.560.46

0.33

0.25

0.99 = sprt(ψn)(edge)

0.820.88

nose

77676.3360

0.93

Fig. 12. (a) Magnetic fluctuation spectrum as a function of time for high "# p discharge #77676 correlated

with stability predictions from GATO at 5 separate discharge times. (b) Normalized (s,!) ballooning

stability plot for discharge #77676 at 3.36 s during the quiescent period. Each point is normalized to the

values of (s,! ) at the nose of the instability boundary for that flux surface and is labeled by the

corresponding square root of the normalized poloidal flux value. [Reprinted courtesy of AIP, Phys. Plasmas

1, 1545 (1994).]




Here, the local (s,!) stability diagrams for each surface are renormalized to the nose

of the stability boundary and superimposed to show the trajectory of the profile relative to

the fixed, normalized stability boundary; the diagram shows relative positions but

absolute scales in this figure are meaningless. The region of improved confinement inside

" ~ 0.3 is well in the transition regime to second stability and extends beyond the nose

of the instability boundary.

E. NCS discharges

NCS discharges exhibit a wide variety of MHD phenomena. These phenomena are

generally well described by MHD calculations. The " values reached in broad pressure,

NCS discharges with H-mode edge pedestals are limited by stability, not confinement;

power balance calculations, with 20 MW input and typical high confinement levels in

these discharges, predict " values well in excess of those achieved. NCS H-mode edge

discharges are predicted to be simultaneously near several stability limits and exhibit a

variety of limiting instabilities, as might be expected from highly optimized plasmas [96].

Infernal modes like those in the high "p discharges can limit performance directly in

some cases [96]. DIII-D NCS discharge #87937, for example, exhibited a centrally

located, predominantly m/n = 4/3 mode on the SXR and magnetic diagnostics. An ideal

infernal n = 3 mode, dominated by the m = 4 poloidal harmonic, is found to be unstable

from stability calculations using GATO for the equilibrium reconstruction in this case

[96]. The n!=!1 and 2 modes are stable; the n = 3 mode is shown in Fig. 5 of Ref. [96].

Other NCS H-mode discharges are often limited by an edge ELM (X event) or a resistive

n!=!1 mode. The former is discussed in considerable detail in Ref. [44] and also in

Ref.![45]. The resistive n = 1 mode is discussed briefly in Sec. V.C.

The closely related AT option, the WNS discharges [87], utilized NB timing to

optimize the shear profile and reduce edge current buildup. The WNS discharges have

strong positive shear near the axis and weak shear at midradius with a slight dip around

"!= 0.6. The edge is essentially a normal NCS H-mode. Stability analysis for WNS

discharge #84713 [87] shows it to have a region of second stability access at the edge

("!> 0.9), similar to VH-mode (discharge #75121) and NCS H-mode discharges.

However, the WNS discharge also has a broad intermediate range 0.2 < " < 0.5 with

second stability access along with a first-regime limited core. In contrast, the standard

NCS H-mode discharges typically have access in the core but not at intermediate radii.

Ideal n = 1 calculations for this WNS discharge [87] find a strongly unstable global kink

mode with no wall but stability with a wall, much like the more conventional NCS

H-mode discharges.




In DIII-D, NCS discharges with an L-mode edge invariably develop an ITB with

runaway pressure peaking and ultimately disrupt at low "N

. This is highly reproducible

and is also seen routinely in TFTR [97], as well as in other L-mode discharge

experiments with strong ITBs [98101]. Analysis of one of the DIII-D discharges,

#87009, has provided probably the most successful verification study for an MHD

instability in an actual discharge. Linear stability calculations coupled with a model for

an equilibrium being driven through an ideal instability boundary [102] have matched the

disruption limit, the non-exponential growth "2, and the detailed poloidal mode structure

against experimental data with unprecedented agreement [45,102].

The predicted Mirnov signal for the disruption precursor also exhibited the apparent

phase reversal on the inboard side that was observed in this discharge and in a

considerable number of other DIII-D discharges. This had constituted a longstanding

puzzle. The phase reversal is simply due to the geometry of the wall, plasma, and mode,

none of which have a simple poloidal structure [45]. The prior MHD bursts that are often

seen as well are documented in the literature [82,87]. These have been identified with

resistive interchange modes and will be discussed in Sec. V.D.

F. Sawteeth

One issue that has not been successfully resolved to the same level, however, is the

relationship between the ideal internal kink and the sawtooth mode. The parameter

surveys in Ref. [54] delimited the basic internal kink, toroidal kink [54,103,104], and

quasi-interchange (QI) [105] stability criteria. However, the inter-relationships and

dependences on the q profile are subtle and the relative roles of these modes in the

observed sawtooth have been, and remain, problematic. There is also a wide variation in

behavior among different machines. TEXTOR, [106] TFTR, [107] and JET [108], in

particular, have reported measurements of q0 considerably below unity even during an

entire sawtooth cycle [107,108] but in DIII-D, measurements yield values of q0

remaining close to unity (±0.05) throughout the sawtooth cycle in most cases [109,110].

Furthermore, q0 apparently always returns to unity after the crash in DIII-D [111].

Nevertheless, two studies for specific DIII-D discharges stand out. In one study

[110], the difference between a standard reference sawtoothing discharge and one with

increased non-axisymmetric error field was investigated. The former exhibited a

conventional Kadomtsev crash, but a Wesson-like crash [105] expected from a QI

mode was observed in the latter. One MSE channel was available and it was anticipated

that this might be sufficient to resolve the differences in the q profiles. However, n = 1

stability calculations for the reconstructed equilibria indicated that both discharges were




unstable to the QI mode but that the results were sufficiently sensitive to the profiles to

prevent any more definite conclusion to be drawn. Nevertheless, the observations were at

least consistent with the GATO stability calculations and were also consistent with the

resonance-detuning model of Thyagaraja [112].

In a more recent study [113], a sawtoothing discharge, #82205 was analyzed in

considerable detail. The puzzle investigated in the study was: what is the actual

operational " limit when q0 < 1 in other words, what is the significant difference

between the ideal internal kink and the kink at high "N

that is responsible for the

operational " limit and that results in a disruption? This was an ITER demonstration

discharge, matching as closely as possible the key conditions expected in the ITER EDA

design except the normalized gyroradius !*. Two equilibria at "N

= 2.1 were

reconstructed and tested for stability one with q0 = 0.95 and one with q0 forced equal to

1.05. Both were equally good fits to the equilibrium data. The analysis found complete

stability except for the internal kink when q0 < 1, consistent with the fact that, except for

the sawtooth, the discharge was stable.

With increasing " , the "N

limit for q0 > 1 occurs at "N ~ "N

op # "N (q0=1.05)

crit= 3.0.

Here, the pressure was scaled by a series of constant factors, keeping the profile and the

surface averaged parallel current density fixed. For a similar sequence with q0 < 1, there

is a transition in growth rate and mode structure to a more virulent instability at "N ~ "N

op .

This is shown in Fig. 13. With q0 < 1, the unstable mode, though still predominantly

m/n!= 1/1, exhibited increasingly larger contributions from higher harmonics outside

q = 1 as " was increased above "Nop , and it became almost indistinguishable from that at

high "N

with q0 > 1. The conclusion is that the "N

limit for q0 < 1 should then be taken as

"Nop [113].

G. Implications for a BPE

The verification studies are a characteristic and unique feature of the DIII-D theory

work. The most important result from this work is that it provides confidence in the ideal

MHD predictions for a BPE. While the studies cited in this section are necessarily

restricted to a limited sample of discharges selected to be representative of distinct types

or phenomena, the conclusions are obviously widely applicable to the much larger

number of similar discharges they represent. The studies have shown that ideal MHD

provides a reliable predictor for ballooning stability limits, infernal modes, axisymmetric

modes, and external kinks [45] the latter, even for the detailed growth and mode

structure [45,102]. In addition, these kinds of studies can furnish invaluable information

needed for further research in active feedback or mode control. The simplest and possibly




the best established example of this is the n = 0 axisymmetric stability study in Sec. III.C,

which led to much improved control systems of direct benefit for any BPE. Also, the new

physics of nonmonotonic q profiles, second stability access, and ITBs revealed in the

record " discharge #69608 and the high "p discharge #77676 were identified as a result

of the detailed post analysis of the discharge equilibria.

βN0.0 1.0 2.0 3.0 4.0 5.0

0q = 1.05

0q = 0.95

OperationalβN Limit

Discharge #82205

ξ edg

e

0.0

0.1

0.2

0.3

0.4

0.5

0.6

γ2 γ2 A/

0q = 1.050q = 0.95

OperationalβN Limit

Discharge #82205

0.015(a)

(b)

0.010

0.005

0.0

Fig. 13. Computed n = 1 mode instability for the ITER demonstration discharge #82205 at 3665 ms

(a) growth rate normalized to a toroidal Alfvén time, and (b) maximum edge normal displacement

!edge = !. "# / "# , as a function of "N . The edge displacement is normalized to the maximum

displacement over the plasma volume. [Reprinted courtesy of IOP, Nucl. Fusion 39, 1557 (1999).]

Regarding specific results of interest for a BPE, the results show one can operate

near the axisymmetric limit predicted from ideal stability calculations in strongly shaped

equilibria. The shape function S is a good quantifier of shape for both axisymmetric and

kink stability. For operation in an AT mode, a key confirmation of the experimental

results provided by the post-analysis modeling is that the uncontrolled pressure profile

peaking in L-mode invariably leads to low " limits ("N~ 2) set by a fast-growing global




n=1 ideal kink mode [45,96,102]. But "N

can be increased considerably by broadening

the pressure by an H-mode transition for example. In that case, several instabilities can

arise but the most serious limit becomes a low to intermediate n edge mode [40,44]. This

has now been corroborated in experiments in JT60-U [101,114] and JET [99,115], where

an H-mode transition was used to modulate the peaking, increase the "N

limit, and

extend overall performance.

With respect to sawteeth, the most practical result is the demonstration that, to a

good approximation, one can determine the operational " limit for a sawtoothing

discharge, or in a numerical stability survey, by taking q0 = 1.05 to avoid the ideal

internal kink [113]. It is also clear from this work that there is just one mode present, not

an independent " limiting mode and a separate internal kink. In a real plasma without a

wall on the plasma boundary, the internal kink is a toroidal kink [54,103] and transforms

continuously into the " limiting mode as " increases [113].

There are also some important caveats to this claim for the operational " limit,

however, which ought to be taken into account in extrapolating to a BPE. The claim is

only valid for discharges with q0 not too far below 1 otherwise a reconstruction with

q0!=!1.05 is not a good approximation. It is also only strictly valid for DIII-D-like

moderate aspect ratio and elongation. Furthermore, this may not hold for strongly ion

cyclotron resonance frequency (ICRF) heated discharges such as in JET where so-called

monster sawteeth, with more serious repercussions to the discharge, can occur. In such

cases where sawteeth are much larger, it becomes arguable whether the consequent

discharge limitations constitute a true " limit as the distinction between a " collapse,

internal reorganization, and minor and major disruptions becomes blurred.

In general, in assessing the relevance of the DIII-D sawtooth results to a future

BPE, the large variation in behavior among different experiments needs to be recognized.

This includes the widely varying measurements of the behavior of q0 during the sawtooth

cycle [106-111] as well as the considerable differences in the consequences of sawteeth,

and m/n = 1/1 modes across machines and even within machines under different

conditions [111]. Differences exist in whether partial or complete reconnection occurs, in

seeding of other instabilities, and in the degree of stored energy loss. For example, while

TFTR has documented high" disruptions from a high n ballooning mode developing as a

nonlinear response to the 1/1 mode [116], this has not been observed in DIII-D. In

DIII-D, the direct stored energy losses are typically small but seeding of a 2/1 tearing

mode by the sawtooth crash can often lead to a later disruption. This will be discussed

further in Sections V.B and V.C.




IV. WALL STABILIZATION AND THE RESISTIVE WALL MODE

The RWM is essentially an ideal mode but with certain non-ideal modifications.

The identification of this mode, and its stabilization, was an especially important triumph

of the synergistic tie between the theory effort and the DIII-D experiment. Confidence in

the ability to verify a posteriori the ideal stability limit was crucial to demonstrating that

wall stabilization is an important factor in the stability of some discharges and was the

key to identifying the role of rotation in maintaining wall stabilization. It is also critical

for successful active feedback. The companion paper by Garofalo [46] discusses several

aspects of the theory contributions to the DIII-D wall stabilization program in detail. That

paper covers the more recent supporting calculations for the experimental demonstration

of wall stabilization, and the active feedback modeling. The discussion here will cover

mainly those aspects not treated in Ref. [46]. The focus in Sec. IV.A will be on the actual

role of the calculations from a more historical viewpoint. Section IV.B will describe the

theory work on the actual stabilization mechanisms, and Sec. IV.C discusses the theory of

feedback stabilization from a general viewpoint. The implications for a BPE are

summarized in Sec. IV.D.

A. Demonstration of passive wall stabilization

Some early high " discharges in DIII-D suggested that the " limits reached were

more consistent with the limit expected assuming wall stabilization " with#wall than with the

no-wall limit " no#wall [117]. This data is summarized in Fig. 14; there are several

discharges at moderate and lower l i that reached "

N above the expected limit with no

wall stabilization. However, the evidence was ambiguous at that time since there was

always another possibility, namely, variations in the safety factor, which was not directly

measured, could yield different stability predictions. For example, discharge #60383,

which disrupted well inside the expected wall stabilized region, was an identically

programmed companion to discharge #60381 on which the calculations in Fig. 3 were

based. Figure 3 shows clearly that the stability is strongly dependent on the pressure

profile. In addition, simple theory suggested that wall stabilization was not possible over

times longer than a characteristic resistive wall diffusion time, even for a rotating plasma

[118,119]. The analysis of discharge #67700, discussed in Sec. III.D pointed strongly to

wall stabilization [93] at the peak " ; despite the absence of MSE data, it was concluded




to be definitely unstable without the wall (Fig. 11). However, the lack of any direct

current profile data and the single time analysis still permitted some ambiguity.

0.70

1

2

3

4

5

Conformalwall at r = 1.5 a

Disruption#60383

No sall

Discharge#60383

β N

0.8 0.9 1.0

i

1.1 1.2

Fig. 14. DIII-D data in (l i,"N ) space compared to predicted n = 1 ideal kink stability boundaries for

model equilibria with a conformal wall at 1.5 times the minor radius and with no wall. Shown also is the

trajectory of discharge #60383. [Reprinted courtesy of E.J. Strait, et al., Proc. of the 12th Int. Conf. on

Plasma Phys. and Control. Nucl. Fusion Research, Nice, 1988 (International Atomic Energy Agency,

Vienna, 1989) Vol. I, p.!83.]

Analysis from this and several other discharges were shown at the 1993 workshop

in San Diego on High " in Tokamaks. These results led to renewed interest in wall

stabilization. In response, at that workshop, Bondeson and Ward presented the first

convincing theory that suggested a resistive wall can stabilize the RWM indefinitely in a

rotating and dissipative plasma. This theory was subsequently published and became

widely cited [120,121].

As a result of the renewed interest, DIII-D discharge #80111 was specifically

designed to test and confirm wall stabilization. The key experimental characteristics were

a strong coupling to the wall from a large plasma volume, high " , and a current ramp-up

to keep l i low; the latter also served to maintain q0 > 1 everywhere so that complications

from the internal kink were avoided. Post-discharge analysis found all acceptable

equilibria, consistent with the equilibrium data, were stable with a wall and unstable with

no wall while the plasma remained rotating, consistent with the new theory [120,121].




Also, as discussed earlier in Sec. III.B, the " = 12.6% discharge #80108 in the same

discharge sequence is interpreted as being wall stabilized as well, although q0 < 1 for this

discharge [60, 71].

The analysis for DIII-D discharge #80111 is described in the companion article in

Ref. [46], and in Refs [60] and [71]. However, the important point that distinguished this

analysis from earlier attempts is the successful process by which unknown variations in

the equilibrium reconstruction were ruled out as explanations for the observed stability.

In the earlier analyses the equilibrium reconstructions left sufficient latitude that an

unambiguous result was rarely possible; even for discharge #67700, a fully convincing

argument could not be sustained. Although the q profile measurement was constrained by

8 channels of MSE in the new experiments, the key unknown was still q0. A large number

of discharge equilibrium reconstructions were produced for discharge times throughout

the high " period with q0 varied. Figure 15 shows the statistical !2 of the equilibrium fit

versus q0 with q0 constrained at these values for the discharge at 704 ms, which was the

time of the peak in " . Statistically acceptable reconstructions are those with

" 2 < "min

2+1, yielding q0 values consistent with the equilibrium data in the range 1.3 < q0

< 1.8. To provide an additional margin and account for the possibility of systematic

errors, the range was increased to 1.1 < q0 < 2.0. This provides the range of growth rate

values depicted in Fig. 4 of Ref. [60]. Given that the extremes are at very best marginally

consistent with the equilibrium discharge data, yet still predict definite instability with no

wall but stability with the DIII-D wall included, the discharge can convincingly be

claimed to be wall stabilized. Moreover, this held over a period of 70 ms, which is at least

10 characteristic wall times ("wall

DIII#D~ 5msec) . Once the plasma rotation slowed,

however, an instability with all the expected characteristics of the RWM appeared

[60,71].

Subsequent experiments in 1998 confirmed these results [122] by measuring the

internal mode structure of the slowly growing RWM after the rotation slowed. The

post-analysis verification studies showed clearly that the mode was the RWM; electron

cyclotron emission (ECE) measurements of the radial displacement profile and Mirnov

data of the poloidal distribution matched the predictions from the GATO code

[45,46,122]. Figure 3 of Ref. [46] in this issue shows this signature result.

These discharges clearly exceeded the semi-empirical "N

crit = 4liscaling for the

optimum no-wall " limit [60,71,122]. Wall stabilization was then found to account for

the remaining anomalous discharges in Fig. 2(b) [60]. The analysis of recent

wall-stabilized discharges in DIII-D has also confirmed the reproducibility of the no-wall




" limit and refined its dependence on l i

. In general, the limit is well described by:

"Ncrit~ kl

i !!, (5)

where k is a constant for a given discharge type. This empirical scaling holds for all the

cases analyzed so far. For the highly optimized, strongly shaped double null (DN)

discharges used to extend wall stabilization to AT scenarios, the coefficient k ~ 4,

consistent with the earlier optimum l i

scaling [57-65]. In contrast, for a set of lower

single-null (SN) discharges, it was found that k ~ 2.5. The latter discharges were designed

specifically to have a reproducible "N

limit with no wall stabilization, "N

no#wall

, suitable

for routine physics studies with reasonable NB power. "N

no#wall

was lowered by reducing !

and applying a current ramp.

CHI2-mseCHI2-totCHI2-mse+CHI2-tot

30

25

20

Acceptableq0 Range

χ2

q0

15

10

5

01.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Fig. 15. Statistical " 2 for the discharge equilibrium reconstructions of wall stabilized discharge #80111 at

704 ms versus q0. The fits in each case were constrained to the specified q0 value. Allowing q0 to vary as a

free parameter yielded a best fit of q0 ~ 1.4. Statistically acceptable equilibrium reconstructions correspond

to the range 1.3 < q0 < 1.8.

The two values of k are completely reproducible for both discharge types. In several

cases, the stability was determined at multiple time slices and the coefficient was found

invariant in time, as well as invariant across similar discharges. The remarkable

constancy of the value of k throughout the discharge evolution is demonstrated for two

particular discharges in Figure 8 of the companion article [46]. Nevertheless, the




coefficient k is clearly a function of the pressure peaking [59,86,123,124] defined in

Eq.!(3) and it may also depend weakly on other profile factors. The dependence of k on

cross section shape, however, is synergistic, as shown in Fig 5 for example; for broad

pressure profiles the dependence on cross section is considerably stronger than for peaked

profiles.

B. Role of rotation in wall stabilization

The wall stabilization verification work led to a whole new research area. The

experiments confirmed the critical role of rotation, but also first identified the puzzling

result that the rotation slows down when " exceeds the limit with no wall, " no#wall [60].

This result was subsequently confirmed and documented in more recent discharges [122].

Stability studies were a crucial element in identifying the role of the RWM in the rotation

slowdown in wall-stabilized discharges. Figure 16 shows the plasma rotational

acceleration d" dt vs #N

/#N

no$wall

for the passively wall stabilized discharges for which

stability analysis has been done. This comprises the two different discharge types: the

lower SN discharges with low "N

no#wall

[Fig. 16(a)] and the strongly shaped DN

discharges [Fig. 16(b)]. In both types, there is a definite correlation between the observed

slowdown (d" dt #0)and the a posteriori calculation that the discharge exceeds the

no-wall limit ("N# "

N

no$wall

) . In Fig. 16, the stability limit was calculated for at least one

time slice and the coefficient k in Eq. (5) determined for each discharge. The formula in

Eq. (5) was then applied subsequently for the other discharge times.

The ideal calculations used to verify wall stabilization and confirm the existence of

the RWM ignore the role of rotation and cannot explain the stability of the RWM for

times longer than the characteristic resistive wall time. In the original pioneering

calculations, Bondeson and Ward [120,121] used the MARS code and assumed a rotating

wall surrounding a stationary plasma. To study the RWM more systematically and in

more realistic detail, the MARS code was extended to include a sheared subsonic toroidal

plasma rotation profile and alternative damping mechanisms [125]. The modified code

then also includes the Coriolis effect though it is small. Centrifugal effects are included in

later studies [126] but are ignorable for subsonic flows typical of most DIII-D

experiments. The major features are the Doppler shift in the complex mode frequency,

additional terms in Ohms law, and addition of a viscous damping term. The theory and

code can then treat both the tearing and ideal kink branches of the RWM.

The MARS code was then used to analyze the effect of sheared rotation and the

effect of different damping mechanisms on wall stabilization and the ideal-plasma RWM

for DIII-D discharge #92544. While insufficient knowledge of the viscosity tensor




implies that truly qualitative comparisons like that for ideal mode limits are not really

possible, the calculations find reasonable agreement with the observed rotation threshold.

The results indicate that the wall stabilization model [120,121] agrees well with the

experimental threshold [125]. Figure 17 shows the dependence of the computed growth

rates on wall position with varying peak rotation values, !0, and a rotation profile !(")

obtained from the experimentally measured profile. The curves are labeled by the fraction

by which !0 is scaled from the experimentally measured value. These calculations used a

self-similar wall (non-conformal) in which the wall-plasma separation is proportional to

the local plasma radius. The actual wall in DIII-D is roughly in the range shaded and

complete stabilization of the RWM is predicted for rotation speeds in excess of the

measured value. The plasma dissipation model was taken to be the sound wave damping

model used in Refs [120] and [121] with the coefficient that controls the damping

strength held fixed but increased significantly above the simplified theoretical estimate.

Low δ SN~

~

92561

96519 ELMing

H-mode

H-mode

ELMing

92544

H-mode

200

–200

0

Acceleration

Deceleration

97802

97798

βN – 2.5 i

High δ DN

β βN Nno wall

/Ew =

100(b)

(a)

–100

0

0.5 1.0 1.5

97934

100219

dΩ/d

t (kH

z/s)

80111ELMing

H-mode

Acceleration

Deceleration

βNno wall – 4 i

dΩ/d

t (kH

z/s)

no wall

Fig. 16. Trajectories of several DIII-D wall stabilized discharges showing the correlation between the

discharges exceeding the predicted no wall " limit and the deceleration of the plasma rotation (d!/dt < 0).

(a)!The SN discharges with current ramp used in wall-stabilization experiments for which k ~ 2.5, and (b)!A

set of strongly shaped DN discharges with k ~ 4. This includes discharge #80111 as well as more recent

wall stabilized AT discharges.




0.00.20.40.60.81.01.2

ApproximateDIII–D wall

1.0–0.5

0.0

0.5

γτw1.0

1.5

2.0

1.1 1.2 1.3Rwall/Rplasma

1.4 1.5 1.6 1.7 1.8

Fig. 17. Stability diagram for an equilibrium with profiles fitted to experimental rotation and density

profiles. Computed growth rates for different levels of rotation frequency and with fixed dissipation are

shown. The curves are labeled in the legend by the fraction of !0 relative to the experimentally measured

value. The threshold for the experimentally measured rotation is well within the actual DIII-D vacuum

vessel location shown shaded. [Reprinted courtesy of AIP, Phys. Plasmas 2, 2236 (1995).]

Several important physics results were obtained from this study. First, the ideal

MHD mode and the RWM represent separate branches of the dispersion relation for the

complex mode frequency but the two branches have very similar eigenfunctions in the

plasma. This validates the use of the ideal eigenfunction in the post discharge analysis in

Refs [45], [46], and [122]. Also, both rotation and energy dissipation are necessary for

stabilization. Plasma rotation separates the ideal plasma mode from the RWM, whereas

dissipation reduces the growth rate of the RWM through coupling to stable modes and

imparts a finite real frequency to the mode. The calculations considered varying rotation

profiles and found that for this discharge the rotation frequency at the q = 2 surface is

more important than the core rotation.

C. Active feedback stabilization of the RWM

The calculations for passive stabilization by plasma rotation provide confidence in

the ability to reproduce the passive limit. This is needed in order to proceed with active

feedback schemes since one needs to know the eigenfunction and to predict the degree of

passive stabilization to decide how well active feedback schemes work. Figure 18 shows

the trajectories of several discharges in comparison to the critical " boundary

C" # ("crit $" no$wall

) /(" with$wall $" no$wall) versus rotation calculated from MARS. These

calculations assumed no feedback and used parameters typical of DIII-D. In the

terminology used in Sec. III, these are examples of post-analysis of the first type. The

overlaid traces correspond to discharges with and without feedback and magnetic




braking. This shows the feedback system actively stabilizing the RWM below the

predicted critical rotation [127]; the discharge without feedback becomes unstable near

the calculated stability boundary, whereas discharges with feedback exceed the calculated

boundary and reach significantly higher beta at lower rotation.

TimeC-coil

No walllimit

Ideal walllimit

No feedback

Cβ

1.0

0.00.0 100 15050

Rotation (km/s)

MARS prediction

≈Zero rotation

Stable (without feedback)

Unstable

I-coil(a)

I-coil

No-wall

I-coil(b)

VALEN(no rotation)

Fig. 18. Trajectories of discharges (a) without feedback, (b) with feedback using internal, and (c) external

coils, and (d) with magnetic braking. Here, " is normalized as C!, the fraction of the no-wall to ideal-wall

interval. The short dashed line indicates the critical " calculated from MARS. Open circles indicate the

onset of the RWM. The inset shows the toroidal rotation velocity profile for the case that underwent

magnetic braking, shortly before the onset of the RWM. [Reprinted courtesy of M. Okabayashi, et al. in

Proc. 20th Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research, Vilamoura, 2004

(International Atomic Energy Agency, Vienna, 2004) Paper EX3-1Ra.]

The theory work on active feedback takes a three pronged approach, consisting of

simplified models at one level [128,129], a more sophisticated approach taking account

of general 3D geometry and engineering features but simplified physics as embodied in

the VALEN code [46,128,130] at a second level, and a third level comprising models that

incorporate general physics principles but simplified engineering and unperturbed 2D

geometry [131133]. A summary of all three theoretical approaches to active feedback is

given in Ref. [134], where an especially useful review of the simplified 1D models is

provided. Reference [134] also summarizes the relative advantages of the different

feedback logic schemes such as mode control, the smart shell, fake rotating shell, and

feedback using poloidal field sensors.

The two simplified physics approaches can be considered to be more engineering

oriented in the sense that they focus on the external circuits and treat the plasma as

another circuit element. The simplified, 1D analytic, lumped parameter, feedback models

[128,129] are used for quick and reasonably accurate results on system performance with




proportional, derivative, and integral gains. They also afford intuition regarding the

stability of different schemes and bandwidth limitations. These models are summarized

by Garofalo [46] and described in greater detail in Refs [128], [129], [134], and [135].

These publications also discuss the 3D VALEN modeling [130] and its guiding role in

the DIII-D experiments at considerable length.

In contrast, the third approach focuses attention on the physics of the plasma

response to the external fields. It is consequently appropriate to summarize the theoretical

aspects of this model in more detail here. Although implemented so far only in 2D, the

full physics model has been instructive for developing a sound theoretical basis and

guiding the DIII-D active feedback program. Based on extension of the ideal MHD

theory, it also promises to provide the most comprehensive physics description of

feedback experiments in the presence of additional plasma rotation.

Chance et al. [131] constructed a general model for feedback as an extension of the

vacuum formulation used in the VACUUM code [19] by adding additional fields from

feedback coils coupled to sensors inside and outside the wall. The resulting formulation is

no longer self-adjoint but self-adjointness is recovered for the special case of an idealized

smart shell. This was implemented by coupling the VACUUM code to the GATO ideal

stability code. The coupled codes then self-consistently incorporate the coil response to

the plasma displacement and the corresponding response of the plasma to the vacuum

fields induced by the active coils. In a detailed study using this idealized smart shell

formulation, the dependence of the relative effectiveness of various coil sets on the

fraction of the poloidal circumference Cf covered by the coils was considered for

different numbers of coils nf [46,131]. The study revealed that the effectiveness increases

with nf. However, for each nf, the effectiveness increases with Cf up to a maximum, after

which it degrades since then the mode scale length cannot be resolved by the finite coil

spanning Cf/nf, thus providing an optimum coil coverage for any given nf,

A more complete model [132,133] based on the approach in Ref. [131]

was subsequently developed and generalized to an arbitrary feedback system. This

theory incorporates the general plasma response to any active feedback logic and

sensor-coil-plasma-wall geometry; the coils and sensors can be inside or outside the

resistive wall. In this formulation, an extended energy principle is constructed as

!W = !Wp + !Wv + DN + Ec = 0!!!. (6)

Here, !Wp and !Wv are the usual ideal MHD contributions from the plasma and

vacuum energy. DN is a norm; in this case, the energy dissipated in the resistive wall is

appropriate since, for the slow RWM, it is expected to be larger than the usual inertial




term. These terms are manifestly self-adjoint. The active coil contribution is embodied in

Ec. Ec is self-adjoint only under special conditions such as for the idealized smart shell

feedback scheme.

For an open loop feedback system, Ec = 0, and the system is self-adjoint so one can

find an associated complete set of eigenmodes. The first, least stable of these is the actual

RWM. The DCON and VACUUM codes have been modified and coupled to find this

eigenmode set from Eq. (6) with Ec = 0. Then, for Ec ! 0, the solutions can be expanded

in this basis set. From the full equation "W = 0, one obtains a set of first order differential

equations for the coefficients of this expansion in terms of an excitation matrix derived

from Ec, and the feedback coil currents Ic. The Ic are subsequently determined from sensor

signals and circuit equations, and embody the feedback logic. The result of solving this

coupled set is the complete response of the plasma system to the active coils.

The response includes contributions from the RWM and all other (stable)

eigenmodes. In contrast, the VALEN approach essentially uses just the RWM term in the

eigenmode expansion, thus ignoring deformations of the RWM structure due to the active

coils. The general theory also incorporates the phenomenon of resonant field

amplification (RFA) (also known as error field amplification or EFA) [46,136138].

Experiments in DIII-D and supporting theory have recently revealed the nonlinear

interactions between plasma rotation, the RWM, and nonaxisymmetric error fields

[137].RFA implies that a marginally stable, almost stationary eigenmode such as the

RWM in a rotating plasma can amplify a pre-existing static error field [136]. This creates

a large drag on the plasma rotation, slowing it until the RWM becomes linearly unstable,

as observed in DIII-D experiments summarized in Fig 16.

The active feedback system in DIII-D also has the capability to dynamically

counteract the error field amplified or not. Recent experiments exploited this to

prevent the rotation slowdown and maintain "N

well above "N

no#wall

by passive rotational

stabilization essentially indefinitely [137,138]. In the theory in Refs [132] and [133], the

response to each mode is inversely proportional to its open loop eigenvalue # = $ + i%.

Thus, if $ and % are small, as is the case for the almost stationary and marginally stable

RWM, the plasma response becomes large.

Recent numerical tool developments have been aimed at including plasma rotation

since DIII-D feedback experiments generally have a sizeable contribution from rotation,

as demonstrated in Fig. 18. To this end, the MARS code has now been extended to

include fields from active coils in addition to plasma rotation.




D. Implications for a BPE

The research performed on wall stabilization and the physics of the RWM in DIII-D

has obvious implications for any future BPE planning to take advantage of the potential

performance gains possible from operation above the no-wall " limit [82,86]. These

overall implications are well summarized in Ref. [46]. With respect to the theoretical

contributions, several specific items should be emphasized. In general, the predictive

capabilities demonstrated as an integral part of the DIII-D program should be invaluable

for the design and in predicting the performance of a future BPE. Figure 16 is the

quintessential example of the integrated nature of the theory contribution to DIII-D. This

figure combines experimental discharge data, d" dt and"N

, with an a posteriori

calculation of "N

no#wall to reveal new physics. The scaling of the no-wall " limit with kl i

in Eq. (5) can be used to predict the " limit for a wide range of discharges, apparently

within a few percent. For ITER FEAT, for example, one can expect k ~ 2.5, based on the

result from similar cross section discharges in DIII-D. The factor k can be increased,

however, by broader pressure profiles [57,86,123,124,139].

There is a considerable gain in the achievable " from passive stabilization by a

nearby resistive wall if the plasma can be made to rotate fast enough. The gains can be

large enough to make it worth investing in systems that can make the plasma rotate

sufficiently fast in a BPE. Alternatively, the actual rotation requirements may be modest

or might even be reduced by a more judicious choice of operating point [46]. In any case,

it is important to pursue experiments in rotational stabilization to determine how the

critical rotation scales, which parameters are important (rotation near q = 2 or 3, or some

average), the question whether the presence of more, higher order, rational surfaces can

substantially improve rotational stabilization, and, of course, the dissipation mechanism.

Active stabilization is being considered in many BPE designs on the basis of the

positive results from DIII-D. Several general findings from the theoretical models and

studies outlined here for DIII-D have been used in designing the DIII-D active feedback

system and are clearly important for any BPE as well. Of particular interest to a future

BPE, a simplified analysis [134] assuming only proportional gain and no rotation

suggests poloidal field sensors are better than radial field sensors. This is due, partly, to

lower coupling with control coils but also to their better time response they act like a

high pass filter to improve the high frequency response. Poloidal sensors are even better

than idealized feedback for this reason. From the more general analysis [132,133],

poloidal field sensors are also less strongly coupled to the stable eigenmodes. They

therefore require considerably lower gains and can stabilize the RWM at higher " with




just one band of control coils at the midplane; with radial field sensors, three bands

are required [132,133]. Sensors on the inside wall also perform better for the same

reason [132,133].

Coils inside the wall, in conjunction with internal sensors, are also clearly better.

Figure 18 shows that the new internal coils installed in DIII-D provide stability at higher

" and lower plasma rotation than was possible with the original external coils. This can

be expected from the general feedback theory described in Sec. IV.C since, in addition to

the interaction through image currents generated in the resistive wall, internal coils

interact directly with the mode through the induced field on the plasma surface [133].

Any future advanced performance BPE will probably benefit from both active and

passive stabilization. One could imagine a system, for example, where feedback

minimizes the inherent error field to maintain the rotation at moderate levels, as described

in Refs. [137] and [138], but which is activated to temporarily suppress a linearly

growing RWM, and then returns to minimizing the error field once the RWM level has

been suppressed and the rotation regained. Calculations so far with the feedback

extension of the MARS code have shown that the effects of rotation and feedback are

even synergistic.

The passive and active stabilization calculations for the DIII-D system have been

applied to a BPE configuration using measured profiles from DIII-D discharge #106029

scaled up to ITER parameters [140]. For this case, the critical rotation determined from

MARS calculations is 1.5 x 104 rad/s. From 1 1/2D transport simulations, it was found

that 33MW of negative ion NB input is required to drive the plasma rotation to a peak

axis value of 2 x 104 rad/s. In the active stabilization calculations [140], the coil

effectiveness versus poloidal coverage for up to three coils sets in three poloidal bands

was evaluated using the GATO and VACUUM approach for the intelligent shell scheme,

as was done in Ref. [131] for DIII-D. Coils were taken to be at 1.4 times the minor

plasma radius, or about 1.2 times the actual design wall radius. It was found that the

three-coil set could achieve about 100% of the effectiveness of an ideal wall with any

degree of poloidal coverage above 10%. With two coil sets, 100% effectiveness is

obtained with 10% poloidal coverage, but the effectiveness drops off with more coverage

as the coils become larger. For a single coil set at the midplane, the effectiveness is

strongly peaked at around 10% coverage, which then provides about 70% effectiveness.

The DIII-D passive and active wall stabilization results have recently been

reproduced in several other tokamaks. In particular, RWM stabilization by plasma

rotation is now an important feature of the world tokamak program. JET [141], JT-60U




[142], and NSTX [143,144] have reported " values above the no-wall limit, as well as

observations of a slowdown in plasma rotation and destabilization of the RWM when "

exceeds the no-wall limit [141,143]. In parallel with the DIII-D active stabilization of

pressure driven modes, a complementary program of feedback control of current-driven

RWMs is also underway on HBT-EP [145]. Internal and external active saddle coils

installed in JET for correction of error fields have recently been employed to study EFA

[141,146]. EFA was also investigated using the active feedback coils in HBT-EP [147]

and a set of external error correction coils similar to the DIII-D C-coil has been installed

on C-Mod that can be used for investigation of EFA [148]. Although the external coils in

JET and C-Mod are not intended to be used for RWM active stabilization, new coils for

RWM control are being designed for NSTX [144].




V. RESISTIVE INSTABILITIES

Resistive modes both tearing and interchange represent a serious limitation to

the performance of tokamaks. Progress in understanding tearing modes is well described

from the experimental point of view in the companion paper by La Haye in this issue

[149]. Here, we concentrate on the theoretical aspects that have been developed in

support of the experiments. The basic theory of linear resistive modes has been a focus of

theoretical development for several years. The linear theory predicts onset conditions for

low " classical (i.e. unseeded) tearing and interchange modes and considerable success

has been achieved recently in developing the theoretical fundamentals and in the

application to actual experiments. This is described below in Section V.A.

Experimentally, however, the linear phase is not observable. The quasi-linear

Rutherford [150] and full nonlinear regimes determine the development of classical

tearing modes as well as of nonlinearly seeded NTMs [151] and the ultimate final states.

Considerable progress has been made recently in developing the numerical tools needed

to study the quasi-linear and nonlinear phases. The comparison of the predictions from

these tools to DIII-D experiments is described in Secs V.B and V.C. Recent success in

modeling the special case of the resistive interchange mode is outlined in Sec. V.D.

A. Tearing mode theory

There are traditionally three approaches to the theory of tearing and resistive

interchange modes: eigenvalue and initial value codes, investigation of neighboring

equilibrium states, and the asymptotic matching approach. All three have undergone

important developments in recent years. Although eigenvalue and initial value codes have

significant differences, they can be considered together for our purposes. The key

distinction is that the eigenvalue codes, of which MARS is the premier example, are

generally linear and the initial value codes CART, NFTC, and NIMROD are nonlinear.

The application of these codes to DIII-D is discussed later in Secs V.C and V.D. The

neighboring equilibrium approach was developed for force free tearing modes [152154]

and later extended to more general cases [155]. This yields stability criteria that reduce to

the standard criteria in the slab and cylindrical limits originally derived by Furth, Killeen,

and Rosenbluth (FKR) [156] and Furth, Rutherford, and Selberg [157]. This approach

subsequently evolved into, and culminated in, the theory of almost ideal MHD




(AIMHD) [158,159]. AIMHD theory, which is currently under development, aims to find

neighboring helical equilibria accessible from an initial 2D equilibrium with appropriate

constraints. This promises to provide a rapid method for identifying the final saturated

state with islands, without needing to (or being able to) follow the intermediate dynamics.

The alternative asymptotic matching approach evolved from the FKR treatment by

considering the matching of the ideal part of the eigenmodes with a resistive inner layer.

The formalism was developed by Glasser, Greene, and Johnson (GGJ) [160]. This has the

advantage over the eigenvalue or initial value approach that it can treat high magnetic

Reynolds numbers, SR ! 108, typical of hot fusion plasmas. In this formalism, the linear

stability parameters "# and $#, corresponding to solutions with tearing and interchange

parity about the rational surface (odd and even in the displacement respectively), are

computed from the ratio of large and small Frobenius expansions of the ideal solution in

the neighborhood of the resistive layer. These provide the matching data for these

solutions to the resistive inner layer solution. In addition, the condition "'> 0 is the

criterion for instability of tearing modes.

The PEST-III code was subsequently developed, on the basis of this method, into a

working, finite " , 2D numerical tool by Pletzer and Dewar [20]. In 1995, Chu and

Greene extended that success to produce a complete, numerically tractable, formulation

for a resistive MHD energy principle with a scheme for extracting the matching data "#

and $# needed to evaluate stability [161]. This also provided a unified theory for tearing

and interchange stability. Nevertheless, the formulations derived in Refs. [20] and [161]

are fundamentally restricted to situations in which the Mercier index µ = "D

I is

between 1/2 and 1.0. Here, DI ! 0 is the ideal interchange stability criterion [160] and

µ = 1/2 corresponds to zero " . For low " with conventional tokamak current profiles,

1/2 " µ < 1 and this formulation works well. Recent successful predictions for DIII-D

equilibria [44,162164] using the PEST-III code are discussed in Sec. V.B. At higher " ,

or for non-monotonic or elevated q profiles, typical of AT equilibria, however, this

restrictive assumption on µ can be violated.

A new, numerically convenient and tractable formulation was obtained in 2000

[165] that eliminates this restriction completely and largely avoids the most serious

infinite quantities that have always plagued the finite " theory. One-dimensional test

cases show considerable promise in that converged results for the matching data are

obtained well beyond µ = 1, and the results reproduce the parity selection rules for

tearing and interchange modes at integer and half integer µ [165]. (Parity selection refers

to the fact that "# and $# exhibit poles with respect to varying µ whenever µ crosses




integer and half integer values respectively, as a result of the corresponding parity mode

being strictly forbidden for that µ). The formulation is now implemented in 2D in the

TWIST-R code, which is presently undergoing benchmarking. In 2D, multiple rational

surfaces are coupled and the dispersion relation obtained from the matching conditions is

replaced by a matrix equation involving !" and #" [161,165,166]. The same numerical

technique was also successfully implemented in solving for the inner layer matching data

[166], thus, in principle, providing a numerical tool for computing linear resistive MHD

stability, including mode onset, linear growth rates, and mode structure, in all situations.

B. Linear stability and the Rutherford approach

Mode onset predictions from linear resistive calculations have historically been

problematic due to a lack of sufficiently accurate equilibrium reconstructions coupled

with a sensitivity of the results to details in the equilibrium profiles. Earlier work for

TFTR, for example, considered the stability of discharges exhibiting slow-growing

m/n = 3/2 modes [139]. However, in that study it was not possible to distinguish whether

the observed modes were ideal infernal modes, or their resistive version, or tearing

modes, since the computed stability was sensitively dependent on a number of key

parameters, especially q0 and SR. No MSE measurements of the internal q profile were

available for the equilibrium reconstruction and q0 was forced above unity to avoid

complications in the stability analysis from the unstable m/n/ = 1/1 ideal mode.

This lack of either a solid predictive or post-analysis capability has now begun to

change with the improvements in equilibrium diagnostics and reconstruction techniques

[167], and the newer theoretical and numerical stability tools described above. Three

cases of predicted mode onset have recently been reported in DIII-D [45,162164]: a

classical tearing mode in discharge #97741, a seeded NTM in discharge #86166 that

decays while " # < 0 but becomes unstable when " # > 0, and an unseeded or spontaneous

NTM at high " in discharge #98549 that became unstable because " # became large as

the discharge approached the ideal " limit.

Chu, et al. [162] showed for the first time, using PEST-III calculations for

reconstructed equilibria of DIII-D discharge #97741, good agreement between the " #

calculations at several times and the onset of a classical tearing mode in a low "

experiment. Subsequent work on the sensitivity of " # from PEST-III calculations to the

equilibrium fitting parameters [163] revealed that proximity to the ideal " limit resulted

in a pole in " # . The sensitivity to, for example, knot positions in the splines used in the

numerical profiles could distinguish the behavior near the ideal limit from the parity

selection poles at integer µ. Brennan et al. [163] were able to show unambiguous




agreement between " # calculations from PEST-III and observed MHD in the high

performance AT discharge #98549 for equilibrium reconstructions around the time of the

onset of a 2/1 spontaneous NTM. Essentially, the spontaneous NTM is a classical tearing

mode linearly destabilized when " # becomes unbounded as the ideal " limit is

approached. There is no need to invoke new unobserved seeding mechanisms in this case.

The subsequent behavior, however, was driven by the nonlinear neoclassical

destabilization and saturation mechanisms [151].

This led to a new model for tearing stability, the pole mechanism [163,164], in

which the approach to the ideal limit at high " resulted in " # becoming large, with

implications for the Rutherford model [150,151]. The Modified Rutherford Equation

(MRE) normally used to describe the quasi-linear growth of the island width w for

NTMs can be written:

(1/")dw /dt = # $ + % p (dG + dNC )[w /(w2

+ wd

2)]+ dP /w

3!!!, (7)

where the first term on the right represents the quasi-linear drive [150], and the other

terms are the nonlinear Glasser stabilization (proportional to dG), neoclassical

(proportional to dNC), and polarization terms (proportional to dP) respectively [151].

Normally, the " # term is considered a constant. In the new model developed in Refs

[163] and [164], it is taken to be time dependent.

While this is a highly simplified, and essentially phenomenological, model,

neglecting many potentially important effects in a real 2D system, such as the coupling of

multiple rational surfaces, it nevertheless has been shown to work remarkably well in

describing DIII-D tearing mode behavior. Also, while the approach of " # to infinity near

a pole would, in principal, invalidate the neglect of inertia in the standard tearing mode

theory, in practice, the growth rates remain a small fraction of the inverse Alfvén time

scale so this assumption remains quite valid. The modes do not actually reach the ideal

point in the cases of interest.

The model was applied [163,164] to the case of a sawtoothing discharge #86166, in

which several successive sawteeth seeded a 3/2 NTM that subsequently decayed before

the next sawtooth. The 3/2 mode decay rates decreased with each new sawtooth, and,

ultimately, the mode grew to large amplitude. The quantitative prediction for the seeded

3/2 island decay rate from the MRE, using the " # values from the reconstructed discharge

equilibria and PEST-III calculations, was compared with the measured decay rates

dw /dt . This is reproduced in Fig. 19, taken from Ref. [164]. The comparison shows




exceptionally good agreement. This showed for the first time, the general validity of the

MRE and role of the linear " # to NTM dynamics.

3/2modegrowth

Sawtoothperiods

086166

n = 1 n = 2

B (G

auss

) Increasing Po/⟨p⟩

∇′ψ

s 2µ

–4

0

4

8

dB/d

t (G

auss

/s)

dB/dt

Time (ms)3000 3200 3400 3600

Theory (MRE)

Experiment

2800 3800

∆′ψs2µ

dw/d

t (s–1

)

2468

1012

–0.04

0.00

0.04

0.08

0.025–0.225

0.2750.525

–0.475

Fig. 19. Theoretical and observed 3/2 island decay and eventual onset after successive seeding from

sawteeth. The final island growth occurs when !"(#(t),t) computed from the reconstructed equilibrium at

each time becomes significantly positive, driving the seeded mode unstable, in agreement with the MRE

prediction. [Reprinted courtesy of AIP, Phys. Plasmas 10, 1643 (2003).]

A general model for tearing modes based on the pole mechanism coupled with the

MRE approach has been developed [164] that appears to be able to explain the variety of

tearing modes so far observed in DIII-D from classical low " tearing modes to seeded

NTMs and unseeded high " tearing modes. The key new feature is the dynamical

variation of " # ($(t)) ; in particular, this becomes large as " approaches the ideal limit.

The model predicts that the rate of approach to the ideal limit determines the eventual

instability that is observed. Below a critical rate NTMs are observed and the model

quantitatively predicts the island growth. With a faster rate, the ideal limit is reached

before the prior tearing mode has grown significantly, and an ideal instability should then

be observed. Experiments in DIII-D were designed and executed to specifically test the

pole mechanism. These are qualitatively consistent with the pole model [164]. Further

quantitative analysis is underway.

One further result from this modeling is the realization of the need to limit the

unphysically singular polarization term at small island widths [164]. This was done in an

ad-hoc manner in the calculations in Ref. [164] by replacing the dp /w3 term in Eq. (7)




by dpw /(wp

4+ w

4). wp is a parameter taken to be of the order of the ion banana width. By

removing the singularity atw = 0 , the model also predicts a lull in the growth rate in

many cases. This lull has since been noted in the experiments [164] and subsequent

analysis of earlier discharges shows that it is ubiquitous but was previously ignored.

Figure 20 shows the results from the predicted and observed island evolution

exhibiting this temporary lull. Here, the evolution of the experimentally inferred island

width is reproduced in Fig. 20(a) from one of the discharges (#109144) of the !" pole

mechanism experiments. The lull where the island growth stagnates between 2700 ms

and 2800 ms (shaded) is clearly seen. Results from integration of the island evolution

equation [Eq. (7)] are shown in Fig. 20(b). This calculation used a model for the

dependence of !"(#(t),t) during the ramp in "N

along with values for the neoclassical and

!pol terms calculated from the equilibrium reconstructions. The lull during the phase

between the initial growth and the later explosive growth is reproduced. The lull results

from the stabilizing effect of the polarization term becoming important as the island size

reaches the ion banana width. Subsequently, the destabilizing neoclassical term becomes

large and the fast growth resumes. [164].

Lull

Lull

0.00

0.00

0.20

0.40

0.60

25000

2

4W (c

m) 6

8

10(a)

(b)

2600 2700Time (ms)

2800 2900

0.05 0.10

w/a

w/10a

ψs ∆′/100

dwdt

∆t (s)0.15

2µ

Fig. 20. Island width w as a function of time for a discharge in the DIII-D !" pole mechanism experiment,

showing the ubiquitous lull (shaded times) in the island growth rate shortly after onset (a) from n!=!1

Mirnov data and (b) from the island evolution equation [Eq. (7)]. The lull in the growth rate observed in (a)

is reproduced in (b). This shows the island growth rate d /dt(w /a) (dotted curve) "s2µ # $ /100 (short

dashed curve), the island size w /a (chain-dash curve), and the island size w /a scaled a factor 10 (solid

curve). [Reprinted courtesy of AIP, Phys. Plasmas 10, 1643 (2003).]




C. Large scale numerical simulation of classical and neoclassical tearing modes

DIII-D was in the vanguard of work comparing experiments to nonlinear resistive

MHD simulations using the CART code [5]. This work was instrumental in

understanding the role of tearing in early DIII-D discharges [168]. The CART code used

reduced MHD and simplified numerics (specifically a Cartesian grid), so the magnetic

Reynolds number SR was generally much lower than the typical experimental values.

Also, some ad-hoc features, such as a cutoff in the growth at the observed mode size

were necessary since there was no physical saturation mechanism for pressure driven

modes. Nevertheless, nonlinear mode coupling was reproduced reasonably well. In a

major success, the code reproduced the mode cascade observed in DIII-D discharges in

which successive modes decayed and were followed immediately by the onset of one

with a lower toroidal mode number [168].

More recently, the MARS code has become a suitable tool for linear stability

analysis, and the NIMROD and NFTC codes for nonlinear simulations. The MARS code

has the distinction of including plasma rotation in addition to various dissipation effects,

including finite resistivity, although recently, the NFTC and NIMROD codes have also

included plasma rotation in nonlinear calculations. One of the common termination

events in NCS H-mode discharges is the onset of a locked n = 1 tearing mode. In one

such discharge [96], MARS calculations found an n = 1, m=2,3 tearing mode unstable if

the ideal wall radius is increased by a factor 1.05, in good agreement with the mode

identification and non-rotating character of the mode; the experimentally observed mode

was locked to the wall at onset with a growth time ~10 ms suggesting it was wall

stabilized. This case was unstable at "N

= 3.4 but a similar case was stable at "N

= 3.8 in

both the calculations and the experiment. This was a single time point comparison but

was based on a well-reconstructed equilibrium.

More sophisticated comprehensive simulations are now available. DIII-D data has

been used to benchmark both the NIMROD and NFTC codes. NIMROD simulations

coupled with the MRE approach applied to equilibria reconstructed from experimental

discharges are providing new understanding. The full-scale simulations and MRE

analysis are complementary approaches. The NIMROD calculations include nonlinear

mode coupling so they can simulate seeding of NTMs, but they are much more time

consuming and are not yet able to treat neoclassical effects consistently. On the other

hand, the MRE approach does not include nonlinear mode coupling.

NIMROD simulations for the sawtoothing seeded NTM discharge #81166 have

confirmed the basic result of the MRE approach discussed in Sec. V.B and shown in




Fig. 19 [164]. From simulations at the earliest and latest time slices performed with near

realistic values of SR ~ 106, the n=1 mode is predicted to be unstable but saturates, while

driving an n=2 m=2 mode. The n/m=2/2 mode has a 3/2 sideband that saturates at

w ~ 2cm, close to the observed value. At the earlier sawtooth crash, the 3/2 mode in the

simulations has a smaller amplitude than at the later crash. At the later crash, the seeded

3/2 mode subsequently grew in both the experiment and the simulations [164].

D. Numerical simulation of resistive interchanges

DIII-D NCS discharges with an L-mode edge commonly show early bursts of MHD

activity well before the final disruption (Sec. III.E). These were shown to correlate with

resistive interchange (RI) modes using the DR criterion and linear MARS code

calculations (including plasma rotation) to identify an expected linear instability [169].

Further analysis [82] showed that the combination of negative pressure gradient, " p < 0,

and negative shear in the q profile, ( " q < 0), typical of L-mode NCS discharges with

strong pressure peaking, is most unstable to the RI mode. While alternative causes for the

MHD bursts, such as rotational shear or fast particle driven modes, were also

investigated, the RI mode remained the leading candidate. This was despite several

theoretical reservations. The principal reservation was the question why the RI mode is

not always seen, since " p < 0 and " q < 0 throughout the L-mode NCS phase. Also

worrisome, however, were the observations that (i) in the linear phase, the RI modes are

predicted to be extremely localized, (ii) non-ideal effects are widely believed to provide

strong stabilization to the RI mode, and (iii) higher n modes should be more unstable than

n = 1, even though n > 1 bursts are rarely, if ever, seen.

Large scale nonlinear extended MHD simulations with the NFTC [25] and the FAR

[170] codes for the "N

= 2 L-mode NCS discharge #87009 showed many of the features

observed and answered some of the theoretical concerns. In particular, the n=1 mode is

nonlinearly dominant, as observed, even though the higher n modes are linearly more

unstable. Also, saturation is delayed if the mode is sustained by sources of heat and

(bootstrap) current, yielding reasonable crash times and amplitudes [25]. In the nonlinear

simulations, the linear mode broadens considerably [170]. Nevertheless, the simulations

also showed several features for which there was insufficient data to confirm, namely, a

transition to a regular tearing mode on the outer rational surface for some parameters.

The initial calculations for discharge #87009 attempted to explain the final observed

disruption from this tearing mode, but the ideal mode has since become the more likely

explanation [45] (Sec. III.E).




Later experiments at lower " , away from the ideal limit, were able to diagnose the

RI mode in DIII-D using ECE data to obtain reproducible and well-resolved internal

mode structures [171]. In these experiments, the RI mode prevented the transition to the

high performance H-mode. Ideal stability calculations using the reconstructed equilibria

were, in this case, crucial to eliminating the presence of any ideal instabilities in

interpreting the observed modes. A re-analysis of the simulation predictions for the early

MHD bursts has shown that many of the predictions that were not understood or observed

in discharge #87009 are exhibited in these more recent lower "N

discharges. In both the

simulations from the NFTC and FAR codes and in the more recent experimental data, the

initially localized resistive interchange mode at the inner rational surface broadens until it

reaches the magnetic axis. It then reconnects either at or just off-axis, consistent with a

double tearing structure in which the innermost island is weaker than the outer island.

The inner island then dissipates forming a single tearing mode. Similar results have

recently been reported in JT60-U [142] where the RI mode was observed in strongly

peaked pressure NCS discharges as a relatively benign bursting phenomenon but later

appeared to couple nonlinearly to a tearing mode at the outer surface, which subsequently

led to a collapse in the stored energy.

One can consider the RI mode as a seed or trigger for the tearing mode as a result

of its modifying the profiles [25]. Instead of leading to a disruption, however, the

modified profiles lead simply to a loss of high performance in the lower "N

experiments.

One likely interpretation of this result is that the simulations for the higher "N

case,

discharge #87009, show the features of the experimental lower "N

case with respect to

the tearing mode development because the simulations used a wall at the plasma

boundary or a pressureless and currentless, highly resistive plasma for the vacuum, and

this increased the ideal limit. Hence, the equilibria used in the simulations were

effectively far from the ideal " limit, just as the later discharges actually were.

E. Implications for a BPE

Tearing mode instabilities both classical (unseeded) and NTMs are a known

challenge for future BPEs since they can limit confinement, limit achievement of desired

high performance equilibrium states, and even result in unwanted disruptions. The RI

mode may be an obstacle to higher performance for some AT scenarios in a BPE. A

predictive capability for these modes is now becoming available. Linear stability onset

can be successfully predicted using the PEST-III and MARS codes. TWIST-R is

expected to be an important future tool for high " and AT scenarios, which often have




µ>1. The MRE approach, coupled with the linear stability " # obtained from PEST-III

(or TWIST-R) has been extremely successful in predicting the tearing mode growth.

The work outlined in Sec. V.B has verified the predictive capability of the MRE

approach. In particular, these studies confirm the appropriate role of the linear

axisymmetric " # matching data in this approach and validates the standard polarization

model in both the sawtoothing seeded case and the high " spontaneous NTM, albeit with

an additional ad-hoc cutoff in the polarization at small island widths. In fact, a key

finding of this work that should be of great interest to future experiments is that Eq. (7),

with the sign of the polarization term taken as stabilizing, does describe the experiments

well. So far, theory by itself has been unable to determine the sign or magnitude of the

polarization term under usual experimental conditions.

Nonlinear simulations are also beginning to make an impact. Recent NIMROD

simulations for ITER are attempting to predict ITER performance with realistic

parameters. The nonlinear simulations for the RI mode in DIII-D, coupled with onset

predictions, are yielding valuable insights here as well.

Careful equilibrium fitting is the key to obtaining agreement with the observed

discharge behavior in each case. In general, with good equilibrium reconstructions, the

" # calculations from PEST-III predict the experimental observations well. In the MRE

approach, both the equilibrium and the NTM terms need to be well known in order to

obtain successful agreement.




VI. SUMMARY

In many respects, DIII-D is a prototype AT. Probably, the key finding from the

DIII-D program is that a smaller, more compact AT can match the performance of larger,

high field machines through increasing " to make up for the lower field. The fusion gain

can be expressed from the fusion power Pfus and input power Pinp as

Q = Pfus/Pinp ~ !* "E B2 ~ Pp ! "E B

2!!!, (8)

where the root mean square (RMS) pressure peaking factor Pp is defined in Eq. (3), and "E

is the energy confinement time. Q can be increased by increasing " and keeping all other

factors constant. But " # "

crit= kl i (I /aB) is limited by stability. Hence, one needs to

optimize " crit against all the important instabilities.

Another useful view can be reached by rewriting Eq. (8), following Ref. [172], as

Q ~ [B2 R2] # Pp k l i (S2/$) (F2/qs

2)!!!. (9)

Here, F is a confinement enhancement factor relative to H-mode (1 " F " 2), and qs is the

boundary q, or for a divertor plasma, q95. The factor in Eq. (9) enclosed in square brackets

is a technological factor and a reactor has a direct cost roughly proportional to this factor.

The rest contains quantities directly related to plasma physics. All but the factor F are

stability related and can be increased by optimization with respect to pressure peaking,

cross section shape, confinement, and safety factor.

The improvement in stability against pressure driven modes from cross section

shaping is embodied largely in the factor (S2/$). Q could be increased by simply

increasing this factor, thereby increasing I/aB and hence " ; this reflects the fact that

n > 0 stability is improved by increasing S. However, (S2/$) is limited by axisymmetric

stability. Optimization from profiles is embodied in the factors Pp k l i and 1/qs

2. The

combination k Pp is a pressure profile factor and is also limited by n > 1 stability. This is

also where the synergistic dependence of the " limit on cross section and profiles enters;

from Fig. 5, for high S, the limiting stable k has an inverse dependence on Pp and the

product is generally optimized by low Pp, whereas at low S, the dependence of k on Pp is

weaker. The current profile factor l i

is limited by n = 0 stability. Stability to current

driven modes provides a limit on how low qs can be. The discharge optimization, as

proposed in Refs [8], [59], [73], and [172], exploits this by finding the best compromise




between n = 0 and n = 1 current and pressure driven modes in DIII-D. The same process

can be applied to any future BPE.

MHD theory played a large role in the optimization effort for fusion gain in DIII-D.

In the high gain deuterium experiments [10,172], the physics factors were maximized to

yield high Q with the moderate size and field of DIII-D. This is the essence of the AT

concept one can optimize the physics and recover high Q in a smaller sized device.

The theory and computational effort associated with DIII-D has several

characteristic and unique features that have been responsible for the success of the DIII-D

stability optimization program. The interaction of the theory effort with the experimental

program is symbiotic. The combination of general analytic theory development, with

numerical methods coupled with simplified models, and backed by large scale

calculations and simulations with realistic conditions, has led to the development

of a successful predictive capability that has yielded unprecedented progress in the

DIII-D program.

One specific feature of the theory effort is the emphasis on quantitative agreement.

This provides a window into new, previously unnoticed, and often important,

phenomena. Some examples of this are the model for a discharge being driven through an

instability boundary [102], the discovery of wall stabilization and the important role of

rotation and RFA, and the verification of the quantitative validity of the MRE. Realistic

boundary conditions are necessary to obtaining the required levels of quantitative

agreement. This naturally includes details of the geometry, current, pressure, and rotation

profiles, along with the wall boundary conditions. In the development of the required

numerical capability, for example, in the resistive MHD codes MARS, TWIST-R, NFTC,

and NIMROD, real boundary conditions and profiles need to be taken into account while

still maintaining tractability.

MHD theory is now recognized as an indispensable guide to any design effort.

Major contributions from the theory and simulation work have been made in all the areas

of stability relevant to a BPE in both the conventional and AT modes of operation. The

early work on the parametric dependence of ideal stability on geometry and profiles,

leading to the identification of the major important AT scenarios, including high l i ,

optimized l i , and NCS, is a major success. The applicability of ideal MHD theory is

quite clearly visible in the reproduction of detailed MHD behavior. Wall stabilization and

both the active and passive stabilization of the RWM have opened new avenues to high

performance that future BPEs can take advantage of; all the major BPE proposals have

incorporated this option, as well as utilizing the specific results from the DIII-D program.




Most recently, with the unprecedented equilibrium reconstruction capability in DIII-D

coupled to the emergence of realistic numerical tools, the tearing mode pole model has

yielded successful simulations and modeled a variety of tearing modes in DIII-D. The

same tools have also provided a new understanding of the role of interchange modes in

limiting performance. In summary, the predictive capability attained in the DIII-D

program is a critical element needed for the success of any future BPE.




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ACKNOWLEDGMENT

Work supported by U.S. Department of Energy Contract No. DE-FC02-

04ER54698. This paper summarizes the work done by the DIII-D program in this area

over the past twenty years or more and includes the contributions of many members of

the DIII-D Team listed in the Appendix of this volume of Fusion Science and

Technology.



GENERAL ATOMICS REPORT A24627 A-1

APPENDIX A: MAJOR NUMERICAL STABILITY ANALYSIS TOOLS

The numerical tools available for MHD stability calculations in DIII-D cover the

full range of linear and nonlinear, local and global, and ideal and extended MHD

effects (i.e. non-ideal extensions). MBC, BALOO, and CAMINO are ideal ballooning

and Mercier interchange stability codes, although extensions in MBC include finite n and

finite Larmor radius (FLR) effects. Sheared rotation is included in BALOO. GATO is an

ideal MHD low n stability code suitable for treating diverted equilibria with a perfectly

conducting, though otherwise realistic, wall and a vacuum region. DCON is a fast,

efficient ideal code based on a generalization [16] of the Newcomb method especially

suitable for finding marginal stability boundaries. ELITE is an intermediate toroidal

mode number, edge stability code with some important new non-ideal extensions [18].

The VACUUM code [19] computes the perturbed vacuum fields in a toroidal annulus

with fairly general axisymmetric boundary conditions. It is coupled to the global stability

codes GATO and DCON and is an important tool in resistive wall mode studies.

CART is a nonlinear reduced MHD code. PEST-III and TWIST-R are linear

asymptotic resistive MHD codes and MARS is an eigenvalue linear resistive MHD code

modified to include several non-ideal extensions such as sheared plasma rotation and

non-resistive damping effects [23]. NIMROD and NFTC are full nonlinear, extended

MHD codes. BOUT is a Braginskii edge turbulence code capable of treating diverted

geometry and open field line regions that has been modified to include a parallel current

in order to treat both current and pressure driven intermediate n MHD instabilities [27].

Benchmarking of these numerical tools has been an important and ongoing effort over the

years. The extensive and mutually overlapping efforts in this area have provided a crucial

element of credibility to the predictions. Most of the codes were either developed or

considerably modified for application to DIII-D. CAMINO, VACUUM, DCON,

PEST-III and NIMROD are exceptions but have been tested and benchmarked against

DIII-D. The MBC, BALOO, and CAMINO codes have undergone many routine

comparisons and are in general agreement. GATO has been extensively benchmarked

against PEST, ERATO, DCON, and KINX [173] for low n, with agreement on marginal

stability points to three figures, and is also in exceptionally good agreement with

experiments in DIII-D, as discussed in this paper. Also, MARS and GATO have been

benchmarked for ideal stability in several prominent cases, including one documented in



A-2 GENERAL ATOMICS REPORT A24627

Ref. [66]. In the intermediate n range, ELITE was benchmarked extensively against

DCON, GATO, and MISHKA [174] for edge modes, and both the marginal points and

absolute growth rates were found to be in remarkably good agreement with those from

GATO and MISHKA [175]. In addition, during development of the finite element

version, the VACUUM code was tested against the native vacuum package in GATO for

simply connected walls and agreement was found in great detail [131]. For linear

resistive modes, the MARS and PEST-III codes were benchmarked and found to be in

agreement [176]. The GATO code was utilized to test and benchmark both the nonlinear

codes NFTC and NIMROD at several stages of their development, and the two latter

codes continue to be benchmarked for linear instability predictions against GATO and

DCON. The BOUT and NIMROD codes are presently undergoing preliminary

benchmarking of their predictions for nonlinear edge instabilities, but these codes differ

significantly in their physics and numerical capabilities, making a complete and

legitimate comparison difficult.

Abla, G. 1 Comer, K. 39 Gilmore, M. 36 Indireshkumar, K. 8Allais, F. 55 Content, D. 24 Giruzzi, G. 44 Isayama, A. 56Allen, S. L. 4 Culver, J. 71 Glad, T. 1 Isler, R. C. 5Anderson, P. M. 1 Cummings, J. W. 1 Glasser, A. H. 3 Ivanov, A. 61Andre, R. 8 Cuthbertson, J. W. 33 Gohil, P. 1 Jackson, G. L. 1Antar, G. 33 Davis, J. W. 71 Gootgeld, A. A. 1 Jacques, A. M. 1Antoniuk, N. 33 Davis, L. 1 Gorelov, I. A. 1 Jaeger, F. 5Astapkovich, A. 60 Davis, W. 8 Goulding, R. H. 5 Jahns, G. L. 1Attenberger, S. 5 DeBoo, J. C. 1 Grantham, F. 1 Jakubowski, M. 39Austin, M. E. 37 DeGentile, J. C. 44 Gray, D. S. 33 Jalufka, N. W. 23Baggest, D. S. 1 deGrassie, J. S. 1 Graznevitch, M. 53 James, R. A. 4Baity, F. W. 5 DeHaas, J. 4 Green, M. T. 1 Janeschitz, G. 52Bakalarski, J. P. 1 Delaware, S. 1 Greene, J. M. 1 Janz, S. 35Baker, D. R. 1 Deranian, R. D. 1 Greene, K. L. 1 Jarboe, T. 38Baldwin, D. E. 1 Diamond, P. H. 33 Greenfield, C. M. 1 Jardin, S. C. 8Barber, D. E. G. 5 Diao, G. 56 Greenough, N. L. 8 Jayakumar, R. J. 4Bastasz, R. 9 DiMartino, M. 1 Groebner, R. J. 1 Jensen, T. H. 1Baxi, C. B. 1 Doan, K. H. 1 Groth, M. 4 Jernigan, T. C. 5Baylor, L. R. 5 Doane, J. L. 1 Grunloh, H. J. 1 Joffrin, E. H. 44Becoulet, M. 44 Doerner, R. P. 33 Gryaznevich, M. 53 Johnson, E. 1Bernabei, S. 8 Doi, I. 72 Günter, S. 52 Johnson, L. C. 8Bialek, J. M. 21 Dokouka, V. 64 Guo, S. C. 41 Johnson, R. D. 1Biglari, H. 33 Dominguez, R. R. 1 Gupta, D. 39 Jong, R. 4Boedo, J. A. 33 Dorland, W. 35 Haas, G. 52 Junge, R. 1Bogatu, I. N. 1 Dorris, J. 27 Hahm, T. S. 8 Kajiwara, K. 6Boivin, R. L. 1 Doyle, E. J. 32 Hanai, S. 14 Kamada, Y. 56Bondeson, A. 40 Duong, H. 31 Hansink, M. J. 1 Kaplan, D. H. 1Bozek, A. S. 1 Edgell, D. H. 26 Harrington, R. J. 29 Katsuro-Hopkins,O. 21Bramson, G. 1 Ejima, S. 1 Harris, J. H. 69 Kawano, Y. 56Bravenec, R. V. 37 Ejiri, A. 58 Harris, T. E. 1 Keith, K. M. 1Bray, B. D. 1 Ellis, R. A. III 8 Harvey, R. W. 12 Kellman, A. G. 1Brennan, D. P. 1 Ellis, R. F. 35 Haskovec, J. 1 Kellman, D. H. 1Brezinski, S. 48 Ernst, D. 8 Hatae, T. 56 Kempenaars, M. A. H. 49Brizard, A. 9 Estrada-Mila, C. 33 Hatcher, R. 8 Kessel, C. 8Broesch, J. D. 1 Evanko, R. G. 1 Hawkes, N. C. 53 Khayrutdinov, R. 64Brooks, N. H. 1 Evans, T. E. 1 Hayden, D. 34 Kim, C. 39Brown, B. 1 Feder, R. 8 Heckman, E. 1 Kim, J. S. 13Brown, R. 1 Feibush, E. 8 Hegna, C. C. 39 Kim, J. 1Buchenauer, D. 9 Fenstermacher, M. E. 4 Heiberger, M. 1 Kim, K. W. 32Budny, R. V. 8 Fenzi, C. 39 Heidbrink, W.W. 31 Kim, Y. 1Burley, B. 1 Ferguson, W. 4 Helton, F. J. 1 Kinoshita, S. 14Burrell, K. H. 1 Ferron, J. R. 1 Hender, T. C. 53 Kinsey, J. E. 25Burruss, J. R. 1 Finken, K. H. 48 Henkel, G. 5 Kirkpatrick, N. P. 1Buttery, R. J. 53 Finkenthal, D. K. 28 Henline, P. A. 1 Klepper, C. C. 5Buzhinskij, O. 64 Fisher, R. K. 1 Hill, D. N. 4 Kohli, J. 1Callen, J. D. 39 Fitzpatrick, J. 31 Hillis, D. L. 5 Konings, J. A. 49Callis, R. W. 1 Fitzpatrick, R. 37 Hinton, F. L. 1 Konoshima, S. 56Campbell, G. L. 1 Flanagan, S. M. 1 Hobirk, J. 52 Kramer, G. J. 8Campo, C. S. 1 Fonck, R. J. 39 Hodapp, T. R. 1 Krasheninnikov, S. I. 33Candy, J. M. 1 Forest, C. B. 1 Hoffman, D. J. 5 Krasilnikov, A. 64Carlstrom, T. N. 1 Fowler, T. K. 30 Hoffmann, E. H. 1 Kruger, S. E. 18Carolipio, E. 31 Fransson, C-M. 19 Hogan, J. T. 5 Kubo, H. 56Carreras, B. 5 Fredd, E. 8 Holcomb, C. 4 Kupfer, K. 7Cary, W.P. 1 Fredrickson, E. 8 Holland, C. 33 Kurki-Suonio, T. 30Casper, T. A. 4 Freeman, J. 1 Hollerbach, M. A. 1 La Haye, R. J. 1Cecil, E. 20 Freeman, R. L. 1 Hollman, E. M. 33 Labik, G. 8Challis, C. D. 53 Friend, M. E. 1 Holtrop, K. L. 1 Lao, L. L. 1Chan, V. S. 1 Fuchs, C. 52 Hong, R. -M. 1 Lasnier, C. J. 4Chance, M. S. 8 Fukuda, T. 56 Hosea, J. C. 8 Latchem, J. W. 1Chang, Z. 39 Fukumoto, H. 14 Hosogane, N. 56 Laughon, G. J. 1Chen, L. 65 Futch, A. H. 4 Houlberg, W.A. 5 Lazarus, E. A. 5Chenglum, Y. 67 Fyaretdinov, A. 62 Howald, A. M. 1 Lebedev, V. 33Chin, E. 1 Gafert, J. 52 Howard, N. 34 Leboeuf, J. -N 32Chiu, H. K. 1 Galkin, S. A. 61 Howell, D. F. 53 Lee, B. 32Chiu, S. C. 1 Gallix, R. 1 Howl, W. 1 Lee, H. 66Choi, M. 1 Garofalo, A. M. 21 Hsieh, C. -L. 1 Lee, J. -H. 32Chu, M. S. 1 Garstka, G. 35 Hsu, W.L. 9 Lee, J. 31Cirant, S. 50 Gentle, K. W. 37 Humphreys, D. A. 1 Lee, P. 1Coda, S. 27 Ghendrih, Ph. 44 Hyatt, A. W. 1 Lee, R. L. 1Colchin, R. J. 5 Gianakon, T. 39 Ikezi, H. 1 Legg, R. A. 1Colleraine, A. P. 1 Gianella, R. 44 Imbeaux, F. 44 Lehecka, T. 32Combs, S. K. 5 Gilleland J. R. 1 In, Y. 13 Lehmer, R. 33

THE DIII-D TEAM (1986-2005)

[The number following each name corresponds to the affiliation shown on page 3 of this Appendix.]AND THEIR AFFILIATIONS

Leikind, B. 1 Nevins, W.M. 4 Rice, B. 4 Synakowski, E. 8Leonard, A. W. 1 Neyatani, Y. 56 Riedy, P. 1 Takahashi, H. 8Leuer, J. A. 1 Nikolski, Y. 1 Robinson, J. I. 1 Takechi, M. 56Lightner S. 1 Nilsen, M. P. 1 Rock, P. 1 Takenaga, H. 56Lin-Liu, Y. R. 68 Nilson, D. E. 4 Rodriguez, J. 34 Tang, W. 8Lippmann, S. I. 1 Nissley, L. E. 1 Rogers, J. 8 Taylor, P. L. 1Lisgo, S. 71 Noll, P. 51 Rognlien, T. D. 4 Taylor, T. S. 1Lister J. 43 Ohkawa, T. 1 Rolens, G. 1 Temkin, R. J. 27Liu, C. 1 Ohyabu, N. 58 Rosenbluth, M. N. 1 Terpstra, T. B. 1Liu, Y. Q. 40 Oikawa, T. 56 Ross, D. W. 37 Thomas, D. M. 1Lloyd, B. 53 Okabayashi, M. 8 Rost, J. C. 27 Thomas, M. P. 1Loarte, A. 52 Okazaki, T. 56 Rothwell, D. A. 1 Thomas, P. R. 44Lodestro, L. L. 4 Olstad, R. A. 1 Rudakov, D. L. 33 Thompson, S. I. 1Lohr, J. M. 1 Omelchenko, Y. 1 Ruskov, E. 31 Thurgood, P. A. 1Lomas, P. J. 51 O'Neill, R. C. 1 Ryter, F. 52 Tooker, J. F. 1Lowry, C. 51 Ongena, J. 46 Sabbagh, S. 21 Trost, P. K. 1Lu, G. 1 Opimach, I. 62 Sager, G. T. 1 Trukhin, V. 62Luce, T. C. 1 Osborne, T. H. 1 Saibene, G. 52 Tupper, M. 1Luckhardt, S. C. 33 Overskei, D. O. 1 Saigusa, M. 56 Turgarinov, S. 64Ludescher-Furth,C. 8 Owens, L. W. 5 Sakurai, S. 56 Turnbull, A. D. 1Luhmann, N. C. Jr.32 Ozeki, T. 56 Sauter, O. 43 Tynan, G. R. 33Lukash, V. 64 Parker, C. T. 1 Sauthoff, N. 8 Ulrickson, M. A. 9Luo, Y. 31 Parks, P. B. 1 Savercool, R. I. 1 Unterberg, B. 48Luxon, J. L. 1 Parsell, R. 8 Savrukhin, P. 62 Vanderlann, J. 1Mahdavi, M. A. 1 Patterson, R. 1 Schachter J. M. 1 VanZeeland, M. A. 6Mailloux, J. 53 Pawley, C. J. 1 Schaffer, M. J. 1 Vernon, R. 39Maingi, R. 5 Pearlstein, L. D. 4 Schaubel, K. M. 1 Visser, S. 1Makariou, C. C. 1 Peavy, J. J. 1 Schissel, D. P. 1 VonGoeler, S. 8Makowski, M. A. 4 Peebles, W.A. 32 Schlossberg, D. J. 39 Wade, M. R. 5Mandrekas, J. 22 Penaflor, B. G. 1 Schmidt, G. 8 Waelbroeck, F. L. 37Manickam, J. 8 Peng, Q. 1 Schmitz, L. 32 Wagner, R. 33Manini, A. 52 Perkins, F. W. 8 Schnack, D. D. 17 Walker, M. L. 1Maraschek, M. E. 52 Perry, M. 24 Schuster, E. 25 Waltz, R. E. 1Martin, Y. 43 Petersen, P. I. 1 Scoville, J. T. 1 Wampler, W. R. 9Matsuda, K. 1 Petrach, P. M. 1 Sellers, D. 1 Wan, B. 65Matsumoto, H. 56 Petrie, T. W. 1 Semenets, Y. 64 Wang, G. 32Matthews, G. 53 Petty, C. C. 1 Seraydarian, R. P. 1 Wang, Z. 67Mauel, M. E. 21 Pham, N. Q. 1 Sevier, D. L. 1 Warner, A. M. 1Mauzey, P. S. 1 Phelps, D. A. 1 Shafer, M. W. 39 Watkins, J. G. 9Mayberry, M. 1 Phelps, R. D. 1 Shapiro, M. 27 Watson, G. W. 31Mazon, D. 44 Phelps, W. 1 Shimada, M. 56 Watson, G. 31McChesney, J. M. 1 Philipona, R. 32 Shoji, T. 56 Welander, A. S. 1McCune, D. C. 8 Phillips, J. C. 1 Shoolbred, K. C. 1 Weschenfelder, F. 48McHarg, B. B. 1 Pigarov, A. Yu. 33 Simonen, T. C. 1 Wesley, J. C. 1McKee, G. R. 39 Piglowski, D. A. 1 Sips, A. C. C. 52 West, W. P. 1McKelvey, T. 1 Pinches, S. D. 52 Skinner, S. M. 1 Whaley, J. 9McLean, A. G. 71 Pinsker, R. I. 1 Sleaford, B. 1 Whyte, D. G. 39Menard, J. E. 8 Pletzer, A. 8 Smirnov, A. P. 63 Wight, J. 1Menon, M. M. 5 Politzer, P. A. 1 Smith, J. P. 1 Wilson, H. R. 53Messiaen, A. M. 46 Ponce, D. 1 Smith, P. 1 Winter, J. 48Mett, R. 1 Porkolab, M. 27 Smith, T. L. 1 Wolf, N. S. 4Meyer, W.H. 4 Porter, G. D. 4 Snider, R. T. 1 Wolf, R. 52Middaugh, K. R. 1 Prater, R. 1 Snyder, P. B. 1 Wong, C. P. C. 1Mikkelsen, D. 8 Pretty, D. G. 69 Solano, E. R. 45 Wong, K. -L. 8Miller, R. L. 1 Pronko, S. G. 1 Söldner, F. 51 Wong, S. K. 1Miller, S. M. 1 Puhn, F. 1 Solomon, W.M. 8 Wood, R. D. 4Mills, B. 9 Punjabi, A. 23 Soon, E. 33 Wròblewski, D. 16Minor, D. H. 1 Raftopopulos, S. 8 Squire, J. 6 Wu, X. 65Mioduszewski, P. K. 5 Ramsey, A. 8 Srivivasan, M. 32 Xu, X. Q. 4Mizuuchi, T. 59 Randerson, L. E. 8 St John, H. E. 1 Yamaguchi, S. 15Moeller, C. P. 1 Rasmussen, D. A. 5 Stacey, W.M. 22 Yin, F. 65Moller, J. M. 4 Rawls, J. 1 Staebler, G. M. 1 Yip, H. H. 1Monier-Garbet P 44 Redler, K. 1 Stallard, B. W. 4 You, K. I. 66Moore, D. 1 Reiman, A. 8 Stambaugh, R. D. 1 Zaniol, B. 54Mossessian, D. 27 Reimerdes, H. 21 Stangeby, P. C. 71 Zeng, L. 32Moyer, R. A. 33 Reis, E. E. Jr. 1 Stav, R. D. 1 Zerbini, M. 50Mui, A. 28 Remsen, D. B Jr. 1 Stockdale R. E. 1 Zhang, C. 65Murakami, M. 5 Ren, C. 39 Strait, E. J. 1 Zhang, D. 65Nagy, A. 8 Rensink, M. E. 4 Street, R. 1 Zhang, J. 33Nave, M. F. A. 47 Rettig, C. L. 32 Stroth, U. 52 Zhou, D. 65Navratil, G. A. 21 Rewoldt, G. 8 Swain, D. W. 5 Zohm, H. 52Nazikian, R. 8 Rhodes, T. L. 32 Sydora, R. D. 9 Zwicker, A. 24Nerem, A. 1

W.

THE DIII-D TEAM (1986-2005) (continued)AND THEIR AFFILIATIONS

[The number following each name corresponds to the affiliation shown on page 3 of this Appendix.]

U. S. LABORATORIES EUROPE1 General Atomics, San Diego, CA 40 Chalmers University, Götteborg, Sweden.2 Argonne National Laboratory, Argonne, IL 41 Consorzio RFX, Padua, Italy3 Los Alamos National Laboratory, Los Alamos, NM 42 Culham Laboratory, Abingdon, UK4 Lawrence Livermore National Laboratory, Livermore, CA 43 Ecole Polytechnique, Lausanne, Switzerland5 Oak Ridge National Laboratory, Oak Ridge, TN 44 EURATOM, CEA, Cadarache, France6 Oak Ridge Institute of Science Education, Oak Ridge, TN 45 EURATOM, CIEMAT, Madrid, Spain7 Oak Ridge Associated Universities, Oak Ridge, TN 46 EURATOM, Ecole Royale Militaire, Brussels, Belgium8 Princeton Plasma Physics Laboratory, Princeton, NJ 47 EURATOM, IST, Lisbon, Portugal9 Sandia National Laboratories, Albuquerque, NM 48 EURATOM, Kernsforschunganlage, Jülich, Germany

10 Sandia National Laboratories, Livermore, CA 49 FOM Inst., Rijnhuizen, The NetherlandsINDUSTRIES 50 ENEA, Frascati, Italy

11 Communications and Power Industries, Palo Alto, CA 51 JET Joint Undertaking, Abingdon, Oxfordshire, UK12 Comp-X, Del Mar, CA 52 Max Planck Institute for Plasma Physics, Garching, Germany13 FARTECH, Inc., San Diego, CA 53 UKAEA Fusion Culham Science Center, Abington, Oxon, UK14 Hitachi Ltd, Japan 54 University of Padua, Padua, Italy15 Mitsumishi Electric Corp., Japan 55 University of Paris, France16 ORINCON Corp, San Diego, CA JAPAN17 SAIC, San Diego, CA 56 Japan Atomic Energy Research Insitute, Naka, Japan18 Tech-X, Boulder, CO 57 Tsukuba University, Tsukuba, Japan19 Tomlab Optimization Inc. Willow Creek, CA 58 National Institute for Fusion Science, Toki, Japan

U.S. UNIVERSITIES 59 Kyoto University, Kyoto, Japan20 Colorado School of Mines, Golden , CO RUSSIA21 Columbia University, New York, NY 60 Efremov Institute, St. Petersburg, Russia22 Georgia Institute of Technology, Atlanta, GA 61 Keldysh Institute, Moscow, Russia23 Hampton University, Hampton, VA 62 Kurchatov Institute, Moscow, Russia24 Johns Hopkins University, Baltimore, MD 63 Moscow State University, Moscow, Russia25 Lehigh University, Bethlehem, PA 64 Troitsk Institute, Troitsk, Russia26 LLE, University of Rochester, NY ASIA AND AUSTRALIA27 Massachusetts Institute of Technology, Cambridge, MA 65 Academia Sinica Institute of Plasma Physics, Heifei, China28 Palomar College, San Marcos, CA 66 Korea Basic Science Institute, Daejeon, Korea29 Rensselaer Polytechnic Institute, Troy, NY 67 Southwestern Institute of Physics, Sichuan, China30 University of California, Berkeley, CA 68 Dong Hua University, Haulien, Taiwan31 University of California, Irvine, CA 69 Australian National University, Canberra, Australia32 University of California, Los Angeles, CA AMERICAS33 University of California, San Diego, CA 70 CCFM, Varennes, Quebec, Canada34 University of Illinois, Champaign, IL 71 University of Toronto, Canada35 University of Maryland, College Park, MD 72 University of Campinas, Brazil36 University of New Mexico, Albuquerque, NM37 University of Texas at Austin, Austin, TX38 University of Washington, Seattle, WA39 University of Wisconsin, Madison, WI

* The affiliation at the time of the most recent collaboration with DIII-D is given.

AFFILIATIONSOF THE DIII-D TEAM MEMBERS*

theory and simulation basis for …theory and simulation basis for magnetohydrodynamic stability in...

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