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Observation of Plasma Toroidal-Momentum Dissipation by Neoclassical Toroidal Viscosity

W. Zhu,1S.A. Sabbagh,1R.E. Bell,2J.M. Bialek,1M.G. Bell,2B.P. LeBlanc,2S.M. Kaye,2F.M. Levinton,3

J.E. Menard,2K.C. Shaing,4A.C. Sontag,1and H. Yuh3

1Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA

2Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA

3Nova Photonics, Princeton University, Princeton, New Jersey 08543, USA

4University of Wisconsin, Madison, Wisconsin 53706, USA

(Received 17 March 2006; published 9 June 2006)

Dissipation of plasma toroidal angular momentum is observed in the National Spherical Torus

Experiment due to applied nonaxisymmetric magnetic fields and their plasma-induced increase by

resonant field amplification and resistive wall mode destabilization. The measured decrease of the plasma

toroidal angular momentum profile is compared to calculations of nonresonant drag torque based on the

theory of neoclassical toroidal viscosity. Quantitative agreement between experiment and theory is found

when the effect of toroidally trapped particles is included.

DOI: 10.1103/PhysRevLett.96.225002 PACS numbers: 52.35.Py, 52.55.Fa, 52.55.Tn, 52.65.Kj

Maximum fusion power production is reached in high

pressure tokamak plasmas by avoiding or stabilizing sig-

nificant magnetohydrodynamic (MHD) instabilities, such

as tearing modes, kink-ballooning modes, and resistive

wall modes (RWMs) [1–3]. Plasma rotation can contribute

to the stabilization of these modes by shielding static,

three-dimensional error fields and by dissipating free en-

ergy available to the instabilities [4,5]. However, stabiliz-

ing rotation is often dissipated in tokamaks by drag

torques, theoretically predicted to occur by several differ-

ent mechanisms including plasma fluid viscosity [6], inter-

action of error fields with tearing mode generated currents

[7], and interaction of the plasma fluid and magnetic field

perturbations [8]. Understanding of the physical mecha-

nisms responsible for plasma momentum dissipation is

needed to determine how the favorable plasma rotation

can be sustained and maximized, or how the plasma rota-

tion profile might be controlled in future tokamaks.

While there has been success in understanding radially

localized damping by tearing instabilities tied to rational

magnetic surfaces [9] (resonant damping mechanisms),

quantitative understanding of nonresonant momentum dis-

sipation observed in experiments has been elusive.

Neoclassical toroidal viscosity [10] (NTV) is a viable

theory of plasma momentum dissipation for these obser-

vations. NTV is caused by interaction of the plasma with

magnetic field components that break the toroidal symme-

try of the magnetic confinement field in a tokamak [11,12].

Using a fluid model, NTV drag can be described as the

force on the plasma fluid as it flows through the nonax-

isymmetric field perturbation. Using a particle model, the

drag can be described in a relatively collisional plasma as a

toroidal force caused by a radial nonambipolar flux of

particles drifting due to the nonaxisymmetric field. In a

collisionless plasma, the effect is dominated by trapped

particle drifts. Experimental comparison to theory has

included semiempirical application to tearing modes in

DIII-D [13] and comparison to the plateau regime formu-

lation using cylindrical approximations in JET [14] and

NSTX [15]. The JET study found qualitative agreement to

the measured global damping, but determined that the

theory underestimated the observed damping by a few

orders of magnitude. A more recent JET study discussed

a rough estimate of the increase in NTV magnitude ex-

pected in the collisionless regime [16]. In the present study,

NTV theory appropriate for all collisionality regimes is

quantitively compared to experimental results from the

National Spherical Torus Experiment [17] (NSTX). A

significant conclusion of the present work is that theory

and experiment agree to order one.

Nonaxisymmetric magnetic field perturbations were

generated in NSTX by applying the field externally, by

amplifying this field by plasma pressure (resonant field

amplification (RFA) due to stable MHD modes [18]), and

by allowing RWMs to become unstable. NSTX is espe-

cially well suited for this study since the relatively small

plasma moment of inertia at the low device aspect ratio,

A ? 1:3 results in greater sensitivity of the plasma rotation

to changes in torque. The device can generate applied

fields with dominant toroidal mode number from 1 to 3,

using six nominally rectangular, toroidally conformed,

dual-turn coils centered on the device midplane with R ?

1:757 m (external to the vacuum vessel) and half-height of

0.483 m. Each coil subtends approximately 60 degrees of

toroidal angle. Plasma rotation is measured at 51 locations

across the outboard major radius at the device midplane by

a charge exchange recombination spectroscopy (CHERS)

diagnostic using emission from C5?at 5290 A˚.

The theoretical formulation of NTV by Shaing, et al. is

expressed in Hamada coordinates (?;?), which are com-

puted from experimental equilibrium reconstructions [19]

including magnetic pitch angle data from a motional Stark

effect diagnostic and used in the calculations in this Letter.

The magnitude of the three-dimensional magnetic field on

each flux surface is decomposed into both poloidal, m, and

toroidal, n, components as

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© 2006 The American Physical Society

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B ? B0

?

1 ?

X

n;m?0

Bnm??;??

?

? B0

?

1 ?

X

n;m?0

?Bnmccos?m? ? n?? ? Bnmssin?m? ? n???

?

:

(1)

The notation n;m ? 0 means that n and m are not simultaneously zero. The flux surface-averaged force on the plasma due

to neoclassical toroidal viscosity in the plateau and collisional transport regimes [10,20] is defined as

??~Bt?~rB

where piis the ion plasma pressure, V is the plasma fluid velocity, VTiis the ion thermal speed, ?iis the ion collision

frequency as defined in Ref. [21], q is the safety factor, R0is the fluxsurface major radius at the midplane, B and Btare the

total and the toroidal magnetic field, respectively, B ? jBj, Bt? jBtj, et? Bt=jBtj, ? is the ion viscous stress tensor,

?ps1? 1:365, and h...i denotes an average over the flux surface. Equation (2) reduces to

h^ et?~r ? ?

VTi

R2

BBt

h^ et?~r ? ?

$ip? 2

????

?

p

pi?V ?~r??

X

n;m?0

BtB

@Bnm

@?

?

?ps1

2???

?

p

3?i? ?ps1

VTi

R0qjm ? nqj

?

;

(2)

$ip?

????

?

p

pi

??q

?

R0R

?1

?

B0Bt

?1

??X

n;m?0

n2?B2

nmc? B2

nms?

?ps1

2???

?

p

3

?i

R0q?? ?ps1jm ? nqj

?VTi

!

;

(3)

which is used for the computation. Here, ??is the plasma toroidal rotation frequency and R is the major radial coordinate.

The equivalent NTV formulation for the collisionless regime can be derived from Ref. [21] as

h^ et?~r ? ?

$i?1=??? BtR

?1

Bt

??1

R2

??1ipi

?3=2?i

??I?;

(4)

where

I??"3=2

???

2

p

?Z1

0

d?2?E??? ? ?1 ? ?2?K?????1X

n

n2

??Hd???2? sin2??=2??1=2An?2?

Fig. 1(a) for coil currents primarily generating an n ? 1

radial field. This field is then decomposed to determine

Bnmcand Bnmsby Eq. (1) with (0 < n < 15), and (?15 <

m < 15). The decomposition of jBj is shown at the plasma

midplane (? ? 0) in Fig. 1(b) by the quantity Bns, where

Bns;c?P

strongly influenced by the n ? 5 component, with n ? 11

also contributing. For a current configuration selected to

primarily generate an n ? 3 radial field at the midplane,

the dominant terms for this quantity are n ? 3 and n ? 9.

Experimentally, the effect of the applied field on the

plasma lags the time of application as the field penetrates

the hot, rapidly rotating core plasma which has reached

rotation frequencies normalized to the Alfve ´n frequency of

0.48 [22]. The nonaxisymmetric field in vacuum is there-

fore shielded in the plasma core by the rotating plasma

using a factor 1=?1 ? ?2

trate completely at radii between 50%–100% of the poloi-

dal flux, and the profile of ? inside this radius is matched to

the core rotation damping profile shape using several dis-

charges and applied to all equilibria tested. The change in

the measured plasma angular momentum profile is com-

pared to the theoretical NTV torque by evaluating the

angular equation of motion d?I???=dt ? ?Tj, where the

torques Tjare due to: (i) NTV, TNTV, (ii) momentum input

from high-power coinjected neutral beams, TNBI, (iii) elec-

tromagnetic forces on rotating magnetic islands (resistive

MHD modes), TJxB, and (iv) fluid viscous forces between

adjacent flux surfaces, T??. MHD theory shows that ??

225002-2

?Hd???2? sin2??=2??1=2Bn?2

??

;

(5)

" ? r=R0, r is the minor radial coordinate, and ?1i?

13:708. The independent integration variable ? is a nor-

malized pitch angle parameter defined in Ref. [21]. The

functions K??? and E??? are the complete elliptic integrals

of the first and second kind, respectively, and the contour

integration is taken over the bounce orbit

2R2arcsin???

are generalized by Bnmcand Bnms. Equation (5) reduces to

I??"3=2

2

n;m?0

which is used for the computation of Eq. (4). Here,

Z1

and

I

Fnms??? ?

Quantitative agreement between the theory and experi-

ment relies on the inclusion of a broad spectrum when

computing Bnmcand Bnms, rather than simply one resonant

component and/or sidebands. The applied nonaxisymmet-

ric field is determined by a Biot-Savart calculation using a

detailed 3D numerical model of the nonaxisymmetric coils

as built, given the measured coil currents. The variation of

jBj as a function of toroidal angle, ?, and R is shown in

Hd? ?

0

d?. The Fourier coefficients An, Bnfor the

magnetic field B as defined in Equation (6) of Ref. [21]

???

p

X

n2?B2

nmc? B2

nms?Wnm;

(6)

Wnm?

0

d?2b?Fnmc????2? ?Fnms????2c

?E??? ? ?1 ? ?2?K????

;

(7)

Fnmc????

d???2?sin2??=2??1=2cos??m?nq???; (8)

I

d???2? sin2??=2??1=2sin??m ? nq???: (9)

mBnms;c. The relevant NTV quantity n2?B2

ns? plotted in Fig. 1(c) shows that the viscosity is more

nc?

B2

??2?0:5. The field is taken to pene-

PRL 96, 225002 (2006)

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is constant on a flux surface, each of which is considered

a thin, inertial toroidal shell with moment of inertia, I.

The plasma mass density, ?, is measured at the plasma

midplane. The total integrated torque over the flux sur-

face due to NTV, TNTV?TNTV?p?TNTV??1=??, where

TNTV?p;?1=???R0Vah^ et?~r??

ume of the inertial shell. Experimentally, TNTVis isolated

by allowing the plasma rotation profile to reach a quasi-

steady state, so that TNBIj0? ??TJxB? T??? TNTV?j0.

After this time, the nonaxisymmetric field is applied. The

differences between TJxBj0, T??j0, TNBIj0, and their values

at later times of interest are significantly smaller than

TNTV?t?. The experiment is conducted during periods

when significant tearing instabilities are absent, so TJxB

is itself small. The plasma rotation profile evolution due to

tearing modes is distinct from NTV-induced rotation

damping so their lack of influence on rotation drag can

be verified [15]. This is illustrated in Fig. 2, where the

rotation damping observed in the present experiments

[Fig. 2(a)] is contrasted with rotation damping due to

TJxB[Fig. 2(b)]. The damping due to NTV is relatively

rapid, global, and the rotation profile decays in a self-

similar fashion. In contrast, the damping due to TJxBis

initially localized near the island, and diffusive from this

radius, leading to a local flattening of ??and a distinctive

momentum diffusion across the rational surface from

smaller to larger R as expected by theory [7,9]. Also,

T??? TNTVcan be shown by first measuring the magni-

tude of ??in similar plasmas that exhibit radially local-

ized tearing modes. Applying the measured value of ??

using the plasma illustrated in Fig. 2(b) at t ? 0:395 s to

the NTV momentum dissipation experiment in Fig. 2(a),

we find that T??< 0:15 TNTVat the peak value of TNTV

between t ? 0:355–0:375 s. Finally, (TNBI? TNBIj0? ?

TNTVis independently verified using the TRANSP code.

The plasma shown in Fig. 3 has ?TNBI? TNBIj0?=TNTV?

0:02 at the peak value of TNTV. With these assumptions, the

$ip;?1=??, and Vais the vol-

equation of motion during times of interest becomes

d?I???=dt ? TNTV, which is evaluated on reconstructed

equilibrium flux surfaces.

The plasma toroidal-momentum dissipation is first eval-

uated at values of ?Nbelow the n ? 1 ideal no-wall beta

limit, ?no-wall

N

, the time evolution of which is evaluated by

the DCON [23] ideal MHD stability code. When ?N<

?no-wall

N

, beta effects such as RFA are insignificant, so the

nonaxisymmetric field that penetrates the plasma can be

modeled simply as the vacuum field modified by shielding

caused by plasma rotation. Comparison of the measured

dissipation of plasma angular momentum caused by the

externally applied nonaxisymmetric fields to the theoreti-

cal NTV torque profile is shown in Fig. 3 for an n ? 3

applied field configuration. The measured value of

d?I???=dt includes error bars that take into account the

uncertainty in ??and ?.

As ?Napproaches and exceeds ?no-wall

sured as the applied nonaxisymmetric field is amplified by

theweakly stabilized RWM and needs to be included in the

calculation of TNTV. The RFA magnitude is defined as the

N

, RFA is mea-

(deg)

0 60120 180 240 300 360

10

14

12

8

6

4

2

0

|B| (G)

(a)

1.1 1.5 1.2 1.3

R(m)

1.4

(b)

(c)

116939

t = 0.37s

R = 1.5m

1.4m

1.3m

R = 1.2,1.1,1.0m

Bns(G)

3

2

1

0

-2

20

-1

0

15

10

5

n = 5

n = 1

n = 11

n = 5

n = 1

n = 11

n = 3

n2(Bnc2+Bns2)(G2)

FIG. 1 (color online).

radius for nonaxisymmetric field coil currents yielding a radial

magnetic field with dominant n ? 1 component, (b) major radial

profile of toroidal mode number spectrum (sine component) of

this field, and (c) flux surface-averaged field component sum-

mation n2?Bnc2? Bns2? relevant to the NTV calculation.

(a) jBj vs toroidal angle and major

0.365s

0.375s

0.385s

0.395s

0.405s

0.425s

0.375s

0.385s

0.395s

0.405s

0.425s

116939

(a)

(kHz)

40

30

20

10

0

0

-40

-30

-20

-10

(kHz)

0.335s

0.325s

0.345s

0.355s

0.365s

0.375s

0.385s

0.395s

0.335s

0.345s

0.355s

0.365s

0.375s

0.385s

0.395s

115600

(b)

0.9 1.0 1.1 1.2 1.3 1.4 1.5

R (m)

0.9 1.0 1.1 1.2 1.3 1.4 1.5

R (m)

0.8

FIG. 2 (color online).

radius, and the difference between initial profile and subsequent

profiles for rotation damping (a) during application of nonaxis-

ymmetric field, and (b) during excitation of rotating tearing

instability.

Toroidal plasma rotation profile vs major

TNTV(N m)

3

4

2

1

00.9

1.11.31.5

R (m)

measured

theory

t = 0.360s

116931

axis

FIG. 3 (color online).

profile to theoretical integrated NTV torque for an n ? 3 applied

field configuration.

Comparison of measured d?I???=dt

PRL 96, 225002 (2006)

PHYSICALREVIEW LETTERS

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Page 4

ratio of the plasma-induced field amplitude (measured field

minus applied vacuum field) to the applied vacuum field

amplitude from magnetic sensors interior to the vacuum

vessel. As the rotation decreases below a critical profile,

the field grows more strongly and rapidly as the RWM

becomes unstable. The evolution of ?N, its relation to the

DCON computed ?no-wall

N

, the timing of the applied field,

and the plasma response to this field are shown in Figs. 4(a)

and 4(b) for an n ? 1 nonaxisymmetric field configuration.

The field amplification used in TNTVis computed in two

ways for comparison: (i) during early periods of RFA, the

field is increased by multiplying the n ? 1–3 components

of the applied field by the measured RFA magnitude and

(ii) during stronger RFA, or if the RWM becomes unstable,

the field of the mode is modeled using the theoretical

eigenfunction computed using DCON. The poloidal

mode decomposition of this eigenfunction is computed

for each n ? 1–3, multiplied by the measured RFA mag-

nitude, then added to Bnmcand Bnmscomputed from the

applied field.The results usingeach technique are shownin

Fig. 4(c) for the earlier time in the dischargewhen the RFA

is relatively weak. The second technique gives a slightly

better result, and is used for the later time [see Fig. 4(d)]

when the RWM becomes unstable. In both frames, TNTVis

also shown using the nonamplified applied field for com-

parison. The RWM case [Fig. 4(d)] clearly shows mea-

sured??damping at the magnetic axis that doesnot match

the theory due to the "1:5dependence. This mismatch is

likely due to momentum diffusion caused by T??that

becomes more prominent away from the peak of TNTV

and late in time during the ??damping evolution. The

difference in peak values between theory and experiment

can be attributed in part to uncertainty in evaluating the

RFA magnitude.

Similar analyses were conducted for 30 different equi-

libria with varying levels of nonaxisymmetric field ampli-

fication. The ratio ?TNTV?=?d?I???=dt? using the peak val-

ues along the radial profile for this ensemble of equilibria

has a mean value of 1.59 with standard deviation of 0.87.

This is significantly closer agreement than found in previ-

ousstudies. Also, the radial peak of the measured and com-

puted profiles vary in alignment by 2:28 cm ? 1:71 cm.

This small misalignment might be due in part to alteration

of the reconstructed equilibrium by the nonaxisymmetric

field and the 10 ms averaging time of the CHERS

diagnostic.

Quantitative agreement between theory and experiment

in this study shows NTV theory as viable physics to be

included in studies of torque balance in tokamak plasmas.

The agreement also suggests it as a possible energy dis-

sipation mechanism to be included in stability analysis of

modes such as the RWM [24]. Its inclusion may help solve

present outstanding issues found in both areas of research,

giving greater confidence to predictive models of plasma

behavior. Further research is planned to investigate the ion

collisionality and aspect ratio dependence of TNTV.

The authors thank Dr. J.D. Callen and Dr. C.C. Hegna

for informative discussions. This research was supported

by U.S. Department of Energy Contracts No. DE-FG02-

99ER54524 and No. DE-AC02-76CH03073.

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[5] A.M. Garofalo et al., Nucl. Fusion 40, 1491 (2000).

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[12] K.C. Shaing, Phys. Rev. Lett. 87, 245003 (2001).

[13] R.J. La Haye et al., Phys. Plasmas 9, 2051 (2002).

[14] E. Lazzaro et al., Phys. Plasmas 9, 3906 (2002).

[15] S.A. Sabbagh et al., Nucl. Fusion 44, 560 (2004).

[16] E. Lazzaro and P. Zanca, Phys. Plasmas 10, 2399 (2003).

[17] S.M. Kaye et al., Nucl. Fusion 45, S168 (2005).

[18] A.H. Boozer, Phys. Rev. Lett. 86, 5059 (2001).

[19] S.A. Sabbagh et al., Nucl. Fusion 41, 1601 (2001).

[20] K.C. Shaing, Phys. Fluids B 5, 3841 (1993).

[21] K.C. Shaing, Phys. Plasmas 10, 1443 (2003).

[22] S.A. Sabbagh et al., Nucl. Fusion 46, 635 (2006).

[23] A.H. Glasser and M.C. Chance, Bull. Am. Phys. Soc. 42,

1848 (1997); W. Newcomb, Ann. Phys. (N.Y.) 10, 232

(1960).

[24] M.S. Chu et al., Phys. Plasmas 2, 2236 (1995).

TNTV(N m)

3

4

2

1

0

0.91.1 1.31.5

R (m)

With RWM

(DCON)

6

4

2

0

8

10

TNTV(N m)

applied

field

only

axis

t = 0.370s

116939

t = 0.400s

axis

0.30 0.35 0.40 0.45

t(s)

0

10

20

30

40

0

1

2

3

4

5

2

1

0

IRWM

(kA)

RWM

RFA

(a)

(b)

With RFA

(c)

(d)

N

| Bp|(n=1)(G)

N>

Nno-wall

applied

field

only

With RFA

(DCON)

FIG. 4 (color online).

nonaxisymmetric field configuration: (a) evolution of ?N, com-

puted period when ?N> ?no-wall

N

nonaxisymmetric field coils; (b) n ? 1 plasma response mea-

sured by poloidal field sensors; (c) comparison of measured

d?I???=dt profile to theoretical integrated NTV torque during

RFA; and (d) similar comparison during RWM evolution.

NTV during amplification of n ? 1

, and maximum current in

PRL 96, 225002 (2006)

PHYSICAL REVIEWLETTERS

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225002-4