Observation of plasma toroidal-momentum dissipation by neoclassical toroidal viscosity.
ABSTRACT 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.
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 , inter-
action of error fields with tearing mode generated currents
, and interaction of the plasma fluid and magnetic field
perturbations . 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  (resonant damping mechanisms),
quantitative understanding of nonresonant momentum dis-
sipation observed in experiments has been elusive.
Neoclassical toroidal viscosity  (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  and comparison to the plateau regime formu-
lation using cylindrical approximations in JET  and
NSTX . 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 . In the present study,
NTV theory appropriate for all collisionality regimes is
quantitively compared to experimental results from the
National Spherical Torus Experiment  (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 ), 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 
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
PRL 96, 225002 (2006)
9 JUNE 2006
© 2006 The American Physical Society
B ? B0
?Bnmccos?m? ? n?? ? Bnmssin?m? ? n???
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
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. , 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 ? ?
h^ et?~r ? ?
R0qjm ? nqj
R0q?? ?ps1jm ? nqj
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.  as
h^ et?~r ? ?
d?2?E??? ? ?1 ? ?2?K?????1X
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
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 . 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 ??
" ? 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. . 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
are generalized by Bnmcand Bnms. Equation (5) reduces to
which is used for the computation of Eq. (4). Here,
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
d?. The Fourier coefficients An, Bnfor the
magnetic field B as defined in Equation (6) of Ref. 
?E??? ? ?1 ? ?2?K????
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
??2?0:5. The field is taken to pene-
PRL 96, 225002 (2006)
9 JUNE 2006
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
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 . 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
, the time evolution of which is evaluated by
the DCON  ideal MHD stability code. When ?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
, RFA is mea-
0 60 120 180 240 300 360
1.1 1.5 1.21.3
t = 0.37s
R = 1.5m
R = 1.2,1.1,1.0m
n = 5
n = 1
n = 11
n = 5
n = 1
n = 11
n = 3
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
0.9 1.0 1.1 1.2 1.3 1.4 1.5
0.9 1.0 1.1 1.2 1.3 1.4 1.5
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
Toroidal plasma rotation profile vs major
t = 0.360s
FIG. 3 (color online).
profile to theoretical integrated NTV torque for an n ? 3 applied
Comparison of measured d?I???=dt
PRL 96, 225002 (2006)
PHYSICAL REVIEW LETTERS
9 JUNE 2006
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
, 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
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
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 . 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.
 A. Bondeson and D.J. Ward, Phys. Rev. Lett. 72, 2709
 A.M. Garofalo et al., Phys. Rev. Lett. 82, 3811 (1999).
 S.A. Sabbagh et al., Phys. Plasmas 9, 2085 (2002).
 R. Betti and J.P. Friedberg, Phys. Rev. Lett. 74, 2949
 A.M. Garofalo et al., Nucl. Fusion 40, 1491 (2000).
 F.L. Hinton and R.D. Hazeltine, Rev. Mod. Phys. 48, 239
 R. Fitzpatrick, Nucl. Fusion 33, 1049 (1993).
 K.C. Shaing et al., Phys. Rev. Lett. 80, 5353 (1998).
 M. Yokoyama et al., Nucl. Fusion 36, 1307 (1996).
 K.C. Shaing et al., Phys. Fluids 29, 521 (1986).
 A.I. Smolyakov et al., Phys. Plasmas 2, 1581 (1995).
 K.C. Shaing, Phys. Rev. Lett. 87, 245003 (2001).
 R.J. La Haye et al., Phys. Plasmas 9, 2051 (2002).
 E. Lazzaro et al., Phys. Plasmas 9, 3906 (2002).
 S.A. Sabbagh et al., Nucl. Fusion 44, 560 (2004).
 E. Lazzaro and P. Zanca, Phys. Plasmas 10, 2399 (2003).
 S.M. Kaye et al., Nucl. Fusion 45, S168 (2005).
 A.H. Boozer, Phys. Rev. Lett. 86, 5059 (2001).
 S.A. Sabbagh et al., Nucl. Fusion 41, 1601 (2001).
 K.C. Shaing, Phys. Fluids B 5, 3841 (1993).
 K.C. Shaing, Phys. Plasmas 10, 1443 (2003).
 S.A. Sabbagh et al., Nucl. Fusion 46, 635 (2006).
 A.H. Glasser and M.C. Chance, Bull. Am. Phys. Soc. 42,
1848 (1997); W. Newcomb, Ann. Phys. (N.Y.) 10, 232
 M.S. Chu et al., Phys. Plasmas 2, 2236 (1995).
t = 0.370s
t = 0.400s
0.30 0.35 0.40 0.45
FIG. 4 (color online).
nonaxisymmetric field configuration: (a) evolution of ?N, com-
puted period when ?N> ?no-wall
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
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9 JUNE 2006