**Relationship between onset thresholds, trigger types, and rotation
shear for the m/n=2/1 neoclassical tearing mode in a high-β spherical
torus**

> S.P. Gerhardt^1^, D.P. Brennan^2^, R. Buttery^3^, R. J. La Haye^4^, S.
> Sabbagh^5^, E. Strait^4^, M. Bell^2^, R. Bell^1^, E. Fredrickson^1^,
> D. Gates^1^, B. LeBlanc^1^, J. Menard^1^, D. Stutman^6^, K. Tritz^6^,
> H. Yuh^7^
>
> ^1^ Princeton Plasma Physics Laboratory, Princeton, New Jersey, USA
>
> ^2^ University of Tulsa, Tulsa, Oklahoma, USA
>
> ^3^ EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon,
> Oxon, OX14 3DB, United Kingdom
>
> ^4^ General Atomics, P.O. Box 85608, San Diego, California 92186-5608,
> USA
>
> ^5^ Columbia University, New York, New York, USA
>
> ^6^ The Johns Hopkins University, Baltimore, Maryland, USA
>
> ^7^ Nova Photonics, Princeton, New Jersey, USA
>
> **Abstract** The onset conditions for the m/n=2/1 neoclassical tearing
> mode (NTM) are studied in terms of neoclassical drive, triggering
> instabilities, and toroidal rotation or rotation shear, in the
> spherical torus NSTX \[M. Ono, et al., Nuclear Fusion **40**, 557
> (2000)\]. There are three typical onset conditions for these modes,
> given in order of increasing neoclassical drive required for mode
> onset: triggering by energetic particle modes, triggering by edge
> localized modes, and cases where the modes appear to grow without a
> trigger. In all cases, the required drive increases with toroidal
> rotation shear, implying a stabilizing effect from the shear.

PACS #s: 52.35.Py, 52.55.Fa

1: Introduction

The neoclassical tearing mode (NTM) \[1\] is a beta-limiting instability
in tokamaks: when a magnetic island forms, rapid parallel transport
causes pressure flattening across the magnetic island, leading to a
reduction in the pressure driven bootstrap current which then further
increases the island size. NTMs are likely to be a dominant performance
limiting instability in large tokamaks like ITER if mitigation
techniques are not implemented \[2\]. Experimental results in NSTX and
MAST \[3\] show that these modes also exist in an ST \[4\] despite more
favorable field-curvature effects \[5\], and must be accounted for in ST
performance predictions as well.

There are many issues in extrapolating our present understanding to
larger and more slowly rotating plasmas, either at conventional
aspect-ratio or in a ST, two of which are addressed in this letter: i)
what instabilities can trigger the nominally metastable NTM, and ii)
what is the role of rotation or rotation shear in setting the onset
threshold? With regard to triggers, many advanced operational scenarios
in the ST or conventional tokamak rely on sawtooth avoidance (with
q~min~\>1) or control \[6\] to avoid triggering these modes; however,
other modes such as fish-bones \[7\] or edge localized modes (ELMs)
\[8\] have been seen to trigger NTMs in conventional aspect ratio
tokamaks. With regard to rotation, its role in setting the 2/1 NTM onset
threshold in DIII-D was presented in Ref. \[8\], which utilized that
device's unique ability to control the plasma rotation through the
co-/counter- neutral beam injection mix. It was found that with all
other parameters fixed, the value of β~N­~ (β~N~=βaB~T~/I~P~) at mode
onset was reduced as the plasma rotation slowed. While that data
analysis indicated that the toroidal rotation shear at q=2, rather than
the rotation magnitude, was likely the relevant parameter, the strong
co-linearity between rotation and rotation shear in the data set made it
difficult to separate the relative contributions.

It is the purpose of this letter to utilize results from the National
Spherical Torus Experiment (NSTX) \[9\] to provide insight into these
issues. NTMs are triggered by both energetic particle modes (EPMs) and
ELMs; in other cases, the modes grow with no apparent trigger. A large
database of NSTX discharges has been formed, containing a wide variety
of toroidal rotation profiles and a factor of three variation in the NTM
onset threshold. These different onset types yield different initial
mode frequencies; however, no trend of the onset threshold with the mode
frequency in the E~r~=0 frame has been found in the database (E~r~ is
the radial electric field), which seems to discard ion polarization
current effects \[10\] as providing a rotation dependence for the onset.
Additionally no strong trend is found between NTM onset threshold and
the absolute value of toroidal rotation. However, there is observed a
strong trend of increasing onset threshold with increasing toroidal
rotation *shear* at q=2. For fixed shear, the EPM triggered cases tend
to be struck at the lowest values of NTM drive, with ELM triggers
occurring at higher drive and the triggerless varieties at the highest
drive. These results are the first ever study of seeding mechanisms for
the 2/1 NTM in an ST plasma, and provide the strongest evidence yet that
rotation shear, rather than rotation itself, plays a key role in
determining the 2/1 mode onset threshold.

2: Definitions and data set selection

The formulation in the NTM problem in this letter is based on the
Modified Rutherford Equation (MRE) \[11\], written as:

$\frac{t_{R}}{r_{s}^{2}}\frac{\text{dw}}{\text{dt}} = D' + \frac{C_{\text{BS}}m_{0}L_{q}}{B_{q}}{dj}_{\text{BS}}\left( \frac{w}{w^{2} + w_{d}^{2}} - \frac{\text{ww}_{\text{pol}}^{2}}{w^{4} + w_{b}^{4}} \right) - \frac{6D_{R}w}{w^{2} + w_{d}^{2}}$
(1)

Here, ∆' is the classical tearing stability index. The term proceeding
the parenthesis represents the drive due to the lost bootstrap current
(δj~BS~) inside the magnetic island. For the analysis described here,
the missing bootstrap current is calculated from the formulas in Ref.
\[12\] as

${dj}_{\text{BS},\text{Sauter}} = \frac{I(y)p_{e}\left( L_{\text{31}}\frac{1}{p_{e}}\frac{\text{dp}_{e}}{dy} + L_{\text{32}}\frac{1}{T_{e}}\frac{\text{dT}_{e}}{dy} \right)}{\sqrt{\left\langle B^{2} \right\rangle}}$
, (2)

where the terms I(ψ), L~31~, and L~32~ are defined in that reference.
The D~R~ term represents the stabilizing effects of field line
curvature, and provides a stronger stabilizing contribution in an ST
than at conventional aspect-ratio.

The terms w~d~, w~pol~, and w~b~ describe stabilizing effects relevant
to small islands (see Ref \[La Haye 2006\] and references therein). Of
particular interest in this rotation-oriented study is the w~pol~ term,
a characteristic size for the polarization current effect \[Wilson
1996\] given by:

$w_{\text{pol}}^{2}{µ\rho}_{i,q}^{2}\frac{\Omega\left( \Omega - w_{i}^{} \right)}{w_{e}^{2}}$
(3)

Here, Ω=ω~Mirnov~-ω~Er=0~ is the frequency of the mode in the E~r~=0
frame and ω~e,i~\* are the diamagnetic drift frequencies of electrons
and ions; this term is predicted to be stabilizing for 0\<Ω\< ω~i~\*,
but destabilizing in the regime Ω\<0, indicating a means for rotation to
impact the stability at onset.

Rotation can enter the NTM problem through at least three additional
routes \[Buttery 2008\]: i) the absolute value of toroidal rotation may
play a role through coupling to error fields and the resistive wall, ii)
differential rotation between surfaces can modify coupling to triggers
and iii) both rotation shear and differential rotation may modify the
classical tearing stability, through either ∆' or inner-layer effects .
Here, the toroidal rotation shear here is defined as
2πL~s~τ~A~(dF~T~/dr) \[Buttery 2008\], where F~T~ is the toroidal
rotation frequency in Hz, L~s~=qL~q~/ε, L~q~=q/(dq/dr), ε is the
difference in magnetic fields strengths between the inboard and outboard
midplane normalized to their sum, and
τ~A~=R~0~(m~o~m~i~n~i~)^1/2^/B~T0~. The derivative with respect to minor
radius is defined as $\begin{matrix}
\frac{d}{\text{dr}} = \frac{1}{a}\frac{d}{d\rho}, & \rho = \sqrt{\frac{y - y_{\text{edge}}}{y_{\text{axis}} - y_{\text{edge}}}}
\end{matrix}$, where ψ is the poloidal flux and a is the minor radius
defined as half the distance between the inboard and outboard separatrix
points on the midplane.

The results presented here come from the high-elongation (κ),
high-triangularity (δ) lower-single-null plasma used for both the
morning reference discharge and many experiments \[Gates 2006\]. In
particular, all discharges in this letter come from a data set with
900kA\<I~P~ (plasma current)\<1000kA, 2.1\<κ\<2.4, 0.6\<δ\<0.8,
0.55\<l~i~ (internal inductance)\<0.8. All quantities are derived from
equilibria constrained by external magnetics, the magnetic field pitch
angle at 12 points along the midplane, and the requirement that the
magnetic surfaces be surfaces of constant electron temperature, as
measured by a 30-point Thomson scattering system. There are no sawteeth,
with q~min~\>1 through the duration of the discharge. All data were
collected during the 2007 run campaign, and combined into a database
with 56 discharges falling within the I~P~, κ, δ range noted above.
Although the shape parameters and I~P~ range are fairly restricted in
this database, a large variety of rotation profiles are present due to a
wide range of applied non-resonant magnetic field n=3 rotation braking
\[13\].

3: Examples of triggers

A common set of features distinguishes these modes in NSTX. The Fourier
analysis of the magnetic signals implies a toroidal mode number equal
to 1. Although the mode frequency immediately at onset may not match the
q=2 rotation frequency inferred from charge exchange and equilibrium
reconstruction, the two frequencies quickly approach each other and the
saturated mode frequency closely tracks the q=2 rotation thereafter.
Flat spots become visible in the electron temperature profile at the q=2
surface, and the rotation profile becomes hollow inside of q=2. These
common characteristics, visible for every case discussed in this letter,
constitute the rules used to identify the mode.

There are, however, multiple mechanisms responsible for the onset of the
mode. NTM growth has been observed in NSTX from at least three
mechanisms: chirping EPMs, ELMs, and "triggerless" cases where the mode
appears to grow with no discernable trigger. Examples of each case are
illustrated in Fig 1, where each row corresponds to a different
discharge and trigger type. The mode frequency and amplitude are
determined by following the zero-crossings of the filtered odd-n Mirnov
signal.

The lowest row shows an example EPM trigger case. The mode is observed
to strike at 0.605 seconds, at the same time as a rapid drop in the D-D
neutron rate. Spectrogram analysis shows that there are chirping MHD
modes present at this neutron rate drop, as well as at the preceding
drops around t=0.55 seconds. Note that the mode frequency at turn-on is
close to, but slightly greater than, the q=2 rotation frequency; other
examples of this onset mechanism have the initial mode frequency
slightly less than the q=2 rotation frequency.

The center row shows a case where the mode is triggered by the ELM at
0.51 sec. When it first strikes, the mode frequency is significantly
less than the q=2 rotation frequency, as might be anticipated when the
trigger-perturbation comes from the plasma edge. These modes often grow
from very low amplitude; the presence of a large ELM at the onset time
and the low initial mode frequency are the distinguishing feature of
this onset mechanism.

The details of the mode triggering in these cases are not understood, as
the eigenfunctions of the underlying instabilities and their non-linear
interactions are not understood. However, it appears possible that these
are examples of "mixed seeding" \[14\], where the modes are weakly
linearly unstable at onset, but the initial island is provided by
another instability. Verification of this hypothesis requires
calculations of ∆', a challenging endeavor \[15\] that is only now
beginning for NSTX plasmas.

The top row illustrates a case where there are no clear NTM triggers
present. Both the D~α~ and neutron emission show no features when the
mode strikes, and there are no trigger signatures on the multi-chord
ultra-soft X-ray (USXR) system. The mode is struck very near the q=2
frequency, and subsequently tracks it closely. NTMs without a trigger
have been observed in the past \[16, 7, 17\]; a likely yet untested
hypothesis in the present case is that the modes are linearly unstable
at onset.

![](media/image1.png){width="4.7756944444444445in"
height="3.2333333333333334in"}

*Fig 1. Time traces of typical trigger types. Each row corresponds to a
single discharge exhibiting a type of mode trigger: EPM trigger in the
bottom row, ELM trigger in the middle row, and "triggerless" in the top
row . The columns illustrate the mode amplitude, frequency of the mode
and q=2 surface rotation, and the neutron emission and divertor D~α~.*

The mode eigenfunction has been studied with the 30-chord USXR detector
system. This system typically sees the inversion layer associated with
the magnetic island, and use of a 2/1 island model and the inverted USXR
data is capable of reproducing the measured emission fluctuations from
chords whose tangency radius is near q=2. However, the emission from the
plasma core is typically inconsistent with a pure even-m eigenfunction
(for instance, the measured emission is odd across the magnetic axis),
implying the presence of a coupled odd-m mode, likely 1/1. Any effect of
this inner mode on the 2/1 mode stability is not reflected in an
analysis based on equation 1. Note that the coupling of m/n NTMs to
m-1/n modes has been observed in TFTR \[16\] and ASDEX-Upgrade \[7\],
and predicted by theory \[18\].

4: Onset threshold vs. rotation and trigger type

As noted in the introduction, the mode frequency at onset may be an
important parameter in determining the onset threshold. Fig. 2a)
illustrates this initial mode frequency, compared to the toroidal
rotation frequency of the q=2 surface. The trends discussed for the
three example cases in Fig. 1 hold up for the larger database. When the
mode is triggered by an ELM, the mode frequency at onset is always
significantly slower than the q=2 surface rotation frequency. The cases
with EPM triggers can rotate either faster of slower than the q=2
surface. Those cases with no visible triggers are typically observed to
grow with a frequency slightly less than the q=2 surface rotation. For
reference, the inset in Fig. 2a shows the mode and q=2 surface rotation
frequencies at a later time, when the mode amplitude has saturated; the
frequency match is then quite good.

Given that the modes most often travel more slowly than the q=2 surface
and the arguments in Sect. 2, it appears that the polarization term
might be a candidate for providing a rotation dependence in the onset
drive. The method used here to assess this physics is essentially that
in Ref. \[19\]; and the NTM-drive at onset is plotted against the E~R~=0
rotation in Fig 2b. However, similar to observation in DIII-D \[8\], the
NTM drive at onset is entirely uncorrelated with the early E~r~=0 frame
rotation. Mechanisms that involve other parameters, such as the absolute
rotation magnitude (impacting wall/error field interactions), or
differential rotation between surfaces and rotation shear (impacting
classical tearing stability), must be responsible for setting the onset
threshold.

![](media/image2.png){width="2.328472222222222in"
height="3.265972222222222in"}

*Fig 2. a) Mode frequency onset compared to the q=2 frequency at onset,
with a similar figure for the mode frequency at saturation in the inset,
and b) the approximate mode frequency in the E~r~=0 frame, normalized to
the ion-diamagnetic drift frequency.*

The NTM drive at mode onset is plotted against the q=2 rotation
frequency in Fig. 3a. Also indicated in the figure are linear regression
coefficients, used as a measure of the trend. In this case, the dataset
as a whole shows no trend (r^2^=0.1); only the ELM triggered modes show
some correlation with rotation magnitude. However, a correlation becomes
immediately clear, and improved for each onset mechanism, when plotted
against rotation shear, as in Fig 3b). Inspection of the raw data and
associated linear regression fits show that for fixed rotation shear,
the EPM cases in orange are typically triggered at lower values of
bootstrap drive. The ELM triggered cases in blue are triggered at
intermediate levels of drive, and the "triggerless" cases in purple
occur at the highest drive levels. The required drive increases as the
local rotation shear is increased, with reasonable correlation
coefficients for the EPM and ELM triggered modes and for the data set as
a whole. There is significantly more scatter for the "triggerless"
cases, possibly due to effects not related to rotation shear such as
proximity to the ideal kink limit \[17\]; however, even in this case,
the correlation is better with rotation shear than rotation. The slopes
of the best-fit lines are, within error bars, equal, implying that
rotation shear is not entering directly through the triggering physics,
and must therefore be influencing the underlying NTM stability.
Importantly, *both* the trigger type and rotation-shear effects are
important in discerning the underlying trend.

![](media/image3.png){width="2.6in" height="3.915277777777778in"}

*Fig 3. Onset bootstrap drive at mode onset vs. a) rotation and b)
rotation shear. Linear correlation coefficients (r^2^) are indicated in
each frame, sorted by trigger type.*

Some simulations and measurements of tearing-mode onset with flow shear
have indicated that differential rotation between adjacent rational
surfaces \[20\] can be stabilizing, or that differential rotation with
respect to the triggering surface reduces the seed island size \[21\].
The onset NTM drive data above has also been plotted against the
differential rotation between q=2 and each of the core rotation, q=3
rotation, and edge rotation; none of these measures are as clearly
correlated with the onset drive as the local rotation shear. Rather, the
present data appears to support the notion that rotation shear couples
to magnetic shear to modify the intrinsic classical stability of the
tearing mode \[22,23,24\].

5: Conclusions

In summary the present results demonstrate multiple mechanisms that can
lead to 2/1 NTM onset in a spherical torus, even with q~min~\>1. For
fixed rotation shear, energetic particle modes can trigger NTMs at lower
values of drive, edge localized modes at intermediate values of drive,
and triggerless modes are struck at the highest drive values.
Importantly, the inclusion of a wide variety of discharges, including
many with magnetic braking, allows the roles of rotation and rotation
shear to be examined. Within each subset of trigger types, it is shown
that the NTM drive at onset, and thus the intrinsic stability, increases
with increasing rotation shear, but is not strongly correlated with the
absolute magnitude of rotation. These results are the clearest evidence
yet that rotation shear, and not the absolute value of rotation, plays
an important role in determining the onset threshold, possibly through
the role of the classical tearing stability in governing mode onset.
Future devices without strongly driven rotation may thus be more
susceptible to 2/1 modes, but modifications to the current profile may
be able to partially compensate for the reduced rotation shear.

6: Acknowledgements

This research was funded by the United States Department of Energy. RJB
was funded jointly funded by UK Engineering and Physical Sciences
Research Council and the European Communities.

7: References

\[1\] R.J. La Haye, Phys. Plasmas **13**, 055501 (2006).

\[2\] T.C. Hender, et al, Nuclear Fusion **47**, S128 (2007).

\[3\] R. J. Buttery, et al, Phys. Rev. Lett **88**, 125005 (2002).

\[4\] Y.K.M. Peng and D.J. Strickler, Nuclear Fusion **26**, 769 (1986)

\[5\] S. Kruger, C.C. Hegna, and J.D. Callen, Phys. Plasmas **5**, 455
(1998)

\[6\] O. Sauter, et al., Phys. Rev. Lett **88**, 105001 (2002).

\[7\] A. Gude, S. Gunter, S. Sesnic, and the ASDEX Upgrade Team, Nuclear
Fusion **39**, 127 (1999).

\[8\] R. J. Buttery, et al, Phys. Plasmas **15**, 056115 (2008)

\[9\] M. Ono, et al., Nuclear Fusion **40**, 557 (2000).

\[10\] H.R. Wilson, et al., Plasma Phys. Control. Fusion **38**, A149
(1996).

\[11\] P.H. Rutherford, Phys. Fluids **16**, 1903 (1973).

\[12\] O. Sauter, C. Angioni, and Y.R. Lin-Liu, Phys. Plasmas **6**,
2834 (1999).

\[13\] W. Zhu, et al., Phys. Rev. Lett **96**, 225002 (2006)

\[14\] D. Brennan, S. E. Kruger, T.A. Gianakon, and D.D. Schnack,
Nuclear Fusion **45**, 1178 (2005).

\[15\] D. Brennan, et al., Phys. Plasmas **9**, 2998 (2002).

\[16\] E. Fredrickson, Phys. Plasmas **9**, 548 (2002).

\[17\] D. Brennan, et al., Phys. Plasmas **10**, 1643 (2003).

\[18\] D. Brennan, et al., Phys. Plasmas **14**, 06108 (2007).

\[19\] R. J. La Haye, C. C. Petty, E. J. Strait, F. L. Waelbroeck, and
H. R. Wilson, Phys. Plasmas **10**, 3644 (2003).

\[20\] D. Chandra, A. Sen, P. Kaw, M.P. Bora, and S. Kruger, Nuclear
Fusion **45**, 524 (2005).

\[21\] R. J. Buttery, et al., Proceedings of the 28^th^ EPS Conference
on Controlled Fusion and Plasma Physics, Portugal, 2001, European
Physical Society Conference Abstract (Institute of Physics, Bristol,
2001), Vol. 25, p.5.011.

\[22\] R. Coelho and E. Lazzaro, Phys. Plasmas **14**, 012101 (2007).

\[23\] A. Sen, et al., *Neoclassical tearing modes in the presence of
sheared flows*, 32^nd^ EPS Conference on Plasma Physics, Tarragona,
Spain (2005)

\[24\] R.J. La Haye, *The stabilizing effect of flow shear on m/n=3/2
magnetic island width in DIII-D*, submitted to Physics of Plasmas.