### Tearing mode / NTM physics

***Goal:** Study neoclassical and classical tearing modes in the unique
low aspect ratio, high beta, and large ion gyro-radius plasmas available
in NSTX, in order to both extrapolate our understanding to future ST
devices and provide data complementary to that from conventional aspect
ratio tokamaks. Investigate the impact of these modes on plasma beta,
rotation, fast particle redistribution, and the potential of low aspect
ratio operation as being theoretically favorable for NTM stability.
Determine scenarios that allow operation without these deleterious
instabilities.*

***Contributes to:***

*- ITPA MDC-3 -- Joint experiments on neoclassical tearing modes*

*- ITPA MDC-4 -- Neoclassical tearing modes -- aspect ratio comparison*

**(i) physics -- results and plans**

The neoclassical tearing mode (NTM) is a magnetic island driven by a
hole in the bootstrap current. These modes are a likely problem for any
high-performance tokamak, leading to confinement degradation (soft *β*
limit) in the case of the m/n=4/3 and 3/2 modes, and potential
mode-locking and disruption for the 2/1 NTM \[[^1]\]. These modes were
studies in early NSTX \[[^2]\] research, and have been observed in MAST
\[[^3]\]. We plan a renewed focus on NTM physics in the coming research
period. Important areas for proposed NTM research include i)
investigation of the onset-*β* and marginal island widths as functions
of kinetic parameters and plasma rotation ii) studies of the NTM
triggers in q~0~\>1 regimes, iii), modeling these modes in low-aspect
ratio geometry including the effects of flow and flow shear, and iv)
studies of the interaction of NTMs with fast particles, including
particle redistribution during mode activity.

The degrading effects of an m/n=2/1 NTM on a high performance discharge
are shown in Fig 2.1.1-1. The mode amplitude is shown in frame a), with
the mode striking at t=0.56 and maintaining it's saturated state until
t=0.71, when it locks; the plasma then disrupts at t=0.75. Frames b) and
c) show that *β~N~* continually drops during the period of the mode, as
does the energy confinement time (both determined by the EFIT code). The
mode has a profound effect on the rotation profile: an immediate
collapse of the core rotation and transition to rigid-body rotation.
Indeed, it is likely that both the short-circuiting of the magnetic
surfaces \[[^4]\] and the reduction of rotation and rotation shear
contribute to the severe confinement degradation. It is clear that in
NSTX as well as conventional tokamaks, these modes must be avoided in
order to maintain the desired high-β state.

![](media/image1.png){width="2.2180555555555554in" height="3.15625in"}

**Fig 2.1.1-1: Illustration of the effects of a 2/1 NTM on discharge
confinement. From top to bottom are the mode amplitude, β~N~, τ~E~, the
rotation frequency at five radii, and the plasma current.**

The Modified Rutherford equation (MRE) provides a convenient basic
framework to consider for the NTM studies described below:

$\frac{t_{R}}{r_{s}}\frac{\text{dw}}{\text{dt}} = r_{s}D' - \frac{6r_{s}D_{R}w}{w^{2} + w_{d}^{2}} + \sqrt{\frac{r_{s}}{R}}\frac{L_{q}}{L_{P}}\beta_{P}r_{s}\left( \frac{w}{w^{2} + w_{d}^{2}} - \frac{w_{\text{pol}}^{2}}{w^{3}} \right)$
(1)

This equation describes the evolution of the full island width (w),
including classical tearing stability (∆'), neoclassical drive due to
the bootstrap current missing inside the island (∝ *β~P~/w*),
stabilizing terms due to toroidal geometry (*D~R~/w* ∝ *β /w*, the
"Glasser, Greene and Johnson" term), and threshold effects (the
transport threshold manifested by *w~d~* \[[^5]\] and the polarization
threshold manifested through *w~pol~* \[[^6]\]) which are important for
small islands. Given that *β* can begin to approach *β~P~* in NSTX, the
GGJ term is expected to be strongly stabilizing \[[^7]\], an effect to
be examined as these studies move forward. Additional terms can be added
to this equation, such as those associated with stabilizing electron
cyclotron current drive (ECCD) \[[^8]\].

Equation 1 predicts that for saturated island sizes larger than *w~d~*
and *w~0~*, the island width should be proportional to *β~P~*. This
trend is indeed observed, as illustrated in Fig. 2.1.1-2, where the
square root of the magnetic fluctuation amplitude at the wall (∝ to the
island width), is plotted against the global *β~P~* for four discharges
where NBI power ramp downs occurred*.* The linear relationship is clear.
This observation, coupled to the close frequency match between the
magnetic fluctuations and the q=2 rotation frequencies and the
observation of flat spots at q=2 in the electron temperature profile,
confirms that the observed mode is indeed a neoclassical island. Similar
observations have been made for 3/2 NTM as well, and the physics
discussed above and below is typically relevant to both modes.

![](media/image2.png){width="2.1840277777777777in"
height="1.5833333333333333in"}

**Fig 2.1.1-2: The measured m/n=2/1 island width**
$\left( µ{\overset{\sim}{B}}_{P,\text{wall}} \right)$ **is proportional
to *β~P~*, as expected for a saturated NTM.**

The small island terms in (1), w~d~ and w~pol~, prevent NTM growth for
arbitrarily small islands when classical tearing is stable (∆'\<0).
Instead, a seed island must be formed whose width exceeds some threshold
value. For the 3/2 NTM, this seed-island is typically provided by
sawteeth in conventional aspect ratio tokamaks; ELMs have been suggested
to play a similar role for the 2/1 NTM in DIII-D and JET. In NSTX, rapid
changes in the beam timing have been observed to trigger the 3/2 NTM.
The NSTX discharge illustrated in Fig. 2.1.1-3, however, shows a large
2/1 NTM growing with no observable trigger. The spectrogram from a
single Mirnov coil in the top frame shows the mode growing at t=0.83,
with no precursor; this smooth growth is clearly indicated in the mode
amplitude plot in the frame below. Neither the H~α~ nor the edge and
core USXR signals show a precursor. The NTM eventually locks at t=0.95,
as evidenced by the rapid drop of the edge USXR signal and spike in
H~α~, followed by a disruption. It is possible that cases like these
begin as classical tearing modes, which transition to neoclassically
driven modes when the island width exceeds the critical island width
\[[^9]\]. This classical physics can be especially important for high-β
plasmas, where a pole in ∆' occurs at the ideal kink limit \[[^10]\].
This discharge had β~Ν~=4.3 when the mode struck, near to if not
exceeding the no-wall limit. Understanding this triggering will be an
important part of NSTX research.

![](media/image3.png){width="2.1243055555555554in" height="3.15625in"}

**Fig 2.1.1-3: Example of an m/n=2/1 NTM with no clear trigger.**

The marginal island width, i.e. the island width where dw/dt=0 for the
smallest β~P~ where NTMs are still metastable, is a key parameter for
understanding the relevant small island physics (w~d~ and w~pol~) and
extrapolating to larger and hotter plasmas. Experiments at conventional
aspect ratio in DIII-D \[[^11]\], JET \[[^12]\] and ASDEX-U \[[^13]\]
suggest that the marginal island width scales empirically with the
poloidal gyroradius. Verifying this empirical scaling at low aspect
ratio is particularly important, as it predicts that an ST reactor or
CTF (or indeed, ITER) will be very susceptible to NTM destabilization,
and that NTM control techniques must be implemented. β~P~ ramp down
experiments will be conducted to establish the marginal island widths
for both the 3/2 and 2/1 NTMs, as a function of parameters such as
rotation and poloidal gyroradius. It is planned to make a first complete
set of marginal island-width experiments in the FY09-FY11 time-frame.
However, the extended field and current capabilities of the upgraded
NSTX will expand the parameter regime in quantities like poloidal
gyroradius, collisionality, q­~min~ and q-shear, providing greater a
wider range of applicability for the results.

Plasma rotation and rotation shear enter the eqn. 1 in a number of
places, for instance through the polarization threshold term and through
modifications to ∆', as well as through mechanisms outside of the MRE,
such as modifications to the coupling between the rational surface and
the surface where triggering instabilities occur. Given that the
momentum input to a future reactor is likely to be much reduced compared
to present experiments, there has been recent interest in how rotation
and rotation shear experimentally impact the onset and saturation of
NTMs. Experiments in JET and DIII-D demonstrated the importance of
rotation in increasing the NTM threshold \[[^14]\]. For the 3/2 NTM,
this effect may be associated with dynamic rotation shielding between
the NTM resonant surface and the q=1 surface, where the seed
perturbation originates. Experiments in DIII-D have also demonstrated a
reduction in the onset β for 2/1 modes when the co-injection beam torque
is reduced to and beyond zero \[[^15]\].

Experiments in NSTX have already begun to study the dependence of the
2/1 mode onset on rotation shear. We have observed a \~20% reduction in
onset global *β~N~* as the rotation at q=2 is reduced. However, given
that the NTM evolution in eqn. (1) is determined by local parameters, we
have undertaken more systematic studies in terms of local bootstrap
drive and rotation shear. Fig. 2.1.1-4 shows the local drive bootstrap
drive for the island at mode onset,
$m_{0}\left\langle J \cdot B \right\rangle L_{q}\sqrt{\left\langle B_{P}^{2} \right\rangle}$
, as a function of the local rotation shear at q=2,
$\left( \frac{d\Omega}{\text{dR}} \right)t_{A}L_{s}$. The discharges in
the scan were chosen from a wide variety of experiments which utilized
n=3 braking and NBI torque to control the rotation. All were high-κ,
high-δ plasmas with I~P~=1MA and B~T~=0.45T. typical of high-performance
regimes in NSTX. There is a clear offset-linear dependence of the
bootstrap-drive on rotation shear when the mode strikes, implying the
importance of rotation shear in allowing higher-β operation.
Importantly, this is the first ever correlation of the local NTM
properties with rotation shear, expanding on the previous DIII-D work.

![](media/image4.png){width="2.8722222222222222in"
height="2.0347222222222223in"}

Figure 2.1.1‑4 Local bootstrap drive and a function of rotation shear at
q=2, for discharges high-κ, high-δ discharge. Both n=3 braking and
variations on the NBI are used to vary the rotation shear.

There may, however, be NTM physics outside the direct purview of eqn
(1). For instance, the tearing modes in TFTR \[[^16]\], AUG \[[^17]\],
and possibly other major tokamaks, are commonly coupled to core
kink-like modes. As these internal kink-like modes resemble the fishbone
instability, it is reasonable to view this as a mechanism through which
NTMs are coupled to the fast ion distribution. Evidence for such
coupling, e.g., strong frequency chirping, has not been seen as yet in
NSTX, though it has been observed in AUG \[[^18]\] and TFTR. However,
NSTX presently operates in a fast ion regime far from typical operating
regimes of present large aspect ratio tokamaks. The upgrade to the
center stack will allow higher field operation, bringing NSTX fast ion
parameters more in line with those of conventional tokamaks. This will
allow experiments to study the interactions of fast ions, including
potential drive mechanism, with NTMs.

These experimental tasks will be executed in parallel with modeling and
simulations of NTM dynamics at low aspect ratio. A three part modeling
effort is planned in this regard, involving the linear dispersion
relation describing the threshold for onset of the mode, the quasilinear
island evolution equation for intuitive study of the effects of low
aspect ratio, and the simulation of mode onset and evolution with
initial value codes. The presence of toroidal flow shear introduces a
number of physics issues into these analyses, which are critical to
understanding the experimental outcome. A series of experimental
discharges that exemplify the physics of flow shear effects in the onset
and evolution of resistive MHD instabilities will be compared to
theoretical predictions.

The effect of a sheared flow field on the coupling between modes can be
described by the solution to the dispersion relation
$\text{det}(D^{'} - D(Q)) = 0$ where $D^{'}$ includes both tearing
($D^{'}$) and interchange parities, $D(Q)$ is the solution vector inside
of the resonant layer, and $Q$ is the normalized growth rate. With flow
shear $Q$ becomes complex in the sense that the growth rate now has both
real and imaginary parts in the inner layer ${QºQ}_{r} + \text{iQ}_{i}$
where $Q_{r} = k \cdot V_{f}$ (ignoring poloidal precession from $w_{}$
effects) represents the toroidal angular frequency at the rational
surface, and $Q_{i}$ represents the mode growth rate. This directly
affects the coupling between modes in toroidal geometry, as addressed at
length in \[[^19]\] and recently published in Ref. \[[^20] as well as
the local stability via the inner layer physics, as described in
\[[^21]\], among several others.

In this research plan, we intend to compute the tearing and interchange
solutions for multiple rational surfaces using the PEST-III code, and
compute the inner layer solutions using the code described in Ref.
\[[^22]\] to solve the dispersion relation. The individual effects can
then be examined and compared with a similar approach to Refs.
\[xix,xxi\] using the NIMROD code.

A further step will be to implement solutions of the MRE (equation 1) in
full low aspect ratio geometry. These studies were begun in the work by
\[[^23]\], where PEST-III was used to calculate ∆', and routines from
NIMROD were used to calculate the bootstrap drive and D~R~. These steps
will allow full calculations of the island growth and decay rates, as
well as the saturated island eigenfunctions, for comparison with the
experimental measurements described above. A primary goal of these
efforts will be to assess the impact of the GGJ term in the context of
experimental measurements.

A more advanced modeling step will be to utilize initial value codes
such as M3D and NIMROD to simulate the non-linear evolution of NTMs.
Compared to the approach in equation (1), these methods are far more
advanced in their treatment of heat conduction around the island, the
effects of rotation shear, and self consistent current profile and
island width evolution.

A thorough theoretical and computational analysis of the effects that
low aspect ratio will have on the resistive stability will be
complimentary to the results from the experiment. Also, a comparison
with results from the high aspect ratio experiments will give us insight
to further develop our theoretical description of the physics of these
instabilities.

Our ability to meet these scientific goals will be augmented through
improved diagnostic coverage during this research period. NSTX plasmas
are overdense, preventing traditional measurements of island widths via
electron cyclotron emission (ECE); measurements to date have relied on
edge magnetics and USXR emission to constrain the island size and
location. However, the BES system currently under construction will
allow core fluctuation measurements with high frequency and spatial
resolution. The CIF-MSE system also has the ability to measure spatially
localized fast magnetic fluctuations at NTM relevant frequencies. This
measurement, combined with USXR emission and edge magnetics, should
allow detailed studies of the island eigenfunction, including the
coupling to internal kink-like modes if present.

**(ii) NTM Control**

The results in Fig. 2.1.1-1 illustrate that 2/1 NTMs must be avoided in
order to maintain optimal performance in NSTX. Experiments in DIII-D,
JT-60, and AUG have demonstrated the ability to suppress both the 2/1
and 3/2 NTM with ECCD replacing the missing bootstrap current. This
method is not available in the overdense NSTX plasma. Hence, passive
control techniques will be employed in the planned research period.

To date, the most successful means of mitigating the 2/1 NTM has been
through the use of combined lithium conditioning and dynamic error field
correction. This is illustrated in Fig. 2.1.1‑5, where the mode activity
is compared between discharges with and without lithium evaporation; the
rotation frequency in the core and at q=2 are also shown, based on CHERS
data and equilibrium reconstruction. Both discharges show a period of
MHD activity during and immediately after the current ramp, as the
q-profile evolves. The control shot 129013 then develops the large 2/1
mode, the plasma rotation is dramatically reduced, and the shot
ultimately disrupts at t=1 sec. after the mode locks. The discharge with
DEFC and lithium evaporation shows no sign of the later rotating
instability, and ultimately succumbs to a fast-growing external mode,
likely a resistive wall mode, at t=1.3 seconds.

![](media/image5.png){width="2.5409722222222224in"
height="1.8694444444444445in"}![](media/image6.png){width="2.7104166666666667in"
height="1.8694444444444445in"}

**Figure 2.1.1‑5: Comparison of low-frequency mode activity in shots
without (129013) and with (129072) with lithium evaporation. Also shown
are the core (green) and q=2 (magenta) rotation frequencies. The two
discharges, designed to determine the effect of lithium evaporation on
plasma performance, have all other controlled parameters the same.**

The stabilization noted in the above figure was achieved with both
lithium evaporation and DEFC; in other more rare cases, simply one of
these two ingredients is sufficient to suppress the mode for the
duration of the discharge. A high priority for near-term studies is to
understand how the current and rotation profiles with lithium and DEFC
change the mode stability, including changes in mode triggering. The
studies offer exciting prospects for NTM control, as the liquid lithium
divertor and NCC should allow a significant extension of these high
performance discharge regimes.

In the five-year period starting in FY2014, an additional 3 NBI sources
will be added to the NSTX NBI system. As discussed in Chapter X, this
upgrade should allow access to fully non-inductive equilibria with
*q~min\ ~*\> 2. This scenario would have the effect of eliminating the
pernicious m/n = 2/1 and 3/2 resonances from the plasma. The effect of
higher order NTMs, such as m/m=3/1 and 5/2, can then be studied.
Additionally, the current profile control allowed with the more
tangential beams may allow stabilization or mitigation of the NTM
through modifications to ∆', in a fashion similar to what was achieved
in COMPASS-D \[[^24]\] with lower-hybrid current drive.

SUMMARY OF RESEARCH PLANS AND TIMELINE

FY09-FY11 Complete characterization of NTM onset, including triggering
physics and the role of rotation, rotation shear, and small island
effects.

FY09-FY11 Utilize β~P~ ramp down experiments to characterize the
dependencies of the small island physics on kinetic parameters at low
aspect ratio

FY09-FY11 Utilize PEST-III for accurate calculations of ∆' in NSTX
geometry. Develop solutions to (1) utilizing low-aspect ratio solutions
for all terms. Asses merits of low-A for NTM stability

FY10-FY12 Implement nonlinear initial value simulations of NTMs with
NIMROD and M3D. Compare measured island seeding and evolution to
simulations.

FY12-FY13 Utilize upgraded NSTX to look for interactions of NTM with
fast particles

FY12-FY13 Utilize previously developed knowledge base and new tools to
develop routine high-β discharges with no deleterious island activity.

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