Controlled neoclassical tearing mode (NTM) healing by fueling pellets and its impact on electron cyclotron current drive requirements for complete NTM stabilization

The m/n  =  2/1 neoclassical tearing mode (NTM) is a major impediment in the development of operational scenarios of present and future tokamaks, as it degrades plasma confinement and, if large enough, often locks to the wall, slows down the plasma rotation and leads to plasma termination [1, 2] (m/n are the poloidal/toroidal mode numbers). NTMs are destabilized by a helical bootstrap current perturbation ($\newcommand{\djbs}{\delta {{\rm j}}_{\rm BS}} \djbs$) arising due to 'missing' pressure gradient ($\newcommand{\gradp}{\nabla {{\rm p}}} \gradp$) at the magnetic island O-point[3–5]. NTM stabilization has been achieved by electron cyclotron current drive (ECCD) in various H-mode scenarios by replacing $\newcommand{\djbs}{\delta {{\rm j}}_{\rm BS}} \djbs$ at the mode rational surface [6–9]. This control method is the leading candidate for active NTM stabilization in ITER burning plasmas. However, in ITER Pre-Fusion Power Operation-1 (PFPO-1), the high electron density ($n_{\rm e}$) at low magnetic field ($\newcommand{\B}{{B}} \B$) imposes a major concern for NTM stabilization via ECCD due to parasitic third harmonic absorption in the edge which lowers ECCD efficiency in the core [10]. Therefore, the avoidance of locked mode–induced disruptions in ITER start up operations would benefit from an active NTM control method that does not rely on high gyrotron power. Such a method may also improve the net electricity output of future reactors and/or could free up gyrotrons for heating and current drive elsewhere in the plasma.

A solution is offered by restoring $\newcommand{\gradp}{\nabla {{\rm p}}} \gradp$ at the island O-point which can be accomplished in practice by fueling pellets. Experimentally observed $\newcommand{\te}{{T}_{\rm e}} \te$ structures of magnetic islands are well described [11–14] by a simple anisotropic diffusion model [15] ($\newcommand{\te}{{T}_{\rm e}} \newcommand{\p}{{{\rm p}}} \newcommand{\pa}{{p}_\alpha} \chi_\perp\nabla_\perp^2\te+\chi_\parallel\nabla^2_\parallel\te=0$, where $\newcommand{\p}{{{\rm p}}} \newcommand{\pa}{{p}_\alpha} \chi_\parallel$ is the parallel electron thermal diffusivity), where $\newcommand{\te}{{T}_{\rm e}} \te$ is flat if $\newcommand{\p}{{{\rm p}}} \chi_\perp=0$ at the O-point, but finite $\newcommand{\p}{{{\rm p}}} \chi_\perp$ results in a boundary layer $\newcommand{\p}{{{\rm p}}} \newcommand{\lchi}{L_\chi} \lchi\propto\chi_\perp^{1/4}$ which decreases the $\newcommand{\djbs}{\delta {{\rm j}}_{\rm BS}} \djbs$ drive of the NTM. Inside the island, $\newcommand{\p}{{{\rm p}}} \chi_\perp$ is reduced to neoclassical level [12–14], in accord with local reduction of ion temperature gradient (ITG) and trapped electron mode (TEM) scale turbulence [16–18]. Under these conditions $\newcommand{\lchi}{L_\chi} \lchi\approx1.2$ cm [14], but in principle $\newcommand{\lchi}{L_\chi} \lchi\approx4$ cm is possible by restoring $\newcommand{\net}{\widetilde{n}} \net$ inside the island. This is viable through turbulence spreading [19], allowing $\newcommand{\net}{\widetilde{n}} \net$ to propagate radially [20] into the linearly stable O-point region, delocalizing the relation between $\newcommand{\net}{\widetilde{n}} \net$ intensity (and $\newcommand{\p}{{{\rm p}}} \chi_\perp$) and $\newcommand{\te}{{T}_{\rm e}} \nabla\te$, as reported in recent magnetic island experiments in DIII-D [21]. As gradients are turbulence drives, it is expected that a surge in gradients outside the island (caused by the pellet) can reduce the $\newcommand{\jbs}{{{\rm j}}_{\rm BS}} \jbs$ drive of the NTM through $\newcommand{\net}{\widetilde{n}} \net$ spreading.

Formation of magnetic islands due to pellets and triggered growth of pre-existing small magnetic islands by pellets have been observed [22–26]. Pellets have been also utilized to study the perpendicular particle transport in the O-point region of islands [27, 28]. NTM suppression by fueling pellets, however, has not been reported before.

In this paper we present controlled partial stabilization of NTMs by fueling pellets for the first time in DIII-D [29] and KSTAR [30] H-mode discharges, investigate possible physics origins of this behavior and discuss consequences on ECCD requirements for complete NTM stabilization. In these experiments, pellet injection from the high-field side allows density ($\newcommand{\deltan}{\delta n} \deltan_{\rm e}$) and temperature perturbations ($\delta T$) to propagate to the NTM rational surface in a few milliseconds, due to favorable polarization drift of the ablated pellet material [31]. DIII-D data shows that these enhance local gradients in the island region, which are turbulence drives, and concomitant low-$\newcommand{\net}{\widetilde{n}} k~\net$ levels in the expected range of the micro tearing mode (MTM) and trapped electron mode (TEM) instabilities. When the pellet–driven perturbations reach the island region, the islands shrink by maximum 20% and then recover in about 40 ms in DIII-D. In KSTAR, the islands shrink by maximum 30% and then recover in about 80 ms. Analyses suggest that classical stability can not change in this time scale and changes in the resistivity or the rotation are not the underlying causes of this behavior. Reduced $\newcommand{\jbs}{{{ j}}_{\rm BS}} \jbs$ drive due to enhanced cross-field transport inside the island offers a quantitative explanation of the reported phenomenon.

Complementing the experimental data, non-linear GENE [32] gyrokinetic simulations with magnetic islands were employed to investigate causality between elevated gradients outside of the island, turbulence spreading into the island and reduced $\newcommand{\jbs}{{{ j}}_{\rm BS}} \delta\jbs$ at the O-point region. These show that increased gradients outside of the island increase the $\newcommand{\net}{\widetilde{n}} \net$ radial correlation and $\newcommand{\net}{\widetilde{n}} \net$ spreading into the O-point region. The layer of finite gradients inside the island ($\newcommand{\lchi}{L_\chi} \lchi$) increases with $\newcommand{\net}{\widetilde{n}} \net$, which in turn reduces $\newcommand{\jbs}{{{ j}}_{\rm BS}} \delta\jbs$. Interestingly, the modified Rutherford equation (MRE) predicts that a transport enhancement inside the island can reduce the required $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ for complete NTM stabilization by 70%.

The paper is structured as follows. DIII-D experimental set-up and observations are described in sections 2 and 3, respectively, with KSTAR data included in section 3.4. Physics origins of the reported behavior are investigated in section 4. Evolution of DIII-D plasma profiles and low-k $\newcommand{\net}{\widetilde{n}} \net$ are discussed in section 5, gyrokinetic simulations in section 6 and implications on ECCD requirements in section 7. Section 8 summarizes the study.

The response of naturally occurring and rotating 2/1 NTMs to fueling deuterium pellets was studied in reproducible, stationary hybrid DIII-D H-mode plasma conditions. The plasma shape was biased up double null (figure 1(a)), the major radius of the magnetic axis was $R _\circ=173$ cm, minor radius was ${a}=60$ cm, elongation was $\kappa=1.83$. A steady plasma current of 1.4 MA was maintained, in the stationary state the neutral beam power was 4.65 MW (figure 1(b)), the line averaged density was $\newcommand{\ti}{{T}_{\rm i}} ~5.5\times10^{19}~{{\rm m}}^{-3}$ (figure 1(c)) (interferometry [33]), the toroidal magnetic field was B_T  =  −1.86 T. The square root of the normalized toroidal flux surface label of the q  =  2 rational surface was $\rho\approx0.68$. At this location the electron temperature ($\newcommand{\te}{{T}_{\rm e}} \te$, Thomson scattering [34]) and ion temperature ($\newcommand{\ti}{{T}_{\rm i}} \ti$, charge exchange recombination, CER [35]) were both approximately 1.5 keV (figure 1(d)). The normalized plasma beta was 1.39 ($\beta_{N}=\beta\,a{B}_ T/{I}_{\rm P}$) (figure 1(e)). The n  =  1 magnetic amplitude (Mirnov) started growing at about 2000 ms and saturated around 18 Gauss (figure 1(f)). The frequency of this mode is 15 kHz initially and 5 kHz in the stationary state and the poloidal mode number is m  =  2. The full width of the magnetic islands in the outboard mid-plane was $\newcommand{\wisl}{{W}} \wisl\approx8$ cm at q  =  2 as seen in electron cyclotron emission (ECE) [36] and ECEI [37] data in the discharges without pellets. In the discharges with pellets $\newcommand{\te}{{T}_{\rm e}} \te$ measurements are possible only with the slower Thomson scattering system as ECE is cut off due to the higher density. In addition, all of these plasmas exhibit an m/n  =  3/2 mode at q  =  1.5, which is responsible for maintaining $q_{\rm min}>1$ which keeps the core sawtooth free.

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During the flat top phase of the discharge when the NTM amplitude has saturated, 1.8 mm deuterium pellets are injected from the high-field side at 45 degree poloidal angle (figure 1(a)), at a rate of 10 Hz, starting at t  =  2318 ms. With typical velocities of 150–200 m s⁻¹, the pellets are ablated within 15 cm of the plasma edge, but the resulting $\newcommand{\deltan}{\delta n} \deltan_{\rm e}$ and $\delta T$ propagate to the island region in a few milliseconds due to polarization drifts [31]. Along the pellet trajectory, the q  =  2 surface is 31 cm from the last closed flux surface. A clear signature of the pellet is a surge in $n_{\rm e}$ (figure 1(c)). Concomitant drops of $\newcommand{\te}{{T}_{\rm e}} \te$ and $\newcommand{\ti}{{T}_{\rm i}} \ti$ leas to no or nearly no change in $\beta$ (figure 1(e)). These events are accompanied by a drop of the 2/1 amplitude by typically about 20% (figure 1(f)), which is evidence that core fueling pellets have stabilizing effect on pre-existing saturated 2/1 NTMs.

Several of the utilized pellets broke into two or more pieces before reaching the vacuum chamber, which resulted in the variation of the induced $\newcommand{\deltat}{\delta T} \deltat$ and $\newcommand{\deltan}{\delta n} \deltan$. Figure 1 shows the evolution of the line integrated $n_{\rm e}$ (g), $\newcommand{\te}{{T}_{\rm e}} \te$ and $\newcommand{\ti}{{T}_{\rm i}} \ti$ outside of the island on the low field side (h), 2/1 NTM magnetic amplitude (i), the raw Mirnov signal (j ) and the divertor $\newcommand{\da}{{D}_\alpha} \da$ signal (k) during the injection of the largest pellet that reached the plasma in one piece. The surge of $n_{\rm e}$ is accompanied by a drop in the local $\newcommand{\te}{{T}_{\rm e}} \te$ and $\newcommand{\ti}{{T}_{\rm i}} \ti$. Notice that the recovery of $\newcommand{\te}{{T}_{\rm e}} \te$ and $\newcommand{\ti}{{T}_{\rm i}} \ti$ takes about 60 ms, which is tracked by the n  =  1 NTM amplitude. Both natural ELMs and the pellets can cause the island to shrink. Additionally, the pellets trigger ELMs, which also interact with the island. These are discussed in sections 3.1–3.3, where phase-locking analysis of the n  =  1 magnetic amplitude is employed to isolate the effect of ELMs and pellets on the 2/1 NTM.

3.1. Effect of natural ELMs on NTMs

On top of the pellet-caused perturbation, faster drops of the NTM amplitude are also observed in figure 1(i), which consistently align with edge-localized modes (ELMs) in the $\newcommand{\da}{{D}_\alpha} \da$ signal in figure 1(k). These drops are not the ELM magnetic component, but reflect the shrinking and recovery of the 2/1 islands in about 10 ms after ELMs [38, 39]. This evolution is also seen directly in the raw Mirnov signal in figure 1(j). Additionally to the island response, some ELMs cause a sharp positive or negative spike in the Mirnov signal (but not in the slower n  =  1 signal), which lasts only for about 0.1 ms. These spikes are interpreted as the magnetic component of ELMs, not a rapid change in the n  =  1 NTM amplitude.

The $\newcommand{\te}{{T}_{\rm e}} \te(t,R)$ contour phase-locked to ELMs in the lower density shots in figure 2(a) shows that the ELM–induced 'cold-pulse' in the edge plasma propagates to the island on sub-millisecond time-scale. In sync, the m/n  =  2/1 island heals and then recovers in less than 10 ms (figure 2(b)). The maximum perturbation of the 'cold-pulse' reaches the island region about 5 ms after the ELM. This data is taken at lower density ($\newcommand{\ti}{{T}_{\rm i}} n_{\rm e}\approx3.2\times10^{19}~{{\rm m}}^{-3}$; without pellets), where the second harmonic ECE is not cut off, hence local measurements of $\newcommand{\te}{{T}_{\rm e}} \te$ are possible. On average, $\newcommand{\te}{{T}_{\rm e}} \te$ drops at q  =  2 by $5\%$ and the n  =  1 amplitude drops by $20\%$. The island structure can be monitored by phase-locking local measurements of $\newcommand{\te}{{T}_{\rm e}} \te$ to the island rotation frequency [14, 40, 41], which transforms the $\newcommand{\te}{{T}_{\rm e}} \te(R,t)$ data (from the lab frame) to $\newcommand{\te}{{T}_{\rm e}} \te(R,\xi)$ (to the island frame). Here $\newcommand{\p}{{{\rm p}}} \xi=m\theta-n\phi$ is the helical phase of the island, $\theta$ and $\newcommand{\p}{{{\rm p}}} \phi$ are the poloidal and toroidal angles, respectively. Figures 2(c) and (d) show the $\newcommand{\te}{{T}_{\rm e}} \te(R,\xi)$ contours across the island before and after the ELM, respectively. Increase of $\newcommand{\te}{{T}_{\rm e}} \nabla\te$ inside the island after the ELM is indicated by the fact that the $\newcommand{\te}{{T}_{\rm e}} \te$ contours are more dense compared to before the ELM. Interestingly, when $1.7~{{\rm W~cm}}^{-3}$ EC power is deposited in the island region in the same discharge, the power balance $\newcommand{\p}{{{\rm p}}} \chi_\perp$ of the background plasma (calculated with TRANSP) increases from 1.7 m² s⁻¹ to 5 m² s⁻¹. At the same time, the drop of $\newcommand{\te}{{T}_{\rm e}} \te$ at q  =  2 as well as the drop of the n  =  1 magnetic signal approximately doubles (not shown), hence the EC heating and/or current drive enhances the NTM response to ELMs. Increased $\newcommand{\te}{{T}_{\rm e}} \nabla\te$ inside the island 5 ms after the ELM, compared to 2 ms before the ELM, is more clearly shown in the $\newcommand{\te}{{T}_{\rm e}} \newcommand{\p}{{{\rm p}}} \te(R,\xi=\pi)$ radial profiles across the O-point in figure 2(e). Ensemble averaging the data over 11 (24) ELMs in the window with (without) EC heating and current drive shows that $\newcommand{\te}{{T}_{\rm e}} \nabla\te$ increases by $\newcommand{\p}{{{\rm p}}} 52~{{\rm eV~cm}}^{-1}\pm 29\%$ ($\newcommand{\p}{{{\rm p}}} 12~{{\rm eV~cm}}^{-1}\pm 41\%$) at the island separatrix in line with the O-point ($\newcommand{\p}{{{\rm p}}} R =207.9~{{\rm cm}}, \xi=\pi$) due to ELMs.

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Similarly to the above described lower density shots, the $n_{\rm e}$ and $\newcommand{\te}{{T}_{\rm e}} \te$ perturbations caused by ELMs both propagate to q  =  2 in the discharges with pellets and cause $n_{\rm e}$ ($\newcommand{\te}{{T}_{\rm e}} \te$) to drop by about 5% (10%) (figure 3(a)). In sync, the n  =  1 magnetic signal drops with a similar character as in the low density shot (figure 3(d)). Here the $n_{\rm e}$, $\newcommand{\te}{{T}_{\rm e}} \te$ and n  =  1 magnetic signal were phase-lock averaged within the flat top phase of a single discharge (DIII-D #174968), where 70 ELMs and eight pellets were available for the analysis. The uncertainty of the drop of the ensemble averaged n  =  1 signal is characterized by the precision of the mean before ELMs and after ELMs for which the $\bar{\sigma}$ standard error is the metric ($\bar{\sigma}=\sigma/\sqrt{N-1}$ where $\sigma$ is the standard deviation and N is the sample size). $\bar{\sigma}$ is shown by the shaded regions in figure 3. The results shown in figure 3 are in quantitative agreement with two repeat discharges (DIII-D #174964 and #174965).

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3.2. Triggered ELMs

3.3. Effect of pellets

3.4. NTM partial stabilization due to pellets in KSTAR

The DIII-D observation of NTM partial stabilization by fueling pellets was also tested in KSTAR H-mode plasmas. The relevant plasma parameters were: $R _\circ=1.8$ m, ${a}=0.5$ m, BT  =  1.8 T, ${I}_{\rm p}= 400$ kA, $q_{95} \approx 6.2$. These plasmas developed an n  =  1 magnetic mode rotating with 10 kHz frequency, which is consistent with a 2/1 NTM. The major radius coordinate of q  =  2 is expected to be in the $R _{q=2}=1.9{{\rm \mbox{--}}}2.0$ m range based on EFIT constrained by the motional Stark effect diagnostic (MSE [44]). Based on Mirnov data, the width of these islands is expected to be $\newcommand{\wisl}{{W}} \wisl=2{{\rm \mbox{--}}}3$ cm, which is significantly smaller than the DIII-D 2/1 islands. Deuterium pellets with cylindrical shape (2 mm diameter and 1.5 mm length) were injected from the high field side at near 20 degree poloidal angle and entered the plasma with an average speed of 211 m s⁻¹. Reduction of the 2/1 island amplitude after pellet injection is seen from the time history of the n  =  1 Mirnov amplitude with the most significant response after the injection at about 11.15 s in figure 4. Here the n  =  1 signal briefly shrinks and recovers after ELMs as in the DIII-D discharge and then shrinks by about 30% for about 80 ms at the time of the pellet injection. Broadband magnetic fluctuations increase during this time window. This data demonstrates that the pellet has a stabilizing effect on the 2/1 mode in the KSTAR plasma, in line with the DIII-D results.

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Due to limited run-time in the KSTAR experiment, some diagnostics data required for further analysis is not available. Investigation of physics origins of the partial stabilization is based upon DIII-D data in section 4 and thereafter.

As mentioned in section 1, partial stabilization of the 2/1 NTM due to pellets can not be explained by (i) a change in the classical stability parameter $\Delta'$, (ii) a rotation drop or (iii) a resistivity surge ($\newcommand{\te}{{T}_{\rm e}} \newcommand{\p}{{{\rm p}}} \newcommand{\e}{{\rm e}} \eta\propto \te^{-3/2}$), but is consistent with (iv) increased cross-field transport ($\newcommand{\p}{{{\rm p}}} \chi_\perp$) at the island O-point.

Some of these effects can be discussed within the framework of the Modified Rutherford equation that prescribes NTM evolution. The simplest form including classical stability (first term), the neoclassical bootstrap current drive (second term) and the ECCD current (third term) reads:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\pa}{{p}_\alpha} \newcommand{\dnc}{{D}_{\rm NC}} \newcommand{\lchi}{L_\chi} \newcommand{\wisl}{{W}} \newcommand{\p}{{{\rm p}}} \displaystyle \label{eq:mre} \frac{{\rm d}\wisl}{{\rm d}t}=\frac{\eta}{\mu_\circ}\Delta'+\frac{\eta}{\mu_\circ}\frac{\dnc\wisl}{\wisl^2+\lchi^2}+\pa\frac{\alpha_\circ}{\wisl}. \nonumber \end{align} \tag{ 1 }$$

Here $\mu_\circ$ is the vacuum permittivity, $\newcommand{\p}{{{\rm p}}} \newcommand{\jbs}{{{ j}}_{\rm BS}} \newcommand{\dnc}{{D}_{\rm NC}} \dnc\propto\jbs\propto\nabla {p}$ is the neoclassical bootstrap current drive and $\newcommand{\lchi}{L_\chi} \lchi$ is the cross-field electron thermal transport length-scale (recall that $\newcommand{\p}{{{\rm p}}} \newcommand{\lchi}{L_\chi} \lchi\propto\chi_\perp^{1/4}$). $\alpha_\circ$ is a constant negative parameter quantifying the magnitude of externally driven $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ at the O-point at complete stabilization. $\newcommand{\p}{{{\rm p}}} \newcommand{\pa}{{p}_\alpha} \pa$ is then a dimensionless parameter between 0 (no $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$) and 1 (complete stabilization).

• (i)  
  Classical stability. The islands shrink about 10 ms after injections, which is much faster then the time-scale required for the current profile to evolve. Therefore, it can be assumed that $\Delta'$ is nearly constant when the island shrinks due to the pellets.
• (ii)  
  Rotation. Former DIII-D experiments [45] reported a decreased $\beta$ threshold for 2/1 NTM onset at lower rotation, implying that 2/1 NTMs are more unstable at lower rotation. The pellets are injected radially, increasing the plasma density, without delivering angular momentum. This reduces the plasma rotation which then decreases the island frequency. Therefore, based on earlier studies, the reduction of island frequency due to the pellets can not explain the observed partial stabilization.
• (iii)  
  Temperature. Drops in $\newcommand{\te}{{T}_{\rm e}} \te$ increase $\newcommand{\e}{{\rm e}} \eta$, which in turn increases the growth rate classically and neoclassically. Considering these two terms, the saturated island width ($\newcommand{\wsat}{{W}_{\rm SAT}} \wsat$) is independent of $\newcommand{\e}{{\rm e}} \eta$ (at $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd=0$). Therefore, a change in $\newcommand{\e}{{\rm e}} \eta$ due to the drop in $\newcommand{\te}{{T}_{\rm e}} \te$ can not explain the shrinking of the island. In addition to the above, a drop in $\newcommand{\te}{{T}_{\rm e}} \te$ in the island O-point region could also drive the NTM more unstable through radiation [46].
• (iv)  
  Turbulent transport. Increased $\newcommand{\net}{\widetilde{n}} \net$ enters the MRE through $\newcommand{\lchi}{L_\chi} \lchi$, the boundary layer inside the island where gradients are maintained [15], which can vary in the 1.2–4 cm range, as reported previously [40]. This reduces the size of the $\newcommand{\te}{{T}_{\rm e}} \te$ flat spot at the O-point region compared to the size of the magnetic island, reducing the destabilizing $\newcommand{\djbs}{\delta {{\rm j}}_{\rm BS}} \djbs$. $\newcommand{\wsat}{{W}_{\rm SAT}} \wsat$ is given by the stable root of $\newcommand{\wisl}{{W}} d\wisl/dt=0$. The relative change of $\newcommand{\wsat}{{W}_{\rm SAT}} \wsat$ (at $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd=0$) due to a change in $\newcommand{\lchi}{L_\chi} \lchi$ is approximately:

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lchib}{\bar{L}_\chi} \newcommand{\lchi}{L_\chi} \newcommand{\wo}{{W}_\circ} \displaystyle \label{eq:frac} R_W=\frac{\lchib^2/\wo^2}{1+\lchib^2/\wo^2}\left(1-\sqrt{R_\chi}\right). \nonumber \end{align} \tag{ 2 }$$

  Here $\newcommand{\wo}{{W}_\circ} \wo$ is the unperturbed island width, $\newcommand{\lchi}{L_\chi} \newcommand{\lchib}{\bar{L}_\chi} \lchi=\lchib$$\newcommand{\lchi}{L_\chi} \newcommand{\lchib}{\bar{L}_\chi}(\chi_{\rm turb.}/\chi_{\rm neocl.}){}^{1/4}$ and $R_\chi=\chi_{\rm turb.}/\chi_{\rm neocl.}\approx10^2$. $\newcommand{\lchi}{L_\chi} \newcommand{\lchib}{\bar{L}_\chi} \lchib$ is the boundary layer in the unperturbed island and the ratio of transport coefficients was measured in earlier work [14]. Under these conditions, equation (2) estimates that $\newcommand{\wsat}{{W}_{\rm SAT}} \wsat$ can shrink by up to about $20\%$ due to a $\newcommand{\p}{{{\rm p}}} \chi_\perp$ enhancement inside the island, which is in quantitative agreement with the observations of section 3. We note that the modified transport may also influence the $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ directly [47, 48], which is neglected here for simplicity.

In the following, profile and turbulence data analyses as well as gyrokinetic simulations with magnetic islands will be in the focus of the paper. DIII-D data and GENE simulations qualitatively support the hypothesis that pellet–induced turbulent transport enhancement near the O-point causes islands to shrink.

In this section, we first present measurements of density profiles and temperature gradients (section 5.1). The pellets lead to increased gradients in the island region, which are turbulence drives, hence $\newcommand{\net}{\widetilde{n}} \net$ increase in the island region is expected. This is followed by GENE linear stability analysis, showing significant increase of MTM and TEM growth rates in the island region due to the pellet (section 5.2). These are consistent with beam emission spectroscopy measurements of $\newcommand{\net}{\widetilde{n}} \net$ inside the island showing increased low-k turbulence after the pellet (section 5.3).

5.1. Plasma profiles

The $n_{\rm e}(t,\rho)$ contour in figure 5(a) shows that in response to the pellet $\newcommand{\deltan}{\delta n} \deltan_{\rm e}$ propagates radially inward until it reaches the island region ($\rho\approx0.59{{\rm \mbox{--}}}0.79$) where it is confined for about 30 ms. This transport behavior is qualitatively similar to that reported in 1/1 RMP islands in LHD [27]. $n_{\rm e}(\rho)$ profiles at four different times are shown in figure 5(b). These four times are:

• t₁: 3 ms before injection (reference). The plasma is in equilibrium and $\nabla n_{\rm e}$ is small inside the island region.
• t₂: 6 ms after injection; maximum of $\newcommand{\te}{{T}_{\rm e}} -\nabla\te$ at q  =  2. When the pellet starts ablating, a peak develops first at the top of the pedestal around $\rho=0.87$ which then grows and propagates inward.
• t₃: 14 ms after injection; maximum of $\newcommand{\ti}{{T}_{\rm i}} -\nabla\ti$ at q  =  2. The contour of $n_{\rm e}(t,\rho)$ (figure 5(a)) and the $n_{\rm e}(\rho)$ profiles (figure 5(b)) show that $\nabla n_{\rm e}$ is reversed inside the island region at this time. This implies a reduced $\newcommand{\jbs}{{{\rm j}}_{\rm BS}} \delta\jbs$ drive of the NTM.
• t₄: 40 ms after injection; maximum of $-\nabla n_{\rm e}$. $\newcommand{\te}{{T}_{\rm e}} -\nabla\te$ and $\newcommand{\ti}{{T}_{\rm i}} -\nabla\ti$ at q  =  2 are about equal to before pellet injection.

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Time traces in figure 5(c) show that $\newcommand{\te}{{T}_{\rm e}} -\nabla\te$ increases first (t₂), followed by an increase in $\newcommand{\ti}{{T}_{\rm i}} -\nabla\ti$ inside the island region. Note that due to the short electron-electron collision time the drop in $\newcommand{\te}{{T}_{\rm e}} \te$ is almost instantaneous, which is detected by the Thomson scattering diagnostic. In contrast, the thermalization of the pellet deuterium ions with the probed carbon impurity takes place on the much longer ion–ion collision time, which explains the delayed carbon impurity $\newcommand{\ti}{{T}_{\rm i}} \ti$ response, as detected by the CER diagnostic. When the inward propagating density peak passes q  =  2, $\nabla n_{\rm e}$ increases temporarily until the $n_{\rm e}$ peak decays.

Note that $\newcommand{\te}{{T}_{\rm e}} \nabla\te$ and $\newcommand{\ti}{{T}_{\rm i}} \nabla\ti$ at q  =  2 relax in about 40 ms as the perturbations propagate further in but it takes a longer time until the local $\newcommand{\te}{{T}_{\rm e}} \te$ and $\newcommand{\ti}{{T}_{\rm i}} \ti$ recover. The n  =  1 amplitude recovers on the time-scale of the gradients.

MSE data is available during these shots, but measurement limitations seem to preclude direct measurement of current profile changes with the pellets. Due to the limited time response of the utilized lock-in amplifiers, the fast change in density and light intensity caused by a pellet leads to a transient spurious polarization measurement, i.e. the inferred pitch angle changes too much, too quickly to be physically real. This masks the actual local magnetic field response at the NTM rational surface.

As gradients are turbulence drives, it is expected that turbulence driven fluctuations increase after the pellet. The type and growth rates of the fastest growing micro-instabilities was calculated with linear GENE gyrokinetic simulations in section 5.2.

5.2. Linearly unstable drift waves at q  =  2 (GENE)

5.3. Evolution of low-k turbulence (BES)

The BES system [49] provides radially resolved measurements of $\newcommand{\net}{\widetilde{n}} \net$ locally by measuring the light emission from the neutral beam at $\newcommand{\p}{{{\rm p}}} \phi=150^\circ$, 3.6 cm above the mid-plane across the 2/1 island region (figure 6(a)). This system is sensitive to localized, long-wavelength $\newcommand{\net}{\widetilde{n}} \net$ of $k_\theta<1~{\rm cm}^{-1}$ which is the expected wavenumber range of the MTM instability in these plasmas. The typical separation between the channels is about 1 cm, the radial and poloidal localization is about 1 cm.

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The BES system detects increased turbulence intensity ($\newcommand{\net}{\widetilde{n}} \net^2$) by 20% inside the island region, after the pellet (compared to $\newcommand{\net}{\widetilde{n}} \net^2$ before the pellet). This is seen in the radial profile of the $\newcommand{\net}{\widetilde{n}} \net$ RMS increase in figure 6(b). Coherence spectra of 50 ms windows between vertically adjacent channels inside the island show that the increase of $\newcommand{\net}{\widetilde{n}} \net$ is in the broad and high frequency $\Delta f=100$–200 kHz range, which is consistent with increased broadband turbulence inside the island resulting from pellet injection (figure 6(c)). Here only ELM-free windows are analyzed, when the modulated neutral beam was on. Example time trances of $\newcommand{\net}{\widetilde{n}} \net^2$ inside and outside the island are shown in figures 6(d) and (e), respectively. The n  =  1 magnetic amplitude (figure 6(f)) starts to decrease when $\newcommand{\net}{\widetilde{n}} \net^2$ is increased inside the island. This suggests a multi-scale interaction between low-k $\newcommand{\net}{\widetilde{n}} \net$ and the 2/1 NTM. However, this data does not fully explain the observations as the the island remains shrunk for about 30 ms (as long as local gradients are modified) even after the BES $\newcommand{\net}{\widetilde{n}} \net^2$ has decayed back to the level of before pellet injection. Possible explanations include (i) increase of intermediate-k turbulence during the whole 30 ms window which is not detected by BES, (ii) the resistive time-scale of the NTM evolution restraining instantaneous island recovery.

To investigate causality between elevated gradients outside of the island, turbulence spreading into the island and reduced NTM drive, we used static magnetic islands in GENE nonlinear gyrokinetic turbulence simulations in three-dimensional, toroidal geometry. This is accomplished by qualitatively testing if increased background gradients can lead to stronger turbulence spreading into the magnetic island O-point region and if this effect shrinks the area where the temperature is flat. The islands are implemented through the parallel component of the vector potential [50]. GENE self-consistently solves the gyrokinetic-Maxwell system of equations on a fixed grid in five dimensional phase space (plus time). Two particle species (deuterons and electrons) were used including electromagnetic effects. The equilibrium magnetic configuration (large aspect-ratio, circular model equilibrium) is characterized by a safety factor of q  =  1.5, magnetic shear of $\hat{s} = 0.16$ and inverse aspect ratio of $\newcommand{\e}{{\rm e}} \epsilon = 0.19$. In our reference simulation R/L_n  =  2.2 and $\newcommand{\te}{{T}_{\rm e}} \newcommand{\rlti}{R/ L_{{T}_i}} R/L_{\te} =\rlti=5$ ($\newcommand{\ti}{{T}_{\rm i}} \newcommand{\p}{{{\rm p}}} \newcommand{\pa}{{p}_\alpha} \newcommand{\rlti}{R/ L_{{T}_i}} \rlti=R/(\ti/\partial_r\ti)$ and similar definitions hold for R/L_n and $\newcommand{\te}{{T}_{\rm e}} \newcommand{\rlte}{R/ L_{\te}} \rlte$). These equilibrium parameters are similar to the 'Cyclone DIII-D base case parameter set' [51] (discharge #81499) as in other gyrokinetic simulations [52, 53], where the most unstable micro-instability is the ion temperature gradient (ITG) mode. For clarity, we point out that the studied NTM in the DIII-D pellet experiment is characterized by m/n  =  2/1 mode numbers and the dominant linearly unstable drift waves at q  =  2 of the background plasma are the MTM and the TEM instabilities, see sections 2 and 5.2, respectively. Conducting non-linear gyrokinetic simulations with magnetic islands and input parameters (${T}_e/{T}_i$, $\nabla{T}_e$, $\nabla{T}_i$, $\nabla n$, geometry, rotation, etc) carefully matched to a specific experiment would require extensive simulation effort which is beyond the scope of this paper. These simulations therefore only qualitatively test if increased background gradients can lead to stronger turbulence spreading into the magnetic island O-point region and if this effect shrinks the temperature flat spot.

The first comparison of GENE turbulence simulations with magnetic islands to DIII-D turbulence data showed that GENE can qualitatively reproduce the measured scaling of $\newcommand{\net}{\widetilde{n}} \net$ modifications inside and outside the island with respect to the island width [17]. Here we attempt to model the effect of the pellet only on the dominant instability drive near the island, $\newcommand{\ti}{{T}_{\rm i}} \nabla\ti$, hence a set of simulations in the experimentally relevant range of $\newcommand{\rlti}{R/ L_{{T}_i}} \rlti=5{{\rm \mbox{--}}}9$ are comducted, where $\newcommand{\rlti}{R/ L_{{T}_i}} \rlti=5$ and 9 correspond to 'before pellet' and 'after pellet', respectively. The simulations are started with uniform gradients where ITG modes are unstable. Next, turbulence, transport and the profile perturbations are evolved self-consistently on a much longer time-scale than the turbulence correlation time, after the system has reached statistical equilibrium.

Snapshots of the instantaneous $\widetilde{n}(r,\xi)$ in the saturated state show that $\newcommand{\net}{\widetilde{n}} \net$ is reduced in the vicinity of the O-point when $\newcommand{\rlti}{R/ L_{{T}_i}} \rlti=5$ (figure 7(a)) but $\newcommand{\net}{\widetilde{n}} \net$ spreads into the O-point region when $\newcommand{\rlti}{R/ L_{{T}_i}} \rlti=9$ (figure 7(b)). Radial profiles of $\newcommand{\net}{\widetilde{n}} \net_{\rm RMS}$ (the RMS of $\newcommand{\net}{\widetilde{n}} \net$) across the O-point (figure 7(c)) show that $\newcommand{\net}{\widetilde{n}} \net$ is increased by $40\%$ outside and decreased by $75\%$ inside the island when $\newcommand{\rlti}{R/ L_{{T}_i}} \rlti=5$. In contrast, when $\newcommand{\rlti}{R/ L_{{T}_i}} \rlti=9$, $\newcommand{\net}{\widetilde{n}} \net$ is increased by a factor of three outside the island, and $\newcommand{\net}{\widetilde{n}} \net$ is nearly unchanged at the O-point due to increased $\newcommand{\net}{\widetilde{n}} \net$ penetration into the island (figure 7(c)). Here we define $\newcommand{\lntilde}{L_{\widetilde{n}}} \lntilde$ as the width of the boundary layer inside the island separatrix, where $\newcommand{\net}{\widetilde{n}} \net$ is equal to $\newcommand{\net}{\widetilde{n}} \net$ of the reference simulation without pellet and without island. Time averaged helical k-spectra $\langle\widetilde{n}(k_\xi)\rangle_t$ in figure 7(d) show that ITG modes are more unstable primarily in the lower-k part of the spectrum. This increases the radial correlation length of $\newcommand{\net}{\widetilde{n}} \net$, which then offers an explanation of increased $\newcommand{\net}{\widetilde{n}} \net$ penetration into the island.

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As expected, increased $\newcommand{\net}{\widetilde{n}} \net$ is accompanied by increased $\newcommand{\te}{{T}_{\rm e}} \nabla\te$ in the boundary layer inside the island separatrix (figure 7(e)). Here we define $\newcommand{\lchi}{L_\chi} \lchi$ as the width of the boundary layer inside the island separatrix, where $\newcommand{\te}{{T}_{\rm e}} \nabla\te$ is maintained at least at a 50% level of the reference simulation.

The scaling of $\newcommand{\lchi}{L_\chi} \lchi$ and $\newcommand{\lntilde}{L_{\widetilde{n}}} \lntilde$ versus $\newcommand{\rlti}{R/ L_{{T}_i}} \rlti$ in figure 8 shows that $\newcommand{\net}{\widetilde{n}} \net$ and $\newcommand{\te}{{T}_{\rm e}} \nabla\te$ gradually increase inside the island separatrix as the background $\newcommand{\ti}{{T}_{\rm i}} \nabla\ti$ is increased. As $\newcommand{\rlti}{R/ L_{{T}_i}} \rlti$ is increased from 5 to 9, $\newcommand{\net}{\widetilde{n}} \net$ at the O-point increases to that of the background plasma (w/o island and w/o pellet) and $\newcommand{\lchi}{L_\chi} \lchi$ widens by a factor of three, reducing the $\newcommand{\te}{{T}_{\rm e}} \te$ flat spot at the O-point to about 70% of $\newcommand{\wisl}{{W}} \wisl$. Therefore, these gyrokinetic turbulence simulations with magnetic islands demonstrate the causality between reduced NTM drive and elevated background gradients through turbulence spreading. This simulation result makes it conceivable that the increased $\newcommand{\net}{\widetilde{n}} \net$ inside the island detected by BES can be responsible for the shrinking of the NTM in the presented DIII-D experiment.

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NTM growth rates calculated from equation (1) are depicted in figure 9. The growth rate in equilibrium before pellet injection is shown by (i), where $\newcommand{\wsat}{{W}_{\rm SAT}} \wsat=8$ cm (matched to experiment), $\newcommand{\p}{{{\rm p}}} \newcommand{\pa}{{p}_\alpha} \pa=0$ (no $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$) and the cross-field thermal transport coefficient at the O-point is $\newcommand{\p}{{{\rm p}}} \newcommand{\mts}{{{\rm m}}^2/{{\rm s}}} \chi_\perp^\circ=10^{-2}\, \rm m^2 \,s^{-1}$. Increasing $\newcommand{\p}{{{\rm p}}} \chi_\perp^\circ$ to the level of the background plasma transport ($\newcommand{\p}{{{\rm p}}} \newcommand{\mts}{{{\rm m}}^2/{{\rm s}}} \chi_\perp^\circ=1 \,\rm m^2 \,s^{-1}$) leads to the shrinking of $\newcommand{\wsat}{{W}_{\rm SAT}} \wsat$ by $20\%$ (ii), in quantitative agreement with the experiment reported in section 3.

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In comparison, (iv) 50% $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ has the equivalent effect on $\newcommand{\wsat}{{W}_{\rm SAT}} \wsat$ as the pellet. This $50\%$ is relative to the amount of $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ needed for complete stabilization w/o pellet (i.e. $\newcommand{\p}{{{\rm p}}} \newcommand{\pa}{{p}_\alpha} \pa=0.3$). Interestingly, with transport enhancement (iii) ($\newcommand{\p}{{{\rm p}}} \newcommand{\mts}{{{\rm m}}^2/{{\rm s}}} \chi_\perp^\circ=1\, \rm m^2 \,s^{-1}$), only $30\%$ of $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ is enough for complete stabilization of the 2/1 island. In this case the island self-stabilizes at $\newcommand{\wisl}{{W}} \wisl\approx 4.5$ cm. This indicates that a fueling pellet can significantly reduce the gyrotron requirements for the stabilization of NTMs.

Increasing $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ leads to smaller $\newcommand{\wisl}{{W}} \wisl$ until the marginally stable point is reached and the island self-stabilizes (marked with circles), as shown in figure 10(a) at three different values of $\newcommand{\p}{{{\rm p}}} \chi_\perp^\circ$. Note that increasing the transport by a factor of 100 has three consequences: (i) $\newcommand{\wisl}{{W}} \wisl$ shrinks by up to 20% when $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd=0$, (ii) W decreases more rapidly as $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ is increased and (iii) the self-stabilizing $\newcommand{\wisl}{{W}} \wisl$ increases from 4 cm to about 4.5 cm. These three beneficial effects combine to reduces the $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ requirement for NTM stabilization to 30%.

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Controlled partial stabilization of core m/n  =  2/1 neoclassical tearing modes (NTMs) by fueling deuterium pellets is demonstrated in H-mode plasmas in the DIII-D and KSTAR tokamaks. In the framework of non-linear NTM theory, this observation can not be explained by classical stability, resistivity surge or rotation drop caused by the pellet, but is consistent with increased cross-field transport at the island O-point. When the pellet–induced thermal perturbation ($\delta T$) reaches the island in DIII-D, local low-k turbulence ($\newcommand{\net}{\widetilde{n}} \net$) increases and the island shrinks. This $\newcommand{\net}{\widetilde{n}} \net$ is in the expected range of the MTM and TEM instabilities (GENE). When the pellet–induced perturbations decay, $\newcommand{\net}{\widetilde{n}} \net$ restores to the normal level and the island recovers. Complementing the experimental data, non-linear GENE gyrokinetic simulations with magnetic islands qualitatively support that increased background temperature gradients can increase $\newcommand{\net}{\widetilde{n}} \net$ spreading into the O-point region. This can enhance the cross-field transport across the island separatrices and thereby reduce the NTM drive. The change in saturated island width resulting from transport enhancement at the O-point in the MRE shows quantitative agreement with the experiment. Furthermore, the MRE predicts significant reduction of the $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ requirement for complete NTM stabilization when the transport is enhanced at the island O-point.

As NTMs are non-linear instabilities with a threshold island width of a few centimeters, NTMs self-stabilize when the island width shrinks below the threshold. After stabilization, the discharge may run NTM free until a large enough trigger destabilizes the NTM again. In such event, the suppression procedure must be repeated as in the'catch and subdue' (C&S) control method [9]. In C&S the gyrotrons track the q  =  m/n surface using real-time mirror steering and the island is suppressed by turning the ECCD on when the NTM amplitude exceeds a threshold. C&S is more cost efficient than 'preemptive' stabilization (where the ECCD is constant on and aiming at q  =  m/n). C&S NTM control with combined ECCD & pellet may offer a path toward further reducing the $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ needed for complete NTM stabilization.

Active NTM control with low gyrotron power is desirable, as it can (i) improve the net electricity output of future reactors, (ii) can free up gyrotrons for heat and current drive elsewhere in the plasma and (iii) can extend the active NTM control solution via ECCD to operational regimes where the current drive efficiency is low. In particular, in ITER startup operations, EC current drive efficiency at q  =  2 at the second harmonic resonance is predicted to be lower due to partial absorption of the EC waves at the third harmonic resonance in the edge. Further experiments with $\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ scans have been proposed in DIII-D to explore the potential of combined pellet—$\newcommand{\jeccd}{{j}_{\rm ECCD}} \jeccd$ control of the 2/1 NTM.

The authors would like to thank Francesca Turco and Christopher T. Holcomb for very useful discussions.

This research was supported by the General Atomics Postgraduate Research Participation Program administered by ORAU. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Award ¹DE-FC02-04ER54698. DIII-D data shown in this paper can be obtained in digital format by following the links at https://fusion.gat.com/global/D3D_DMP. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility 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 views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

The GENE gyrokinetic computations have been carried out at the MARCONI-Fusion supercomputer at CINECA, Italy.

KSTAR experiments were supported by Korea Ministry of Science and ICT under NFRI R&D programs (NFRI-EN1901-10 and NFRI-EN1941-5).