Core electron temperature turbulence and transport during sawtooth oscillations in the DIII-D tokamak

Sawteeth are one of the concerning instabilities in ITER and future burning plasma experiments. Sawtooth dynamics and its interaction with broadband plasma turbulence has been a challenge for predictive simulations of core transport in future fusion devices. This study provides new observations of core turbulence behavior during sawtooth oscillations in DIII-D hydrogen L-mode neutral beam injection heated plasmas in an inner wall limited configuration. A strong correlation of electron temperature and density turbulence levels with the sawtooth oscillation phase has been observed at locations inside the T e inversion radius and/or safety factor q = 1 magnetic surface. The T e turbulence amplitude in the core during the sawtooth ramp exhibits a critical T e gradient behavior inside but not near the T e inversion radius/q = 1 magnetic surface. The most unstable mode calculated from the trapped gyro-landau fluid turbulence simulations reveal a change from low-k ion-type to low-k electron-type modes from pre- to post- sawtooth crash time periods.


Introduction
Sawteeth [1] are one of the concerning instabilities in ITER and future burning plasma experiments as they periodically flatten the core pressure profile, can trigger NTMs [2], and expel energetic particles to degrade plasma performance [3].Large sawteeth may destabilize ELMs [4], which are a serious threat to ITER operation.A sawtooth cycle includes a fast Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.collapse phase during which electron temperature (T e ) profile inside a certain plasma minor radius (the inversion radius) drops abruptly, and a sawtooth ramp phase during which T e profile recovers until the next sawtooth crash.There is still no widely accepted theory for the sawtooth phenomenon that is consistent with experimental observations [5].The leading theories explain the fast core profile collapse at the sawtooth crash because of magnetic reconnection near safety factor q = 1 magnetic surface (e.g.[6]).Turbulence may play an important role in a magnetic reconnection process [6][7][8][9].Multiple theories (e.g.see a summary in [10]) have been proposed to explain the trigger mechanism of the fast sawtooth crash behavior including resistive two-fluid MHD, collisionless kinetic effects, widespread magnetic turbulence as the island reaches a critical width, etc.Some models suggested that sawteeth may be triggered by a critical level of turbulence [11].Currently, most turbulence simulations codes ignore the region inside the sawtooth inversion radius.Sawtooth dynamics and its interaction with broadband plasma turbulence has been a challenge for predictive simulations [12,13] of core transport in future fusion devices.
Turbulent enhancement of particle and heat fluxes during sawtooth oscillations at the edge has been reported on the TEXT tokamak [14].Core T e turbulence was measured by correlation electron cyclotron emission (CECE) radiometer on TEXT-U with some degree of time resolution showing that enhancement in fluctuations at the sawtooth crash is correlated to a steepening of the electron temperature gradient created as the sawtooth heat pulse propagates outward [15].2D images of electron temperature fluctuations during the sawtooth crash have been reported in TEXTOR measured with a 2D electron cyclotron emission imaging system and compared with various physics models of sawtooth crash [16].Recently, methods have been developed for analyzing beam emission spectroscopy data to obtain the plasma density evolution associated with rapid sawtooth crash events at the DIII-D tokamak [17].
This study provides new observations of core turbulence behavior during sawtooth oscillations for understanding sawtooth dynamics and testing and validating core turbulence simulations.This work reports on recent observations of the core T e and density (n e ) turbulence during sawtooth oscillations in a DIII-D hydrogen L-mode plasma with an inner wall limited configuration, measured by CECE [18] and Doppler backscattering (DBS) [19], respectively.The remainder of the paper is structured as follows.Section 2 describes details of experiments and diagnostics, and section 3 presents T e and n e turbulence in the core and magnetic turbulence changes during sawtooth oscillations.In section 4, a critical T e gradient behavior of T e turbulence amplitude during the sawtooth ramp phase is demonstrated, and section 5 shows T e turbulence amplitude and feature change from pre-to post-sawtooth crash.Finally, a summary of results and future work is presented in section 6.

Description of experiments and diagnostics
This study was performed on the DIII-D tokamak in hydrogen L-mode, neutral beam injection (NBI) heated plasmas in an inner wall limited configuration with plasma density n e ∼ 3-4 × 10 19 m −3 , plasma current I p ∼ 0.5-0.8MA and toroidal magnetic field B t ∼ 1.3-2.1 T. Figure 1 plots time history of line-averaged plasma density (figure 1(a)), core electron temperature showing sawtooth events (figure 1(b)), NBI power ∼ 3 MW with short beam pulses up to 6 MW (figure 1(c)), and magnetic equilibrium showing contour of square root of toroidal magnetic flux (ρ) calculated from the EFIT equilibrium solver [20].In the studied time window of ∼2200-2400 ms, the T e inversion radius is near ρ = 0.330 observed from the ECE signals.The unity safety factor (q = 1) magnetic surface location is at a similar location from EFIT calculated at times other than the sawtooth crashes.Note that during the sawtooth crashes the magnetic topology is significantly altered due to magnetic reconnection [6].
The major diagnostics are indicated on figure 1(d).The CECE locations are plotted as a red bar, DBS locations are marked as a purple curve, and ECE channels are shown by the blue symbols.The CECE system [21] on DIII-D is a radial array of 8-point correlation ECE radiometer which measures electron temperature fluctuations [18,[22][23][24] over a radial range of ∼10-15 cm (∼0.2-0.3 in ρ) in these plasmas.The system is sensitive to low wavenumber k ⊥ < ∼1-2 cm −1 .The DBS system [19] on DIII-D measures density turbulence amplitude and flow velocity [25][26][27], for low to intermediate wavenumbers (generally k ⊥ < 15 cm −1 ).The ECE system [28] provides high resolution T e radial profile measurements.Magnetic fluctuations are presented from the measurement by magnetic probe (Mirnov coil) arrays [29] outside the plasma around the DIII-D machine.

T e and n e turbulence in the core and magnetic turbulence changes during sawtooth oscillations
It is observed that the core T e turbulence spectrum and amplitude vary with time during sawtooth oscillations, but differently depending on the measurement location relative to the T e inversion radius and/or the q = 1 magnetic surface location.Figure 2 illustrates the T e turbulence spectra from CECE at the eight radial locations from ρ = 0.166-0.364during three full sawtooth cycles including four crashes marked by the vertical dashed lines.It should be noted here that the analysis in this paper is based on these three full sawtooth cycles in the same shot.Similar results have also been observed for different sawteeth with different amplitude and frequency in the same shot and other shots during the same day experiment with varied density/temperature.For these sawteeth, the T e inversion radius/ q = 1 magnetic surface location is near ρ = 0.330 as described earlier in section 2. Figure 2 shows that the T e turbulence spectra have the most power at low frequencies (mostly < 50 kHz).This narrow frequency bandwidth ensured a low system noise level at a short analysis time window, estimated by [22] Te,rms Te where N = 2B vid dt is the number of data points, dt is the analysis time window, B vid is the frequency bandwidth, B IF = 130 MHz is the system IF bandwidth.Figure 2 has a time resolution ∼3.2 ms at a 50% overlap in spectral analysis times to resolve the spectrum evolution during sawtooth cycles, and the noise level for the relative T e turbulence amplitude Te,rms /T e is plotted later in figure 3 as the horizontal dashed lines.Figure 2 demonstrates that the first three channels (figures 2(a)-(c)) at the inner locations of ρ = 0.166-0.217which are well inside the T e inversion/q = 1 radius have the clearest trend of spectral variation, which has a strong correlation with the sawtooth oscillation phase with two features: a sharp drop of T e turbulence at all frequencies immediately after the sawtooth crashes, and a generally gradual increase of T e turbulence (appears to be dominantly at low frequencies <20 kHz) leading to the next sawtooth crash.These features become less apparent as the radius increases.At ρ = 0.364 (figure 2(h)) where CECE location is slightly outside the T e inversion/q = 1 radius, T e turbulence seems to vary little during the ramp phase and shows intermittent increases immediately after the crash.Figure 3 plots the relative T e turbulence amplitude Te,rms /T e integrated over 0-50 kHz from the spectra illustrated in figure 2. The representative error bars are calculated as [18] δ Te,rms from the standard error of the cross-correlation function between bandwidth-limited Gaussian white noise signals [30].
The horizontal dashed lines indicate the noise level calculated using equation (1).It has to be noted that the T e turbulence spectrum is calculated from the correlation between two radially adjacent ECE signals [18,22], there are spurious correlations at sawtooth crashes when the two ECE signals respond to the fast T e change in about a couple of hundred microseconds.The frequency component of this spurious correlation is generally <20 kHz as can be seen in figure 2. Therefore, some of the data points overlapping with the sawtooth crash times indicated by the vertical dashed lines are invalid.further confirms the observations from figure 2. First, the T e turbulence amplitude variation has a strong correlation with the sawtooth oscillation phases clearly for the three further in locations well inside the T e inversion/q = 1 radius (figures 3(a)-(c)) with two apparent features: during the ramp phases, the T e turbulence amplitude gradually increases until the next sawtooth crash, and then decreases immediately after the sawtooth.At a further out location and inside the T e inversion/q = 1 radius (figure 3(d)), both features can be seen but less apparent.At locations near the T e inversion/q = 1 radius (figures 3(e)-(g)), the two features are not clearly seen, and T e turbulence seems to vary little with the sawtooth, except the drop of Te,rms /T e immediately after the crash near ∼ 2310 ms.At a location slightly outside the T e inversion/q = 1 radius (figure 3(h)), T e turbulence appears to vary little during the ramp phase and shows some intermittent increases immediately after the crash.The response of density turbulence in the core to the sawtooth oscillations has been observed by DBS. Figure 4 plots the n e turbulence amplitude frequency spectrum and root-meansquare (RMS) level at two locations: ρ = 0.222 (figures 4(a) and (b)) which are inside the T e inversion/q = 1 rad, and ρ = 0.326 (figures 4(c) and (d)) which is near the T e inversion/q = 1 rad.At ρ = 0.222 (figures 4(a) and (b)), a low perpendicular wavenumber k ⊥ ∼ 3.6 cm −1 is probed.The frequency spectrum especially at low frequencies (<∼300 kHz) show no apparent correlations with the sawtooth phase.The feature of having no apparent correlations with the sawtooth phases of n e turbulence at ρ = 0.326 is similar to the T e turbulence at similar locations illustrated in figures 2(e)-(g) and 3(e)-(g).It is noticeable in figure 4(b) that there is a rapid sharp increase of the RMS level at the sawtooth crash, as will be seen in magnetics later in figure 5, suggesting an electromagnetic feature.
It is noticeable from figure 4(b) that the core density fluctuation usually starts to get larger from the middle of every cycle, while T e fluctuation starts ramping up just after crash in figures 3(a)-(c).This difference could be due to different drive/damping mechanisms of n e and T e turbulence and different evolution of density and temperature profiles after crash.This deserves further research with other available highresolution profile data such as ion temperature and density in the core.It is interesting to note that 1/L Te being above the critical value (see figure 6) first appears near 30%-40% of the sawtooth period.A critical Te gradient behavior is observed in (a) but not in (b).The red curve in (a) is a piecewise linear fit to the data.The grey points in (a) are during sawtooth crash which are not included in the fit.These points are either below the noise level or have raw data overlap with the sawtooth crash which are invalid as discussed in section 3. The inversion radius is near ρ ∼ 0.330.In (a), the data points with negative value of the inverse Te scale length are from a short time window immediately after sawtooth crashes when the temperature gradient is slightly negative in the core.In addition, for calculating the normalized scale length, the interested reader can use either the major radius of magnetic axis R = 170.6 cm or the minor radius at outboard midplane a = 62.7 cm.
Magnetic fluctuations ( Bθ ) during sawtooth oscillations measured by one of the magnetic probes at the outboard midplane are depicted in figure 5. Figures 5(a) and (b) are frequency spectrum and RMS level time history of Ḃθ (=dB θ /dt), respectively.The data in figure 5 indicates a clear increase in magnetic fluctuations at the sawtooth crash, while the inter-ELM Ḃθ seems to have a subtle response but does not clearly vary during the sawtooth ramp phase.During the time of sawtooth crash (about a couple to few hundred microseconds, marked as the vertical dashed lines and indicated by T e at core in figure 5(c)), there is a rapid short time broadband increase of Bθ level shown by the peaks in figure 5(b) and brighter color in the contours in figure 5(a) at low frequencies (less than 20 kHz as seen in figure 5(a)).It is consistent with sawtooth theories (e.g.[6]) that MHD instabilities develop during the sawtooth crash leading to a magnetic reconnection.The toroidal magnetic probe array showed these are n = 1 modes, as has been previously observed in DIII-D [31].As described earlier, in figure 4(b) there is a rapid sharp increase of the n e turbulence RMS level at the sawtooth crash.The observation of sharp increase of Ḃθ at sawtooth crash in figure 5 suggests that the rapid sharp increase of the n e turbulence RMS level at the sawtooth crash in figure 4(b) could be of an electromagnetic feature.

A critical T e gradient behavior of T e turbulence amplitude during the sawtooth ramp phase
Figure 6 presents a feature of the T e turbulence amplitude dependence on the T e gradient during the sawtooth ramp phase.It illustrates that the T e turbulence amplitude in the core during the sawtooth ramp phase exhibits a critical T e gradient behavior inside but not near the T e inversion/q = 1 radius.Figure 6(a) plots the relative T e turbulence amplitude Te,rms /T e as a function of the inverse T e scale length 1/L Te at ρ = 0.190, well inside T e inversion/q = 1 radius (ρ ∼ 0.330).The piecewise linear fit in red indicates that Te,rms /T e starts to increase as 1/L Te is greater than a critical value, ∼ 0.012 seen in figure 6(a).1/L Te has been observed to drop abruptly at the sawtooth crash, then generally increase until the next crash.Figure 3(a) indicates that Te,rms /T e stays at a low level for some time after a sawtooth crash, then starts to increase when 1/L Te is greater than the critical value.This critical T e gradient behavior in figure 6(a) is observed in all the three inner channels with data shown in figures 2(a)-(c) and 3(a)-(c) where two apparent features appear: during the ramp phase, the T e turbulence amplitude gradually increases until the next sawtooth crash, and then decreases immediately after the sawtooth.In contrast, at radially outward CECE locations (data shown in figures 2(d)-(h) and 3(d)-(h)), CECE data show no apparent critical T e gradient behavior, even though 1/L Te variation has similar or larger variation range than that shown in figure 6(a).For example, figure 6(b) plots Te,rms /T e versus 1/L Te at ρ = 0.333, which is near the T e inversion/q = 1 radius.In contrast to figure 6(a), Te,rms /T e varies little with 1/L Te , even though 1/L Te has a larger variation range than that in figure 6(a).It is noticeable that for the locations where Te,rms /T e shows no apparent critical T e gradient behavior, Te,rms /T e did not have a clear variation during the ramp phase as illustrated in figures 2(d)-(h) and 3(d)-(h), compared to the three inner channels with a critical T e gradient behavior observed having T e turbulence amplitude gradually increasing until the next sawtooth crash as illustrated in figures 2(a)-(c) and 3(a)-(c).The result in figure 6 may suggest different turbulence modes observed by CECE at the locations well inside and near the T e inversion/q = 1 radius.At the locations well inside the T e inversion/q = 1 radius, the modes could be TEM [32] or MTM [33], which can be driven by T e gradient (∇T e ), given the correlation with the T e gradient along with the low wavenumber sensitivity of the CECE diagnostic (discussed in section 2).radial profile from ECE versus the normalized radius ρ averaged over a 3 ms time window pre-and post-crash respectively, showing flattening inside the T e inversion/q = 1 radius marked as the vertical dashed line.The profile flattening during the sawtooth crash within a couple of hundreds of microseconds is consistent with the magnetic reconnection picture.Figure 7(b) shows the profiles of Te,rms /T e in the corresponding cases, showing a decrease after sawtooth crash inside the inversion radius (mostly at ρ <∼ 0.28, and most apparent at the innermost 3 locations), but little change near the inversion radius.The dramatic decrease to near the noise level at the innermost three locations is part of the picture of the critical T e gradient behavior observed in figure 6(a).Figures 7(c) and  (d) are the linear growth rate of the most unstable mode from the trapped gyro-landau fluid (TGLF) [34] turbulence simulations as a function of ρ and wavenumber (k y ρ s ) for the two cases respectively.Positive and negative growth rates indicate modes propagating in the electron and ion diamagnetic directions, respectively.It shows a change from ion-to electrontype modes at the T e turbulence measurement wavenumber ranges of CECE (k y ρ s < ∼2 in the plots) from pre-to postsawtooth crash.Considering the critical T e gradient behavior shown in figure 6(a), the picture of T e turbulence change during a sawtooth crash appears to be that immediately after a sawtooth crash, the T e turbulence detected could be TEM mode and it gets stronger during the sawtooth ramp as the T e gradient grows and exhibits the critical T e gradient behavior.While at the highest T e gradient immediately before a sawtooth crash, T e turbulence has evolved to be dominated by an ion mode.A detailed time-dependent analysis is needed to determine the time of TEM to ion mode transition and characterize the cause of the mode transition.This transition could be correlated with changes in peaking of electron and ion temperature and density profiles, variations in collisionality and T e /T i ratio, etc.A previous work by Angioni et al [35] reported a switch from TEM to ITG dominated turbulence in an ASDEX L-mode plasma when density peaking was observed to increase with decreasing collisionality.

T e turbulence amplitude and feature change from pre-to post-sawtooth crash
It is noticeable in figure 7(b) that Te,rms /T e at pre-crash time is relatively flat across radius compared to that at postcrash.Figure 8 plots the square of the T e turbulence RMS level T2 e, rms which is proportional to the T e fluctuation energy at preand post-crash times using the data in figures 7(a) and (b).The T2 e, rms profiles look very different: it peaks near ρ ∼ 0.2 and decreases radially outward at pre-crash, while increases radially outward at post-crash.This profile change within a couple of hundreds of microseconds before and after a sawtooth crash appears consistent with the picture in which the fluctuation energy propagates radially outward during the sawtooth crash [36], but also consistent with changes in turbulence drive/damping mechanisms, e.g.electron temperature profile, collisionality, T e /T i , etc.Further study of the turbulence propagation during the sawtooth crash is underway and will be published in the future.

Summary
New observations of core electron temperature and density turbulence during sawtooth oscillations are reported in DIII-D hydrogen L-mode neutral beam injection heated plasmas in an inner wall limited configuration.A strong correlation of electron temperature and density turbulence level with the sawtooth oscillation phase has been observed at locations inside the T e inversion radius/q = 1 magnetic surface.It is demonstrated that there is an increase of electron temperature turbulence level leading to the next sawtooth crash, and a rapid sharp increase of the n e turbulence level and broadband magnetic turbulence at the sawtooth crash.These results suggest that turbulence is involved in and might be the triggering mechanism of the fast sawtooth crash (e.g.[11] predicts the mechanism leading to the fast crash is the onset of the magnetic stochasticity due to the magnetic component of fluctuations).More work is needed in this area to compare with theories in detail.In addition, the electron temperature turbulence level demonstrates a critical gradient behavior inside but not near the T e inversion radius/q = 1 magnetic surface.This suggests the turbulence which increases leading to a sawtooth crash could be TEM or MTM modes, which can be driven by the T e gradient.Finally, A change from ion-to electron-type modes is observed at lowwavenumber from pre-to post-sawtooth crash by TGLF turbulence simulations.This has not been understood and is a potential path for future simulation research.It is also noted that there is a dramatic shape change of the radial profile of the T e fluctuation energy at the sawtooth crash in a couple of hundreds of microseconds, including a sharp drop of electron temperature and density turbulence level immediately after the sawtooth crash inside the T e inversion radius.This suggests a consistency with the picture of fluctuation energy propagating radially outward during the sawtooth crash.The study of the turbulence propagation during the sawtooth crash will be addressed in a future work.

Figure 1 .
Figure 1.Time history of (a) line-averaged plasma density, (b) core electron temperature, and (c) NBI power for the DIII-D shot 183587.(d) is contour of square root of toroidal magnetic flux (ρ) at 2300 ms, with locations of CECE, DBS, and ECE diagnostics.

Figure 4 .
Figure 4. Density turbulence amplitude as a function of time and frequency measured by DBS at (a) ρ = 0.222 and (c) ρ = 0.326 for shot 183 591.(b) and (d) are the time history of the RMS level of ñe corresponding to (a) and (c), respectively.(e) is the Te time history measured by ECE near plasma center ρ = 0.The vertical dashed lines indicate sawtooth crashes.

Figure 5 .
Figure 5. (a) Magnetic fluctuation amplitude Ḃθ as a function of time and frequency measured by a pickup coil at the outboard midplane, (b) the corresponding RMS level time history of (a), and (c) the time history of electron temperature measured by ECE near plasma center ρ = 0.

Figure 6 .
Figure 6.Relative Te turbulence amplitude as a function of inverse Te scale length at (a) ρ = 0.190, (b) ρ = 0.333 for shot 183587 from 2235-2345 ms including 3 full sawtooth cycles as shown in figure 3.A critical Te gradient behavior is observed in (a) but not in (b).The red curve in (a) is a piecewise linear fit to the data.The grey points in (a) are during sawtooth crash which are not included in the fit.These points are either below the noise level or have raw data overlap with the sawtooth crash which are invalid as discussed in section 3. The inversion radius is near ρ ∼ 0.330.In (a), the data points with negative value of the inverse Te scale length are from a short time window immediately after sawtooth crashes when the temperature gradient is slightly negative in the core.In addition, for calculating the normalized scale length, the interested reader can use either the major radius of magnetic axis R = 170.6 cm or the minor radius at outboard midplane a = 62.7 cm.

Figure 7
Figure7compares the measured T e turbulence amplitude profiles and calculated mode growth rates from simulations between pre-to post-sawtooth crash.Figure7(a) plots the T e

Figure 7 .
Figure7compares the measured T e turbulence amplitude profiles and calculated mode growth rates from simulations between pre-to post-sawtooth crash.Figure7(a) plots the T e

Figure 8 .
Figure 8.A comparison of radial profiles of the square of the Te turbulence RMS level between a 3 ms window pre-and postsawtooth crash near 2310 ms for shot 183587.