Perturbative Thermal Diffusivity from Partial Sawtooth Crashes in Alcator C-Mod

Transcription

1 PSFC/JA Perturbative Thermal Diffusivity from Partial Sawtooth Crashes in Alcator C-Mod A.J. Creely 1, A.E.White 1, E.M. Edlund 2, N.T Howard 3, A.E. Hubbard 1 1 MIT Plasma Science and Fusion Center, Cambridge, MA, USA. 2 Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA 3 Oak Ridge Institute for Science and Education (ORISE), Oak Ridge, TN 37831, USA November 2015 Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge MA USA This work is supported by the US DOE under grants DE-SC and DEFC02-99ER54512-CMOD. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.

2 Perturbative Thermal Di usivity from Partial Sawtooth Crashes in Alcator C-Mod A.J. Creely 1,A.E.White 1,E.M.Edlund 2, N.T Howard 3, A.E. Hubbard 1 1 MIT Plasma Science and Fusion Center, Cambridge, MA, USA. 2 Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA 3 Oak Ridge Institute for Science and Education (ORISE), Oak Ridge, TN 37831, USA Abstract. Perturbative thermal di usivity has been measured on Alcator C- Mod via the use of the extended-time-to-peak method on heat pulses generated by partial sawtooth crashes. Perturbative thermal di usivity governs the propagation of heat pulses through a plasma. It di ers from power balance thermal di usivity, which governs steady state thermal transport. Heat pulses generated by sawtooth crashes have been used extensively in the past to study heat pulse thermal di usivity [1], but the details of the sawtooth event typically lead to nondi usive ballistic transport, making them an unreliable measure of perturbative di usivity on many tokamaks [2]. Partial sawteeth are common on numerous tokamaks, and generate a heat pulse without the ballistic transport that often accompanies full sawteeth [2]. This is the first application of the extended-timeto-peak method of di usivity calculation [3] to partial sawtooth crashes. This analysis was applied to over 50 C-Mod shots containing both L- and I-Mode. Results indicate correlations between perturbative di usivity and confinement regime (L- vs. I-mode), as well as correlations with local temperature, density, the associated gradients, and gradient scale lengths (a/l Te and a/l n). In addition, di usivities calculated from partial sawteeth are compared to perturbative di usivities calculated with the nonlinear gyrokinetic code GYRO. We find that standard ion-scale simulations (ITG/TEM turbulence) under-predict the perturbative thermal di usivity, but new multi-scale (ITG/TEM coupled with ETG) simulations can match the experimental perturbative di usivity within error bars. Perturbative di usivities extracted from heat pulses due to partial sawteeth provide a new constraint that can be used to validate gyrokinetic simulations. PACS numbers: Fa Keywords: Perturbative Transport, Thermal Di usivity, Sawtooth Submitted to: Nucl. Fusion

3 Partial Sawtooth Perturbative Thermal Di usivity 2 1. Introduction Understanding thermal transport in tokamak plasmas will be key to the design and operation of future devices. Thermal transport in tokamaks is an incredibly complicated process. In addition to conduction and convection, plasma turbulence plays a major role in determining the cross-field electron thermal transport. In particular, drift-wave turbulence in the tokamak arises due to linear instabilities, and many of the modes theoretically exhibit a threshold in the gradient driving the instability: the mode is linearly stable below the threshold, or critical gradient, and unstable above [4]. Nonlinearly, the saturated turbulence also exhibits a critical gradient, which can be di erent from the linear critical gradient [5, 6]. It is possible to experimentally characterize the critical gradient for a given type of turbulence, as well as to analyze the associated property of transport sti ness [7, 8]. Sti ness is the incremental change in flux for an incremental change in gradient, above the critical gradient [9]. In the Alcator C-Mod tokamak [10] it is observed that there is little change in electron temperature profile shape with additional input power, which is consistent with high sti ness [11]. A better understanding of electron heat transport and electron temperature profile sti ness is required to improve prediction for scenarios with strong electron heating, such as burning plasmas in ITER and other future devices. To this end, measuring electron thermal transport and the associated sti ness has been an active topic of experimental research for many years. Experimentally, it is worthwhile to reduce the complexity of the electron heat transport analysis by using a simple di usive model, with an e ective thermal di usivity, e. This model does not capture all of the intricacies of thermal transport, but is able to o er a straightforward method of characterization. The power balance thermal di usivity, e, governs steady state heat transport and is calculated based on the electron energy confinement time [1]. It is defined as, e = Q e (1) n e rt e where Q e is the electron heat flux, rt e is the electron temperature gradient, and n e is the electron Figure 1: A cartoon illustrating a heat pulse propagating through a tokamak plasma. The four panels represent four radial locations in the plasma, with the first closest to the center. It is assumed that the heat pulse is generated in the center of the plasma. The y-axes are temperature and the x-axes are time. number density [1]. Note that later in this paper, the subscript e may be dropped, but all di usivities refer to electron thermal di usivities. In addition to measuring the power balance di usivity, is is also possible to measure a di usivity based on the thermal transport of perturbations to the steady state temperature profile in the form of heat pulses [12]. Figure 1 illustrates a heat pulse propagating through a tokamak plasma. The four panels represent four radial locations in the plasma, with the first closest to the center. It is assumed that the heat pulse is generated in the center of the plasma, and can be tracked in space and time as it propagates towards the edge. Since the di usivity measured by the propagation of heat pulses governs perturbations, rather than steady state heat transfer, it is defined as [3], e = e (2) n e In past literature the thermal di usivity measured from heat pulse propagation is often termed the heat pulse di usivity [1]. The slope of the heat flux plotted against temperature gradient is sometimes also termed the incremental di usivity [1]. Since these

4 Partial Sawtooth Perturbative Thermal Di usivity 3 quantities are directly comparable [1, 2, 13, 14, 15, 16, 17, 18], this paper will call both quantities the perturbative di usivity, and will use the superscript to denote the perturbative di usivity. This di erence in definition and e ect makes clear that perturbative and power balance thermal di usivity should not be compared directly. Figure 2 illustrates the di erence between power balance and perturbative thermal di usivity. The red line indicates the relationship between the heat flux and the temperature gradient multiplied by the density. The slope of the orange line is the power balance thermal di usivity. The slope of the green line is the heat pulse thermal di usivity. These quantities are identical only if the heat flux and temperature gradient are linearly related with no o set [1]. The historical context of this di erence in definition is described in more detail in the Appendix, and in [19-37]. In addition, the relationship between perturbative thermal di usivity and profile sti ness means that experimental measurements of the perturbative di usivity could be used to validate the gyrokinetic code GYRO [38], along with more traditional experimental quantities such as the ion and electron power balance thermal di usivities and fluctuation measurements. In this work, we will use the measured perturbative thermal di usivity to investigate the importance of multiscale (coupled ITG/TEM and ETG turbulence) simulations [39] for modeling the electron temperature profile sti ness [40]. In gyrokinetic simulations, the electron temperature profile sti ness is measured via scans a L Te of the slope of the electron heat flux, Q e, against above the critical gradient, where a is the minor radius and L Te is the electron temperature gradient scale length, L Te = Te rt e [40]. This process is described in greater detail later in this paper. The results of these scans are then related to the perturbative thermal diffusivity when multiplied by density, temperature, and minor radius, as Figure 2: A diagram illustrating the di erence between power balance and heat pulse, or perturbative, thermal di usivity. The red line indicates the relationship between the heat flux and the temperature gradient multiplied by the density. The slope of the orange line is the power balance thermal di usivity. The slope of the green line is the heat pulse thermal di usivity. These are only the same if the heat flux and temperature gradient are linearly related with no o set Te = GY RO ne T e a (3) One can therefore compare perturbative thermal di usivity calculated from partial sawtooth-generated heat pulses to the perturbative di usivity calculated via GYRO, and in doing so provide another metric for validation of GYRO. Perturbative thermal di usivity calculated from ECH-generated heat pulses has been related to sti ness in the past at ASDEX, [14] as well as compared specifically to scans from ion-scale (ITG/TEM turbulence only) gyrokinetic simulations at DIII-D [18]. Experimentally, once a di usive heat pulse has been generated, one can calculate the perturbative thermal di usivity by tracking the propagation of this heat pulse in a number of ways. Reviews of perturbative thermal transport studies, including heat and cold pulses generated by both sawteeth and other methods, such as modulated ECH and impurity injection, can be found in [1] and [13]. Since Alcator C- Mod does not have an ECH system, but does typically operate with sawtoothing plasmas, this study will focus on measuring the perturbative thermal di usivities from sawtooth crashes. While cold pulses have been used in C-Mod extensively to study perturbative heat transport [41], heat pulses from sawteeth have only been examined in a cursory fashion at C-Mod [42]. Heat pulses generated from the sawtooth instability were for many years used to measure perturbative di usivity [12], but later work revealed that they were often accompanied by non-di usive ballistic transport and were therefore not suitable for measuring this perturbative di usivity on many machines [2]. A detailed history of perturbative transport studies using sawtooth-generated heat pulses is given in the Appendix. In past work it was noted that partial or compound sawtooth crashes did not lead to the same ballistic transport as full sawteeth [2]. It should therefore be possible to apply well-established analysis techniques to the propagation of heat pulses from partial sawteeth, in order to measure the perturbative thermal di usivity (which is related to sti ness), and then to compare against predictions from nonlinear

5 Partial Sawtooth Perturbative Thermal Di usivity 4 gyrokinetic simulations. The Appendix provides further description of the di erences between full and partial sawteeth, including a definition of ballistic transport. One of the motivations for a renewed interest e in calculating from sawtooth crashes is the relatively recent discovery of the I-mode (improved mode) confinement regime [43]. I-mode plasmas are characterized by high energy confinement, similar to that found in H-mode, but with particle confinement similar to that of L-mode and a natural absence of Edge Localized Modes (ELMs) [43]. I-mode has been observed on Alcator C-Mod, ASDEX Upgrade, and DIII-D [44]. This study will investigate di erences in perturbative thermal di usivity between I-mode and L- mode. It has been shown that the I-mode confinement regime at Alcator C-Mod is predicted by GYRO to have higher temperature sti ness than L-mode [45, 46], and thus one would expect to find higher perturbative thermal di usivities in I-mode compared to L-mode. This study will use the extended-time-to-peak method, originally used by B.J.D. Tubbing et al. in e JET [3] to calculate from partial sawteeth. The next section will outline the details of the method used to calculate e, as well as describing the machine and diagnostics used in this study. The method of calculating perturbative thermal di usivity from heat pulses generated by partial sawteeth was first used to further investigate the properties of I- mode and better understand its thermal confinement. More than 50 shots that contain both L- and I-mode phases were analyzed, and the relationship between the perturbative di usivity and various global and local plasma parameters, as well as the confinement regime, was investigated. The experimental e is compared with long-wavelength (ion-scale turbulence) nonlinear gyrokinetic simulations and also with multiscale (coupled ion-scale and electron-scale turbulence) nonlinear gyrokinetic simulations. 2. Experimental Methods 2.1. Machine and Diagnostics All data analyzed in this study was taken on Alcator C-Mod [10] which is a compact (R = 0.67 m, a = 0.22 m), high field (B = T), diverted tokamak with Boron coated Molybdenum and Tungsten plasma facing components. Temperature measurements were taken with a Grating Polychrometer (GPC) installed on Alcator C-Mod. The GPC uses a di raction grating to split the electron cyclotron emission (ECE) spectrum. The GPC on C-mod has 9 channels at di erent frequencies (and thus di erent radial locations, spaced by about 2 cm), and up to 100 khz sampling [42], though it is typically run at 20 khz (as it was for the experiments in this study). GPC is a standard diagnostic at Alcator C-Mod, and is used to measure the 2nd harmonic X-mode electron cyclotron emission for electron temperature profile measurements using well established techniques [47]. In all plasmas considered here, the emission is optically thick for every GPC channel and the propagation of a heat pulse can be tracked from the core region to the edge region with high fidelity Perturbative Thermal Di usivity Calculation Method In order to calculate the perturbative electron thermal di usivity, e, from the partial sawtooth-generated heat pulses in various discharges, this study will use the extended time-to-peak method, as originally described in [3]. The full derivation will not be presented here, but can be found in [3] and [48]. This method was chosen due to its more general validity than the basic time-to-peak method, and the irregular spacing of partial sawteeth, which precludes the use of Fourier analysis. Previous comparisons have confirmed the general agreement of the Fourier and extendedtime-to-peak methods [15, 49], as further described in the Appendix. The result of the calculation in [3] gives, e =4.2 a cv where v = p apple a a s dtpeak dr (4) 1 (5) is the radial velocity of the peak of the heat pulse in m/s, and = 10(a s) d log(a) (6) dr describes the damping of the heat pulse as it propagates radially (unitless). Variables are defined as follows: minor radius a, minor radius corrected for elongation a c, radius from plasma center r, Shafranov shift s, elongation apple, time that the pulse peak reaches a given radius t peak, and heat pulse amplitude A in ev. Based upon the assumptions made in its derivation and the region of the plasma in which it was applied, there is no reason that the extended-time-topeak method should not be valid on Alcator C-Mod. The derivation is solved in cylindrical coordinates, and then corrected for a toroidal geometry and a shaped plasma [3]. The method is applied outside of the sawtooth mixing radius, as intended. It has been successfully used on other shaped tokamaks, and accounts for di erences in machine size and shaping. It must be noted that this methods calculates a radially averaged

6 Partial Sawtooth Perturbative Thermal Di usivity 5 di usivity, within the region of heat pulse propagation, which is responsible for minor disagreements with other methods of perturbative di usivity calculation, such as Fourier analysis [15]. For Alcator C-Mod, a = 0.22, apple = 1.6, and s = In this study on Alcator C-Mod, the radial extent of the pulse measurements included r/a = 0.6 to 0.9 depending on the shot. This constraint is based on the range in which the heat pulse was outside of the partial sawtooth mixing radius, but also of su cient magnitude to measure. Due to the somewhat noisy nature of the GPC data and small di erences between individual sawteeth, this study will average over several heat pulses in a given discharge in order to calculate the final thermal di usivity. Instead of averaging the time traces of several pulses together, as was done in some past studies [3], this study will calculate the thermal di usivity of a number of individual pulses, and then average these di usivities together. Between 3 and 15 pulses were used for each shot, depending on the number of usable partial sawteeth. While this method of averaging leads to a slightly larger uncertainty when estimating the peak for a given pulse, it allows one to calculate the variance between pulses, and thus give a more accurate value for the total uncertainty. To avoid selecting maxima generated by noise as opposed to true maxima, time traces were smoothed slightly using the MATLAB smooth function, which is based on a local regression using weighted linear least squares and a 2nd degree polynomial model, [50]. Obviously incorrect fits were rejected by eye. Uncertainty was calculated by adding, in quadrature, the experimental uncertainty in the pulse height and peak time to the statistical uncertainty (standard error) between various pulses Comparisons of Thermal Di usivities from Partial Sawteeth, Full Sawteeth and Power Balance Based on the past work presented in the Introduction and the Appendix, this study chose to use only partial sawtooth-generated heat pulses when analyzing L- and I-mode plasmas and comparing to GYRO simulations, to avoid the possible contamination of the data due to the ballistic e ect. In this section, however, we compare the di usivity measured from partial sawteeth, full sawteeth and power balance to illustrate the di erences between the quantities and to further justify our choice. Similar to past work, we found that the di usivities calculated from full satweeth are larger than power balance values. We also found that di usivities calculated from full satweeth are larger than di usivities calculated from partial sawteeth. We discuss these di erences within the context of past work, and with respect to evidence for (a) (b) Figure 3: Partial and full sawteeth on Alcator C-Mod GPC data. This data is from the nine channels of the grating polychrometer (GPC). The time trace in (a) shows first a full sawtooth crash and the accompanying ballistic heat pulse, and then a partial sawtooth crash, and the accompanying di usive heat pulse. The end of the time trace also shows another full sawtooth. Part (b) shows channels 5 through 8 of the GPC for the same shot and time period, with the background temperature subtracted o for each channel. This plot depicts in greater detail the propagation of the heat pulses, analogous to the cartoon in Figure 1.

7 Partial Sawtooth Perturbative Thermal Di usivity 6 Table 1: Comparison of experimental partial sawtooth heat pulse, full sawtooth heat pulse, and power balance (from TRANSP) di usivities. All values are radially averaged. The three values disagree, as expected. The full sawtooth-generated heat pulses result in larger values of measured di usivity, due to the ballistic transport. The power balance di usivity is generally lower than the perturbative di usivity, but this is not always the case. Detailed comparison of perturbative and power balance di usivity is beyond the scope of this study. Shot Confinement Partial (m2 /s) Full (m2 /s) e (m 2 /s) L-Mode 1.13 ± ± Ohmic (LOC) 2.67 ± ± Ohmic (SOC) 1.70 ± ± L-Mode 1.74 ± ± I-Mode 2.03 ± ± L-Mode 1.61 ± ± ballistic transport associated with full sawteeth heat pulses in Alcator C-Mod. From the GPC data on C-Mod, partial sawteeth were identified manually based on a significant drop in temperature slightly o -axis accompanied by a negligible change in on-axis temperature. Figure 3 shows the di erences between full and partial sawteeth on Alcator C-Mod GPC data, as well as representative heat pulses. Qualitatively, the time traces shown in Figure 3 show the combination of sharp rise and gradual decay in temperature for full sawteeth that was incompatible with computational di usive models in past work [2]. The partial sawtooth in Figure 3 reveals a much more gradual rise, again consistent with the analysis performed in past work which indicates di usive transport [2]. This indicates that the di erences between partial and full sawteeth at C- Mod are consistent with the description of di erences on other devices [2]. Based on these di erences, and supported by modeling, past work has concluded that partial sawteeth should lead to di usive transport while full sawteeth often do not [2]. In order to connect with past work studying heat pulse propagation from both full and partial sawteeth, we compared the perturbative thermal di usivity calculated for partial and full sawteeth. Table 1 shows the perturbative thermal di usivity calculated from partial and full sawteeth for five C-Mod shots, one of which has both L- and I-mode phases. These five shots are taken from a larger data set of more that 50 C-Mod shots, which are described in more detail in Section 3. In all cases from C-Mod, the full sawteeth always lead to a perturbative di usivity larger than partial sawteeth perturbative di usivity. This is consistent with previous work [2]. This suggests that, at C- Mod, the full sawtooth crashes generate heat pulses that propagate based on non-di usive ballistic e ects, as often observed in other tokamaks, and the partial sawteeth generate heat pulses that can be described with a di usive model [2]. Table 1 also shows the power balance thermal di usivity calculated via the TRANSP code [51]. The power balance value was averaged over the same time period and radial range as measured by the perturbative analysis. As expected, the perturbative and power balance thermal di usivities di er for most of the cases presented here, with little clear correlation between the two. As described in the Introduction (see Figure 2), the perturbative and power balance thermal di usivities are measures of di erent quantities and are not generally expected to have the same value. We present the comparisons here only for historical context and completeness. The results here, taken in the context of past work, indicate that partial sawteeth on Alcator C- Mod lead to di usive transport, and thus are valid for perturbative thermal transport studies, while full sawteeth lead to non-di usive transport and are thus unreliable indicators of thermal di usivity. For this reason the rest of this study will exclusively use partial sawtooth-generated heat pulses in it s calculations of perturbative thermal di usivity. 3. Comparison of perturbative electron thermal di usivities in L- and I-Mode on Alcator C-Mod This section will detail the perturbative di usivity analysis done on more than 50 Alcator C-Mod shots using partial sawteeth-generated heat pulses. Each of these shots contained at least one stable L-mode phase (in time) and at least one stable I-mode phase. In other words, each of these shots moved from a stable period of one confinement regime to a stable period of another confinement regime. Pulses that occurred during transitions from one confinement regime to another were not analyzed. The shots were all run with ion B rb drift away from the active X-point (unfavorable rb drift), which generally enables a more robust I-mode operating window [44]. The line averaged densities during

8 Partial Sawtooth Perturbative Thermal Di usivity 7 thermal di usivity is often higher in the I-mode phase than in the L-mode phase of a given shot. This is not, however, always the case, as there are a fair number of points below the solid line, though many of these points are within uncertainty of the line. The general trend of higher perturbative thermal di usivity in I-mode is consistent with the higher sti ness found in high confinement regimes, as was discussed above. These results are qualitatively consistent with the past nonlinear gyrokinetic simulations of L- and I-mode plasmas at C-Mod [46]. Quantitative comparisons will be described in detail later in Section Relationship with Global Plasma Parameters Figure 4: Comparison of heat pulse thermal di usivity from partial sawteeth in L- and I-mode periods of a given shot. Each point is data from one shot, and the two axes are the di usivity from the L-mode phase of that shot and the di usivity from the I-mode phase of the shot. Shots with multiple L- or I-mode phases are represented by multiple points. The red line indicates equal di usivities in both modes. Representative error bars are shown. the periods of analysis ranged from approximately 0.6 to /m 3. Plasma current ranged from approximately 0.9 to 1.3 MA. All shots ran with an on-axis magnetic field of 5.4T Di erences Between L-Mode and I-Mode The results of the comparison of di usivities in L- and I-mode are summarized in Figure 4. In this figure, each point represents L- and I-modes from a single shot. The y-axis of the point is the thermal di usivity calculated from heat pulses in the I-mode phase of the shot, and the x-axis of the point is the thermal di usivity calculated from heat pulses in the L-mode phase of the shot. Shots with multiple distinct phases of L-mode or I-mode are represented by multiple points. A total of 79 points are shown, taken from 56 shots (many shots had multiple distinct L- or I- mode phases). The red line indicates equal di usivity in L- and I-modes. Points above the line indicate that the perturbative di usivity is higher in I-mode than in L-mode. Points below the line indicate a higher di usivity in L-mode. Representative error bars are shown to avoid cluttering the plot. While many of the points comparing L- and I- mode in Figure 4 are within experimental uncertainty of having equal thermal di usivity, more are above the red line than below it, indicating that the perturbative In addition to comparing L- and I-mode within a given shot, this study also searched for correlations between perturbative di usivity and both global and local plasma parameters. These correlations may shed some light on the factors within the plasma that play a role in determining the perturbative di usivity. Global plasma parameters that were considered included: plasma current, line-averaged density, central temperature, stored energy, and total RF heating power. Among these, central temperature, RF heating power, and stored energy seem to have little to no correlation with perturbative thermal di usivity. Both density and plasma current showed a fairly clear correlation with perturbative di usivity. Since density and plasma current are correlated with one another, however, one of these correlations may be due to the hidden variable of the other. For this reason, correlations of density at fixed current and current at fixed density were made with the perturbative thermal di usivity. Such analysis revealed that density at fixed current still shows some correlation, while current at fixed density does not. The data therefore indicates that density correlates more strongly with perturbative thermal di usivity, while plasma current only correlates with di usivity through its correlation with density. A straightforward linear regression of di usivity with density gives a p-value of and an R 2 of 0.42, indicating a reasonable correlation. The p-value is the probability of no correlation (the null hypothesis), and R 2 measures the amount of variation of the dependent variable that can be explained by variation in the independent variable [50]. Figure 5 shows a plot of density with perturbative di usivity, using data from all currents. In this plot each point represents one phase of stable L- or I-mode. The blue triangles represent L-mode phases, and the red circles represent I-mode phases. Since shots are now broken down into individual phases of either L- or

9 Partial Sawtooth Perturbative Thermal Di usivity 8 Figure 5: Plot of the perturbative thermal di usivity and the line averaged density of both L- and I-mode plasmas. Each point represents either an L-mode or an I-mode phase of a given shot. In other words, shots are now broken down into individual phases to plot, so that there are more points than the previous plot, even though it is representing the same data. Blue triangles represent L-mode and red circles represent I- mode phases. I-mode, there are more points on this plot, even though it is representing the same data Relationship with Local Plasma Parameters In addition to considering global plasma parameters, correlations between perturbative di usivity and local plasma parameters were also considered. In particular, local temperature, local temperature gradient, local density, local density gradient, and pulse amplitude were analyzed. Some plots show limited correlation, but are included to fully represent the analysis completed. Figures 6a, 6b, and 6c show the perturbative thermal di usivity plotted against, respectively, the local plasma temperature, temperature gradient, and a/l Te (where a is the minor radius and L Te = T e rt e ) averaged over the radial range where the perturbative di usivity was measured. A correlation is observed between the local plasma temperature and the perturbative thermal di usivity. Linear regression gives a p-value of and an R 2 of Temperature gradient shows a similar positive correlation, which may to some extend be related to the generally higher perturbative di usivity in I-mode as compared to L-mode, as I-mode tends to have a higher temperature and temperature gradient than L- mode. Linear regression gives a p-value of and an R 2 of Though the details of turbulent transport are beyond the scope of this study, a/l Te is commonly associated with turbulence drive terms [52]. There seems to be some separation between the L- and I-mode data, with I-mode generally having a lower a/l Te. In addition, it appears that there may be some value of a/l Te, above which the perturbative thermal di usivity is limited to below approximately 3 m 2 /s, but the data in this region is insu cient to draw any strong conclusions. Linear regression revealed little to no linear correlation, but further analysis of this dependence is a topic of possible future work. Figures 7a, 7b, and 7c show the perturbative thermal di usivity plotted against, respectively local density, density gradient, and a/l n (where a is the minor radius and L n = ne rn e ) averaged over the radial range where the perturbative di usivity was measured. Local density showed a nearly identical trend to lineaveraged density, and a linear regression gives a p- Value of and an R 2 of As with density, the density gradient seems to be somewhat correlated with the perturbative di usivity. In addition, with the exception of a few points, it seems that there may be some density gradient below which there is a larger range of perturbative di usivities, and above which there is a smaller range. Linear regression gives a p- Value of and an R 2 of a/l n has been associated with turbulence drive in past work [52], but linear regression revealed little to no linear correlation. A final normalization was made by dividing the thermal di usivity by local temperature to the 3/2 power, and plotting against a/l Te and a/l n,shown in Figures 8a and 8b. This normalization by the gyro- Bohm factor T 3/2 is commonly performed in other perturbative transport studies, as in [8]. While these plots do not reveal anything vastly di erent than the others, it does seem that the normalizations tend to group the I-modes closer together, while the L-modes remain fairly spread out. One possible explanation for this is the greater temperature range spanned by the I-mode shots. Linear regression gives a p-value of and an R 2 of 0.45 for a/l Te and p-value of and an R 2 of 0.13 for a/l n. In addition to local background plasma properties, analysis was performed on the amplitude of the heat pulse generated by the partial sawtooth crash. It may be postulated that it is only the generally larger amplitude of heat pulses generated by full sawteeth as compared to those generated by partial sawteeth, as opposed to the MHD activity, that is responsible for ballistic transport. Figure 9, however, reveals that there is little to no correlation between perturbative di usivity and pulse amplitude. This is further evidence that partial sawteeth lead to a qualitatively di erent heat pulse compared to full sawteeth, and

10 Partial Sawtooth Perturbative Thermal Di usivity 9 (a) (a) (b) (b) (c) Figure 6: Plots of perturbative thermal di usivity against (a) electron temperature, (b) temperature gradient, and (c) a/l Te, all averaged over the radial range of the di usivity measurement. Symbols are used as above. (c) Figure 7: Plots of of perturbative thermal di usivity against (a) electron density, (b) density gradient, and (c) a/l n, all averaged over the radial range of the di usivity measurement. Symbols are used as above.

11 Partial Sawtooth Perturbative Thermal Di usivity 10 (a) Figure 9: Plot of the perturbative thermal di usivity and heat pulse amplitude, averaged over the radial range of the di usivity measurement. Symbols used as above. (b) Figure 8: Plot of the perturbative thermal di usivity, normalized by the gyro-bohm factor of T 3/2, against (a) a/l Te and (b) a/l n. Symbols used as above. further supports the validity of the results presented in this study. 4. Comparison of Experimental Perturbative Thermal Di usivities with GYRO Perturbative Di usivities All GYRO results presented in this paper were taken from [39] and [46], where the simulation set-ups are described in detail. The simulations are local, nonlinear, flux-tube simulations. Two C-Mod plasmas are simulated: a shot with both L-mode and I- mode phases (C-Mod shot ) and an L-mode shot ( ). The first shot has only ion-scale simulations available, while the second has both ionscale and multi-scale simulations available. Inputs to GYRO are taken directly from the experiment. Using standard methods employed in gyrokinetic code validation work at C-Mod, the experimental value of the ion temperature gradient is varied within experimental error bars to obtain a simulation that matches the ion heat flux as described in [39] and [46]. Ion-scale simulations, which do not include electronscale turbulence [46], and also multi-scale simulations, which include both ion- and electron-scale turbulence [39], are presented here. Comparison of the code results with experimental ion heat fluxes from power balance are not shown here, but are discussed in [39, 46]. While the ion heat flux-matched simulations allow for direct comparisons with the experimental power balance transport levels via standard methods [39, 46], we require a series of simulation scans, in which the electron temperature gradient input parameter is varied, in order to compare experimental heat pulse di usivities with the incremental heat flux from GYRO. This method has used before to compare the experimental perturbative thermal di usivity calculated form ECH-generated heat pulses in DIII-D to GYRO simulations [18]. The simulation scans are used to map out the flux gradient space and identify the critical gradient and profile sti ness. Here we define temperature profile sti ness as, the degree of sensitivity of the heat flux to the driving gradient, consistent with the definition applied to gyrokinetic simulation studies of JET plasmas [40]. This definition uses the fact that the turbulence is driven unstable by logarithmic temperature gradients that are above a critical threshold. In a plot of electron heat flux, Q e, against normalized temperature gradient scale length, a/l Te, there will be a critical gradient, a/l T e,crit, below which little heat flux is driven (an x-intercept). When the output heat flux from the scans is normalized

12 Partial Sawtooth Perturbative Thermal Di usivity 11 Table 2: Table comparing the thermal di usivities calculated from GYRO simulations of two discharges ( and ) with experiment. The table shows the GYRO power balance thermal di usivities, GY RO, compared with experimental values, Exp. The table also shows the GYRO incremental (perturbative) di usivity, GY RO, compared with the experimental perturbative di usivity from partial sawtooth analysis, Partial. The Partial are the same as in Table 1. In contrast, the Exp values are not the same as in Table 1, but are instead taken at the location of the GYRO run, as opposed to being radially averaged. All GYRO results are from [39] and [46]. Shot Confinement GYRO Model Exp (m2 /s) GY RO (m2 /s) Partial (m2 /s) GY RO (m2 /s) L-Mode Ion-scale 0.84 ± ± I-Mode Ion-scale 0.31 ± ± L-Mode Ion-scale 0.83 ± ± L-Mode Multi-scale 0.83 ± ± Q e /n e (kev m/s) I-mode L-mode T e(kev/m) Figure 10: GYRO scans of the L- and I-mode phases of Alcator C-Mod shot used to calculate the incremental di usivity. The slope of the line above the critical gradient is the incremental thermal di usivity, which is directly comparable to the experimental perturbative di usivity. For the I-mode, the value is GY RO = 0.8m2 /s, and for the L-mode, GY RO = 0.3m 2 /s. See [46] for details of the simulations. by density, Q e,gy RO /n e, and plotted against rt, the slope of the line above the critical gradient will have units of m 2 /s, and can be compared directly with the experimental perturbative di usivity. As an example, Figure 10 shows the results of performing 7 GYRO runs scanning the input a/l Te in the L- and I-mode confinement phases of shot For the I-mode, the value is GY RO =0.8m2 /s, and for the L-mode, GY RO =0.3m2 /s. These values have been calculated from the GYRO scan results shown in Figure 10 by applying Equation 2. Table 2 shows all three sets of ion-scale GYRO simulations and the one set of multi-scale simulations. GYRO scans around Qi matched simulations (White PoP 2015) The table shows the GYRO power balance thermal di usivities GY RO compared with experimental values Exp. The table also shows the GYROperturba- tive di usivity, GY RO (Equation 2), compared with the experimental perturbative di usivity from partial sawtooth analysis, Partial (Equation 4). For shot , one simulation is for the L mode phase and the other simulation is for the I-mode phase of the shot, and include only ion-scale turbulence [46]. As shown in Table 2,the power balance electron thermal di usivity calculated form GYRO, GY RO, is clearly lower than the power balance experimental value, Exp.This under-prediction is believed to be due to the use of ion-scale GYOR simulations, which neglect the ETG physics [46]. In addition, the perturbative thermal di usivity calculated from GYRO, GY RO, is lower than the heat pulse experimental value, Partial,which means that GYRO under-predicts the electron temperature profile sti ness as well. For shot , the ion-scale simulation results for the L-mode plasma under-predict both experimental power balance thermal di usivities and the experimental perturbative thermal di usivities. In contrast, the multi-scale simulation results for the L-mode plasma in agree with both the experimental power balance and perturbative thermal di usivities, within the experimental and code (±10%) uncertainties. 5. Conclusion and Future Work This study has presented the first extensive use of heat pulses generated by partial sawtooth crashes to calculate the perturbative thermal di usivity in a tokamak. In Alcator C-Mod, heat pulses generated by partial sawteeth avoid the ballistic transport often associated with full sawteeth and are therefore a better measure of perturbative di usivity. Generally speaking, this method can be applied to any tokamak plasma that contains a su cient number of partial

13 Partial Sawtooth Perturbative Thermal Di usivity 12 sawtooth crashes. Based on the results presented here, perturbative di usivity measured with partial sawteeth is related to sti ness di erences between L-mode and I-mode. We find a strong correlation between the level of perturbative di usivity and the plasma density. Local temperature, temperature gradient, density, and density gradient also seem to show some correlation with perturbative thermal di usivity. For the first time, the perturbative thermal di usivity calculated form partial sawtooth-generated heat pulses has been compared with perturbative thermal di usivities calculated from nonlinear gyrokinetic simulations. These comparisons at C-Mod provide valuable insight into turbulent transport. Past work had shown that the nonlinear ion-scale GYRO simulations can match the power balance ion heat flux, but robustly under-predict the electron heat transport in C-Mod L-mode plasmas [39, 46]. In the new work here, we have shown that the ion-scale simulations also under-predict the experimental perturbative electron thermal di usivity. This indicates that the ionscale simulations under-predict the electron temperature profile sti ness, as had been suggested previously [39, 46], but was not shown directly until now. We attribute these disagreements with experiment to the missing ETG physics in the ion-scale simulations. Multi-scale simulations are more complete because the ion-scale turbulence (IGT and TEM) is simulated simultaneously with the electron-scale turbulence (ETG). Using nonlinear multi-scale GYRO simulations [39] it was shown that the experimental ion and electron heat fluxes from power balance could both be matched within uncertainties. The work presented in this paper finds that the multi-scale simulations also match experimental electron perturbative thermal di usivity within uncertainties. This strongly suggests that the past disagreements with ion-scale simulations were in fact due to the missing ETG physics, and indicates multi-scale simulations that capture the cross-scale coupling of the turbulence are required to model both the steady state (power balance) and perturbative electron thermal transport and sti ness in C-Mod L-mode plasmas. Unfortunately, realistic mass, multi-scale simulations like those used at C-Mod [39] are too computationally expensive to run routinely to model experimental plasmas, so the comparisons are at this point limited to the one experimental L-mode condition. Future work comparing the multi-scale model with experiment at C- Mod will likely be done using the trapped Gyro-Landau fluid model TGLF [53], which requires far less computing time and scans on the input temperature gradient can be run easily to map out the predicted perturbative thermal di usivity. Currently, the cross-scale coupling between ITG/TEM and ETG scale turbulence is being incorporated into TGLF, and we will use this tool in future work. Overall, the use of perturbative thermal di usivities calculated from partial sawtooth crashes at C-Mod will be part of future work to validate gyrokinetic simulations in detail (along with power balance analysis and turbulence measurements). There are several caveats with the experimental analysis used here that future work will explore. First, the radial resolution of the GPC diagnostic is limited, and the use of FRCECE data on Alcator C-Mod would allow for greater radial resolution. Analysis is in progress now, and the preliminary results show that the perturbative di usivities calculated with FRCECE agree to within error bars with the values calculated using GPC. Extensive use of the FRCECE data will be reported in future papers. While this paper focussed on comparisons between sti ness in I-mode and L-mode, it would be good to perform analysis of H-mode and compare with I-mode as well. While the extended-time-to-peak analysis used here has been compared in detail with Fourier analysis on other machines [15, 49], we plan to perform these same comparisons using the higher resolution FRCECE data in the future. In addition, it would be very valuable to perform a direct comparison of perturbative thermal di usivity calculated form modulated ECH and partial sawteeth in the same shot (to further establish using the partial sawtooth heat pulses as a routine experimental tool). C-Mod does not have this capability due to the lack of an ECH system, but data from tokamaks such as ASDEX Upgrade and DIII-D may allow for such direct comparisons. There are plans now to expand analysis to additional machines, for cross-machine comparisons. 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Appendix: History of Perturbative Di usivity Measurements via Sawteeth-Generated Heat Pulses In order to describe the historical context of the analysis presented in this work, this section will summarize the history of measuring perturbative thermal di usively via heat pulse analysis. As was stated above, one of the earliest sources of thermal perturbations, or heat pulses, that was utilized in experiment to measure the thermal di usivity was the sawtooth crash on the ORMAK tokamak [12]. Other sources of heat pulses include modulated electron cyclotron heating (ECH) [19] and impurity injection, which generates a cold pulse [20]. Thermal di usivity could be calculated via the time-to-peak method by tracking the propagation of the peak of heat pulses generated by the sawtooth crash or another actuator [12]. It was discovered, however, that the heat pulse e di usivity,, exceeded the power balance di usivity by factors of up to 15 [12]. This discrepancy was later partially mitigated on ORMAK, through the use of other methods involving more detailed calculations based on the same data [21]. Subsequent application of these methods on the ISX-B tokamak found agreement between power balance and heat pulse di usivities [22], while studies on Doublet III [23] and TFTR [24] found significant enhancement of the heat pulse di usivity over the power balance di usivity. A method based on Fourier analysis of regularly spaced sawtooth heat pulses was initially developed for Doublet III [23] and was then applied to TFTR with similar results [24]. This discrepancy on the larger machines with better diagnostics led to some speculation that the power balance di usivity was transiently enhanced during the propagation of the heat pulse [24]. Further investigation of these discrepancies led to the suggestion that the heat pulse di usivity was actually a di erent physical property than the power balance di usivity, and that steady state and perturbative thermal transport were characterized by di erent parameters [3]. This realization suggested that the heat pulse di usivity should be defined, as was described above, as, e = e n e (A.1) which makes it equivalent to the incremental, or perturbative, di usivity (the di usivity governing an incremental increase in heat flux), as opposed to the steady state power balance di usivity [3]. Figure 2, included in the Introduction, illustrates the di erence between these two quantities. This new understanding of the di erence between power balance and perturbative di usivity was accompanied by the development of the extended-time-topeak method of measuring the heat pulse di usivity, which tracks both the propagation of the pulse peak and its radial damping [48]. It should be noted that the extended-time-to-peak method was later compared to the Fourier analysis method, both theoretically and experimentally, and the two were found to generally