Perturbative thermal diffusivity from partial sawtooth crashes in Alcator C-Mod

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1 PAPER Perturbative thermal diffusivity from partial sawtooth crashes in Alcator C-Mod To cite this article: A.J. Creely et al 0 Nucl. Fusion 000 Manuscript version: Accepted Manuscript Accepted Manuscript is the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an Accepted Manuscript watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors This Accepted Manuscript is 0 IAEA, Vienna. During the embargo period (the month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND.0 licence after the month embargo period. After the embargo period, everyone is permitted to use copy and redistribute this article for non-commercial purposes only, provided that they adhere to all the terms of the licence Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. View the article online for updates and enhancements. This content was downloaded from IP address... on /0/0 at :0

2 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NF-0.R 0 0 Perturbative Thermal Diffusivity from Partial Sawtooth Crashes in Alcator C-Mod A.J. Creely, A.E. White, E.M. Edlund, N.T Howard, A.E. Hubbard MIT Plasma Science and Fusion Center, Cambridge, 0 MA, USA. Princeton Plasma Physics Laboratory, Princeton, NJ 0, USA Oak Ridge Institute for Science and Education (ORISE), Oak Ridge, TN, USA Abstract. Perturbative thermal diffusivity 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 diffusivity governs the propagation of heat pulses through a plasma. It differs from power balance thermal diffusivity, which governs steady state thermal transport. Heat pulses generated by sawtooth crashes have been used extensively in the past to study heat pulse thermal diffusivity [], but the details of the sawtooth event typically lead to nondiffusive ballistic transport, making them an unreliable measure of perturbative diffusivity on many tokamaks []. Partial sawteeth are common on numerous tokamaks, and generate a heat pulse without the ballistic transport that often accompanies full sawteeth []. This is the first application of the extended-timeto-peak method of diffusivity calculation [] to partial sawtooth crashes. This analysis was applied to over 0 C-Mod shots containing both L- and I-Mode. Results indicate correlations between perturbative diffusivity 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, diffusivities calculated from partial sawteeth are compared to perturbative diffusivities calculated with the nonlinear gyrokinetic code GYRO. We find that standard ion-scale simulations (ITG/TEM turbulence) under-predict the perturbative thermal diffusivity, but new multi-scale (ITG/TEM coupled with ETG) simulations can match the experimental perturbative diffusivity within error bars for an Alcator C-Mod L-mode plasma. Perturbative diffusivities 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 Diffusivity, Sawtooth Submitted to: Nucl. Fusion

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4 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NF-0.R 0 0 Partial Sawtooth Perturbative Thermal Diffusivity to refer to both quantities, with the superscript HP denoting the perturbative diffusivity. The perturbative and power balance thermal diffusivities, therefore, measure different quantities and should not be compared directly. Figure illustrates the difference between power balance and perturbative thermal diffusivity. 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 diffusivity. The slope of the green line is the heat pulse thermal diffusivity. These quantities are identical only if the heat flux and temperature gradient are linearly related with no offset []. The historical context of this difference in definition is described in more detail in the Appendix, and in [-]. In addition, the relationship between perturbative thermal diffusivity and profile stiffness means that experimental measurements of the perturbative diffusivity could be used to validate the gyrokinetic code GYRO [], along with more traditional experimental quantities such as the ion and electron power balance thermal diffusivities and fluctuation measurements. In this work, we will use the measured perturbative thermal diffusivity to investigate the importance of multiscale (coupled ITG/TEM and ETG turbulence) simulations [] for modeling the electron temperature profile stiffness []. In gyrokinetic simulations, the electron temperature profile stiffness is measured via scans of the slope of the electron heat flux, Q e, against a L Te above the critical gradient, where a is the minor radius and L Te is the electron temperature gradient scale length, L Te = Te T e []. 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 Q e (a/l Te ) = χhp GYRO ne T e a () One can therefore compare perturbative thermal diffusivity calculated from partial sawtooth-generated heat pulses to the perturbative diffusivity calculated via GYRO, and in doing so provide another metric for validation of GYRO. Perturbative thermal diffusivity calculated from ECH-generated heat pulses has been related to stiffness in the past at ASDEX, [] as well as compared specifically to scans from ion-scale (ITG/TEM turbulence only) gyrokinetic simulations at DIII-D []. Experimentally, once a diffusive heat pulse has been generated, one can calculate the perturbative thermal diffusivity 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 [] and []. 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 diffusivities from sawtooth crashes. While cold pulses have been used in C-Mod extensively to study perturbative heat transport [], heat pulses from sawteeth have only been examined in a cursory fashion at C-Mod []. Heat pulses generated from the sawtooth instability were for many years used to measure perturbative diffusivity [], but later work revealed that they were often accompanied by non-diffusive ballistic transport and were therefore not suitable for measuring this perturbative diffusivity on many machines []. 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 []. 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 diffusivity (which is related to stiffness), and then to compare against predictions from nonlinear gyrokinetic simulations. The Appendix provides further description of the differences between full and partial sawteeth, including a definition of ballistic transport. One of the motivations for a renewed interest in calculating χ HP e from sawtooth crashes is the relatively recent discovery of the I-mode (improved mode) confinement regime []. 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) []. I-mode has been observed on Alcator C-Mod, ASDEX Upgrade, and DIII-D []. This study will investigate differences in perturbative thermal diffusivity 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 stiffness than L-mode[, ], and thus one would expect to find higher perturbative thermal diffusivities 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 JET [] to calculate χ HP e from partial sawteeth. The next section will outline the details of the method used to calculate χ HP e, as well as describing the machine and diagnostics used in this study. The method of calculating perturbative thermal diffusivity from heat pulses generated by partial sawteeth was first used to further investigate the properties of I-

5 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NF-0.R Page of 0 0 Partial Sawtooth Perturbative Thermal Diffusivity mode and better understand its thermal confinement. More than 0 shots that contain both L- and I-mode phases were analyzed, and the relationship between the perturbative diffusivity and various global and local plasma parameters, as well as the confinement 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 of an Alcator C-Mod L-mode plasma. regime, was investigated. The experimental χ HP e. Experimental Methods.. Machine and Diagnostics All data analyzed in this study was taken on Alcator C-Mod [] which is a compact (R = 0. m, a = 0. 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 diffraction grating to split the electron cyclotron emission (ECE) spectrum. The GPC on C-mod has channels at different frequencies (and thus different radial locations, spaced by about cm), and up to 0 khz sampling [], though it is typically run at 0 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 nd harmonic X-mode electron cyclotron emission for electron temperature profile measurements using well established techniques []. 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 Diffusivity Calculation Method In order to calculate the perturbative electron thermal diffusivity, χ HP 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 []. The full derivation will not be presented here, but can be found in [] and []. 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 [, ], as further described in the Appendix. The result of the calculation in [] gives, χ HP e =. a cv HP α () where v HP = κ a a s ( dtpeak dr ) () is the radial velocity of the peak of the heat pulse in m/s, and α = (a s) d log(a) () 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 κ, 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 []. The method is applied outside of the sawtooth mixing radius, as intended. It has been successfully used on other shaped tokamaks, and accounts for differences in machine size and shaping. It must be noted that this methods calculates a radially averaged diffusivity, within the region of heat pulse propagation, which is responsible for minor disagreements with other methods of perturbative diffusivity calculation, such as Fourier analysis []. For Alcator C-Mod, a = 0., κ =., and s = In this study on Alcator C-Mod, the radial extent of the pulse measurements included r/a = 0. to 0. 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 sufficient magnitude to measure. In order to confirm the validity of this formula, including its numerical pre-factor, on Alcator C- Mod, the numerical calculations performed in [] and [] were repeated using C-Mod parameters. In our calculation, performed using the same method, but with the machine and measurement location parameters used in this study, a numerical pre-factor of between. and. was calculated, depending on the exact values of the mixing radius and measurement location. This is less than % different than the published value, consistent with the published uncertainty of % []. In order to remain consistent with past work, and since this difference is less than the uncertainty already included in the measurements below, the published value of. was retained. In addition, repeating this calculation verified that the formula should be equally applicable to both full and partial sawtooth-generated heat pulses.

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7 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NF-0.R Page of 0 0 Partial Sawtooth Perturbative Thermal Diffusivity Table : Comparison of experimental partial sawtooth heat pulse, full sawtooth heat pulse, and power balance (from TRANSP) diffusivities. All values are radially averaged. The three values disagree, as expected. The full sawtooth-generated heat pulses result in larger values of measured diffusivity, due to the ballistic transport. The power balance diffusivity is generally lower than the perturbative diffusivity, but this is not always the case. Detailed comparison of perturbative and power balance diffusivity is beyond the scope of this study. Shot Confinement χ HP Partial (m /s) χ HP Full (m /s) χ PB e (m /s) 0 L-Mode. ± 0.. ± Ohmic (LOC). ± 0.. ± Ohmic (SOC). ± 0.. ± L-Mode. ± 0.. ±.. 00 I-Mode.0 ± 0..0 ± L-Mode. ± 0.. ± 0.. to further justify our choice. Similar to past work, we found that the diffusivities calculated from full satweeth are larger than power balance values. We also found that diffusivities calculated from full satweeth are larger than diffusivities calculated from partial sawteeth. We discuss these differences within the context of past work, and with respect to evidence for 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 off-axis accompanied by a negligible change in on-axis temperature. Figure shows the differences between full and partial sawteeth on Alcator C-Mod GPC data, as well as representative heat pulses. Qualitatively, the time traces shown in Figure show the combination of sharp rise and gradual decay in temperature for full sawteeth that was incompatible with computational diffusive models in past work []. The partial sawtooth in Figure reveals a much more gradual rise, again consistent with the analysis performed in past work which indicates diffusive transport []. This indicates that the differences between partial and full sawteeth at C- Mod are consistent with the description of differences on other devices []. Based on these differences, and supported by modeling, past work has concluded that partial sawteeth should lead to diffusive transport while full sawteeth often do not []. In order to connect with past work studying heat pulse propagation from both full and partial sawteeth, we compared the perturbative thermal diffusivity calculated for partial and full sawteeth. Table shows the perturbative thermal diffusivity 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 0 C-Mod shots, which are described in more detail in Section. In all cases from C-Mod, the full sawteeth always lead to a perturbative diffusivity larger than partial sawteeth perturbative diffusivity. This observation is consistent with previous work []. Combined with the results of previous work [] and the qualitative shape of the heat pulses described above, this observation suggests that, at C-Mod, the full sawtooth crashes generate heat pulses that propagate based on nondiffusive ballistic effects, as often observed in other tokamaks, and the partial sawteeth generate heat pulses that can be described with a diffusive model []. Table also shows the power balance thermal diffusivity calculated via the TRANSP code []. 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 diffusivities differ for most of the cases presented here, with little clear correlation between the two. As described in the Introduction (see Figure ), the perturbative and power balance thermal diffusivities are measures of different 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 diffusive transport, and thus are valid for perturbative thermal transport studies, while full sawteeth lead to non-diffusive transport and are thus unreliable indicators of thermal diffusivity. For this reason the rest of this study will exclusively use partial sawtooth-generated heat pulses in it s calculations of perturbative thermal diffusivity.. Comparison of perturbative electron thermal diffusivities in L- and I-Mode on Alcator C-Mod This section will detail the perturbative diffusivity analysis done on more than 0 Alcator C-Mod shots using partial sawteeth-generated heat pulses. Each of these shots contained at least one stable L-mode phase

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12 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NF-0.R 0 0 Partial Sawtooth Perturbative Thermal Diffusivity Table : Table comparing the thermal diffusivities calculated from GYRO simulations of two discharges (00 and 0) with experiment. The table shows the GYRO power balance thermal diffusivities, χ PB GYRO, compared with experimental values, χpb Exp. The table also shows the GYRO incremental (perturbative) diffusivity, χ HP GYRO, compared with the experimental perturbative diffusivity from partial sawtooth analysis,. The χhp are the same as in Table. In contrast, the χpb Exp values are not the same as in Table χ HP Partial Partial, but are instead taken at the location of the GYRO run, as opposed to being radially averaged. All GYRO results are from [] and []. Shot Confinement GYRO Model χ PB Exp (m /s) χ PB GYRO (m /s) χ HP Partial (m /s) χ HP GYRO (m /s) 00 L-Mode Ion-scale 0. ± ± I-Mode Ion-scale 0. ± ± L-Mode Ion-scale 0. ± ± L-Mode Multi-scale 0. ± ± 0.. Q e /n e (kev m/s). I-mode L-mode T e(kev/m) Figure : GYRO scans of the L- and I-mode phases of Alcator C-Mod shot 00 used to calculate the incremental diffusivity. The slope of the line above the critical gradient is the incremental thermal diffusivity, which is directly comparable to the experimental perturbative diffusivity. For the I-mode, the value is χ HP GYRO = 0.m /s, and for the L-mode, χ HP GYRO = 0.m /s. See [] for details of the simulations. to compare the experimental perturbative thermal diffusivity calculated form ECH-generated heat pulses in DIII-D to GYRO simulations []. The simulation scans are used to map out the flux gradient space and identify the critical gradient and profile stiffness. Here we define temperature profile stiffness 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 []. 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, GYRO scans around Qi matched simulations (White PoP 0) a/l Te, there will be a critical gradient, a/l Te,crit, below which little heat flux is driven (an x-intercept). Whentheoutputheatfluxfromthescansisnormalized by density, Q e,gyro /n e, and plotted against T, the slope of the line above the critical gradient will have units of m /s, and can be compared directly with the experimental perturbative diffusivity. As an example, Figure shows the results of performing GYRO runs scanning the input a/l Te in the L- and I-mode confinement phases of shot 00. For the I-mode, the value is χ HP GYRO = 0.m /s, and for the L-mode, χ HP GYRO = 0.m /s. These values have been calculated from the GYRO scan results shown in Figure by applying Equation. Table shows all three sets of ion-scale GYRO simulations and the one set of multi-scale simulations. The table shows the GYRO power balance thermal diffusivities χ PB GYRO compared with experimental values χ PB Exp. The table also shows the GYRO perturbative diffusivity, χ HP GYRO (Equation ), compared with the experimental perturbative diffusivity from partial sawtooth analysis, χ HP P artial (Equation ). For shot 00,onesimulationisfortheLmodephaseand theothersimulationisforthei-modephaseoftheshot, and include only ion-scale turbulence []. As shown in Table,the power balance electron thermal diffusivity calculated form GYRO, χ PB GYRO, is clearly lower thanthepowerbalanceexperimentalvalue, χ PB Exp. This under-prediction is believed to be due to the use of ion-scale GYRO simulations, which neglect the ETG physics []. In addition, the perturbative thermal diffusivity calculated from GYRO, χ HP GYRO, is lower than the heat pulse experimental value, χ HP Partial, which means that GYRO under-predicts the electron temperature profile stiffness as well. For shot 0, the ion-scale simulation results for the L-mode plasma under-predict both experimental power balance thermal diffusivities and the experimental perturbative thermal diffusivities. In contrast, the multi-scale sim-

13 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NF-0.R Page of 0 0 Partial Sawtooth Perturbative Thermal Diffusivity ulation results for the L-mode plasma in 0 agree with both the experimental power balance and perturbative thermal diffusivities, within the experimental and code (±%) uncertainties.. 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 diffusivity 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 diffusivity. Generally speaking, this method can be applied to any tokamak plasma that contains a sufficient number of partial sawtooth crashes. Based on the results presented here, perturbative diffusivity measured with partial sawteeth is related to stiffness differences between L-mode and I-mode. We find a strong correlation between the level of perturbative diffusivity and the plasma density. Local temperature, temperature gradient, density, and density gradient also seem to show some correlation with perturbative thermal diffusivity. For the first time, the perturbative thermal diffusivity calculated form partial sawtooth-generated heat pulses has been compared with perturbative thermal diffusivities 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 [, ]. In the new work here, we have shown that the ion-scale simulations also under-predict the experimental perturbative electron thermal diffusivity. This indicates that the ionscale simulations under-predict the electron temperature profile stiffness, as had been suggested previously [, ], 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(itg and TEM) is simulated simultaneously with the electron-scale turbulence (ETG). Using nonlinear multi-scale GYRO simulations [] 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 diffusivity within uncertainties for an Alcator C-Mod L-mode plasma. This strongly suggests that the past disagreements with ionscale 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 stiffness in C-Mod L-mode plasmas. Unfortunately, realistic mass, multi-scale simulations like those used at C-Mod [] 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 [], which requires far less computing time and scans on the input temperature gradient can be run easily to map out the predicted perturbative thermal diffusivity. 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 diffusivities 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 diffusivities 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 stiffness 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 [, ], 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 diffusivity 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. Further comparisons to nonlinear gyrokinetic simulations are of interest, and the development of a synthetic diagnostic to mimic heat pulse propagation would be interesting

14 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NF-0.R 0 0 Partial Sawtooth Perturbative Thermal Diffusivity to pursue, but is beyond the scope of this paper.. Acknowledgements This work is supported by the US DOE under grants DE-SC000 and DEFC0-ER-CMOD.. References [] Lopes Cardozo, N.J., Plasma Phys. Control. Fusion, (). [] Fredrickson, E.D. et al., Phys. Plasmas, 0 (000). [] Tubbing, B.J.D. et al., Nucl. Fusion, (). [] Horton, W., Rev. Mod. Phys., (). [] Dimits, A.M. et al., Phys. Plasmas, (000). [] Ernst, D.R. et al., Phys. Plasmas, (00). [] Hillesheim, J.C. et al., Phys. Rev. Lett., 00 (0). [] Ryter, F. et al., Plasma Phys. Control. Fusion, A (00). [] Citrin, J. et al., Phys. Rev. Lett., 0 (0). [] Greenwald, M. et al., Phys. Plasmas, 0 (0). [] Greenwald, M. et al., Nucl. Fusion, (). [] Callen, J.D. and Jahns, G.L., Phys. Rev. Lett. (). [] Ryter, F. et al., Plasma Phys. Controlled Fusion, (0). [] Ryter, F. et al., Phys. Rev. Lett., (00). [] Mantica, P. et al., Nucl. Fusion, 0 (). [] Lopes Cardozo, N. J. and de Haas, J.C.M., Nucl. Fusion, (0). [] Lopes Cardozo, N.J. and Sips, A.C.C., Plasma Phys. Control. Fusion, (). [] Smith, S.P. et al., Nucl. Fusion, 0 (0). [] Jahns, G.L. et al., Nucl. Fusion, (). [0] Kissick, M.W. et al., Nucl. Fusion, (). [] Soler, M. and Calen J.D., Nucl. Fusion, 0 (). [] Bell, J.D. et al, Nucl. Fusion, (). [] Jahns, G.L. et al., Measurement of Thermal Transport by Synchronous Detection of Modulated Electron Cyclotron Heating in the Doublet III Tokamak, GA Technologies Rep. GA-A (). [] Fredrickson, E.D. et al., Nucl. Fusion, (). [] Fredrickson, E.D. et al., Phys. Rev. Lett., (0). [] Fredrickson, E.D. et al., Nucl. Fusion, (). [] DeLuca, F. et al., Nucl. Fusion, 0 (). [] Parail, V.V. et al., Nucl. Fusion, (). [] Gentle, K.W. et al. Plasma Sci. Technol., (00). [] Munsat, T. et al., Phys. Plasmas, 0 (00). [] Qin, D. et al., Plasma Sci. Technol., (00). [] Mantica, P. and Ryter, F., C. R. Physique, (00). [] Park, H.K. et al., Phys. Rev. Lett., 00 (00). [] Park, H.K. et al., Phys. Rev. Lett., 00 (00). [] Udintsev, V.S. et al., Plasma Phys. Control. Fusion, (00). [] Chapman I.T., Plasma Phys. Control. Fusion, 00 (0). [] Yamada, M. et al., Rev. Sci. Instrum., (). [] Candy, J. and Waltz, R.E., J. Comput. Phys., - (00). [] Howard, N.T. et al., Phys. Plasmas, (0). [] Citrin, J. et al. Nucl. Fusion, 00 (0). [] Rice, J.E. et al., Nucl. Fusion, 000 (0). [] O Shea, Peter Joseph Larkin, Thesis, Massachusetts Institute of Technology (). [] Whyte, D.G. et al., Nucl. Fusion 0, 00 (0). [] Hubbard, A.E. et al., Nucl. Fusion,, 0 (0). [] White, A.E. et al., Nucl. Fusion, 0 (0). [] White, A.E. et al., Phys. Plasmas, 0 (0). [] Hutchinson, I.H., Principles of Plasma Diagnostics, 00. [] Lopes Cardozo, N.J. et al., Nucl. Fusion, (). [] Jacchia, A. et al., Phys. Fluids B, (). [0] See for documentation concerning MATLAB and its functions. [] See for full documentation concerning the TRANSP code. [] Garbet, X. et al., Plasma Phys. Control. Fusion, (00). [] Staebler, G.M. et al., Phys. Plasmas, 00 (00). Appendix A. History of Perturbative Diffusivity 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 diffusively 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 diffusivity was the sawtooth crash on the ORMAK tokamak []. Other sources of heat pulses include modulated electron cyclotron heating (ECH) [] and impurity injection, which generates a cold pulse [0]. Thermal diffusivity 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 []. It was discovered, however, that the heat pulse diffusivity,χ HP e, exceeded the power balance diffusivity by factors of up to []. This discrepancy was later partially mitigated on ORMAK, through the use of other methods involving more detailed calculations based on the same data []. Subsequent application of these methods on the ISX-B tokamak found agreement between power balance and heat pulse diffusivities [], while studies on Doublet III [] and TFTR [] found significant enhancement of the heat pulse diffusivity over the power balance diffusivity. A method based on Fourier analysis of regularly spaced sawtooth heat pulses was initially developed for Doublet III [] and was then applied to TFTR with similar results []. This discrepancy on the larger machines with better diagnostics led to some speculation that the power balance diffusivity was transiently enhanced during the propagation of the heat pulse []. Further investigation of these discrepancies led to the suggestion that the heat pulse diffusivity was actually a different physical property than the power balance diffusivity, and that steady state and perturbative thermal transport were characterized by different parameters []. This realization suggested that the heat pulse diffusivity should be defined, as

15 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NF-0.R Page of 0 0 Partial Sawtooth Perturbative Thermal Diffusivity was described above, as, χ HP e = n e Q e T e (A.) which makes it equivalent to the incremental, or perturbative, diffusivity (the diffusivity governing an incremental increase in heat flux), as opposed to the steady state power balance diffusivity []. Figure, included in the Introduction, illustrates the difference between these two quantities. This new understanding of the difference between power balance and perturbative diffusivity was accompanied by the development of the extended-time-topeak method of measuring the heat pulse diffusivity, which tracks both the propagation of the pulse peak and its radial damping []. 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 be in good agreement [, ]. Studies on JET found agreement between the heat pulse thermal diffusivity calculated from sawtooth-generated heat pulses and the incremental diffusivity calculated based on a fit of the power balance diffusivities from many shots []. More extensive studies on JET of the dependencies of the heat pulse thermal diffusivity from sawteeth found similar results []. The concept of perturbative diffusivity and the new extended-time-to-peak method of measuring heat pulse diffusivity could not, however, explain the enormous enhancement of heat pulse diffusivity over power balance diffusivity by up to factors of 0 on TFTR []. The sawtooth event itself was found to be the cause of the discrepancy: the sawtooth crash led to additional enhanced transport beyond the mixing radius, which was termed ballistic transport, caused by the macroscopic MHD perturbations to the plasma equilibrium []. This ballistic transport was nondiffusive and was thought to explain the enhancement of the measured heat pulse diffusivity []. This suggested that sawtooth-generated heat pulses were unsuitable for the calculation of heat pulse thermal diffusivity, as they precluded a diffusive model of heat transport even beyond the mixing radius. This study did not, however, invalidate the extended-time-to-peak method of measuring diffusivity, so long as it was not applied to sawtooth-generated heat pulses. Over the next decade or so, various studies debated the validity of using sawtooth-generated heat pulses to study perturbative thermal diffusivity on various machines, including possible spatial or temporal limits on the ballistic transport [,,, ]. A thorough review of heat pulse thermal diffusivity research up to this point can be found in []. Most work with sawtooth-generated heat pulses ceased, however, after further evidence of the dominance of ballistic transport was presented for both TFTR and DIII-D []. Computational modeling revealed that the shape and propagation of the heat pulses generated by sawtooth crashes are inconsistent with a diffusive process, even one that has a highly nonlinear dependence on the temperature gradient []. In addition, measurements of heat pulses generated by partial sawteeth agreed with diffusive simulations, further supporting the theory that it was the sawtooth event itself that led to the ballistic transport [] (the absence of ballistic transport in partial sawtoothgenerated heat pulses forms the basis for the work presented in this paper). This same study revealed that theradialextentoftheballisticeffectwasatleasttwice the inversion radius, leaving very little radial range over which propagation of full sawteeth heat pulses may be considered diffusive. Later studies confirmed these conclusions and further characterized the ballistic transport []. Although a few studies have continued to use full sawtooth-generated heat pulses to measure perturbative diffusivity[, ], it is generally accepted that ballistic transport makes this approach unreliable [, ]. It is for this reason that ECH modulation and impurity injection are generally the preferred sources of heat (or cold) pulses for measuring perturbative thermal diffusivity []. The cause of the non-diffusive ballistic transport, and thus the reason that sawtooth-generated heat pulses cannot be used to measure a meaningful diffusivity is believed to be the MHD effects associated with the sawtooth crash []. A more detailed description of the physics of the sawtooth event is beyond the scope of this paper, but sources such as [,, ] may provide more information regarding recent research into this area. Collectively, the MHD nature of sawtooth crashes and the corresponding ballistic transport has been postulated as one reason that the perturbative χ HP e calculated from sawteeth is typically greater than the power balance χ PB e by a factor of to 0 [], even considering the difference in definition between the two quantities. In this study, the results from [] are taken as sufficient evidence to treat full sawtooth-generated heat pulses as an unreliable measure of thermal diffusivity, while partial sawtooth-generated heat pulses are more reliable. Given the difficulties posed by the use of full sawteeth for calculating perturbative thermal diffusivity, it is worthwhile to explore the use of partial or compound sawtooth crashes for perturbative thermal diffusivity measurements. Partial sawteeth are not be accompanied by the same MHD effects as full sawteeth and thus do not lead to the

16 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NF-0.R 0 0 Partial Sawtooth Perturbative Thermal Diffusivity same ballistic transport [, ]. A partial sawtooth crash is caused by a partial magnetic reconnection that is qualitatively different than that of a full sawtooth crash []. The same computational models that rule out full sawtooth-generated heat pulses as non-diffusive suggest that partial sawtooth-generated heat pulses can be modeled as diffusive []. In addition to computational models, one can compare the amount of energy released during partial and full sawteeth to evaluate the validity of using partial sawteeth for perturbative transport studies. While detailed approaches estimating the full energy release from sawteeth, such as in [], are beyond the scope of this study, a more qualitative approach based on examining temperature profiles, as used in [], may be applied. This rough estimate, applied to the temperature profiles shown above, indicates that partial sawteeth result in significantly smaller redistributions of energy than full sawteeth, and are therefore typically more valid for use in perturbative studies. Based on the evidence presented here, heat pulses generated from partial sawteeth are used to measure the perturbative thermal diffusivity of tokamak plasmas in this study.