Gyrokinetic analysis of inter-edge localized mode transport mechanisms in a DIII-D pedestal

The edge transport barrier (ETB), or pedestal, is a thin region in an H-mode plasma that is characterized by inhibited transport¹ due to the suppression of the deleterious combination of ion temperature gradient (ITG) and trapped electron mode (TEM) instabilities. The formation of a pedestal with sufficiently high temperature will be a requirement for the success of ITER.

Several instabilities have emerged as viable candidates for the mechanisms behind observed anomalous density and energy transport in pedestals.² These instabilities include electron temperature gradient (ETG) modes,^3–7 kinetic ballooning modes (KBMs),^8–11 and microtearing modes (MTMs),^5,10,12–17 along with the possibility of residual ITG^6,7,18/TEM¹⁹ fluctuations. Gyrokinetic simulations predict that the transport from these micro-instabilities along with neoclassical transport can often be combined to account for observed transport levels within experimental uncertainties.^5,7,18–25

Several studies have examined magnetic fluctuations in the pedestal region via external Mirnov coils.^26,27 Recently, studies have indicated that the MTM is the cause of prominent magnetic fluctuations observed in magnetic spectrograms^2,22,28–31 and a major source of inter-edge localized mode (ELM) electron heat transport in the steep gradient region of the pedestal.^{2,5,6,31–36} Figure 1 is a magnetic spectrogram of DIII-D shot 174082, with strong fluctuations at the $≈100$ and $≈400 kHz$ regions (circled in green) and low-frequency (⁠$≈50 kHz$⁠) magnetic “chatter” (circled in black).

Recent advances in experimental diagnostics on DIII-D have provided internal magnetic fluctuation data via a radial interferometer–polarimeter.^28,29 This technique provides line-integrated calculations of magnetic fluctuations across a chord of the plasma, instead of the traditional technique via external pickup coils.³⁷ This tool allows for a more direct comparison with gyrokinetic simulations. An initial comparison along these lines will be reported in Ref. 32.

Given this evidence in favor of MTM as a major pedestal transport mechanism, an outstanding question is the role played by the KBM, which is proposed by the EPED model as the salient inter-ELM transport mechanism limiting the pedestal pressure profile.³⁸ In this paper, we consider both MTM and MHD-like modes (such as KBM) as contributors to pedestal transport and the observed magnetic fluctuations.

To this end, we analyze DIII-D shot 174082. In DIII-D discharge 174082, magnetic fluctuation diagnostics produce data that can be clearly identified as MTMs. Specifically, the measured signals between ∼120 and ∼420 kHz in shot 174082 align well with predicted values for MTMs with the equilibrium profiles under consideration, estimating the electron diamagnetic frequency using the formula $ωe*=kyρscs(1/LTe+1/Ln)$⁠, where the electron temperature and density gradient scale lengths are $LTe,ne$⁠, respectively, ρ_s is the ion gyroradius, c_s is the plasma sound speed, and k_y is the binormal wavenumber. We demonstrate this correspondence directly with simulations using the gyrokinetic code GENE.^3,39 The experimentally observed frequencies match closely the values identified in our gyrokinetic simulations of MTMs, with global simulations indicating that the MTM is the fastest growing mode.

We also carry out several numerical and thought experiments to probe the possible role of MHD modes. By modifying the equilibrium beta, simulations show that an MHD instability can become the fastest growing global linear mode. These simulations exhibit significantly higher transport ratios for normalized particle flux than MTMs and have ion-diamagnetic-directed frequencies. They also have frequencies in the range of the low-frequency activity observed in magnetic spectrograms for this discharge.

We compare several metrics involving ratios of fluctuation amplitudes and transport quantities for both MTMs and MHD modes. This analysis suggests that the available data are consistent with the simultaneous activity of both MHD modes and MTMs provided that the former is limited largely to the particle transport channel. This is the first time gyrokinetic analysis has been carried out for MTMs and MHD modes finding experimental signatures (notably, frequencies in the magnetic spectrogram) of both in the same discharge.

This paper is outlined as follows: First, the details of the experimental configuration under investigation will be discussed in Sec. II, with emphasis on the methodology of reconstructing profile data. Second, the gyrokinetic simulations performed using these experimental profiles will be laid out in detail in Sec. III, including the results of local linear scans in collisionality and wavenumber, global linear wavenumber scans, and local nonlinear simulations at the peak in $ωe*$⁠. Third, simulation results will be compared with experimental transport and fluctuation measurements in Sec. IV.

The simulation of anomalous transport in pedestals requires profiles of temperature and density for all relevant particle species, the radial electric field, and a kinetic reconstruction of the magnetic equilibrium. To ensure that all of these effects are accurately obtained from the experimental discharge, extensive work has been performed using experimental diagnostics and modules found in the software suite OMFIT.⁴⁰ Kinetic equilibria were calculated manually from inter-ELM-averaged profiles and compared with the results from the CAKE code,⁴¹ which provides self-consistent kinetic equilibria from automatic profile fitting and the EFIT Grad–Shafranov solver.⁴² All profiles used in this study were then mapped onto the resulting kinetic equilibria. This study focuses on the baseline (174082) equilibrium from a study on the effects of pellet fueling on a neutral beam-heated plasma.⁴² This shot is heated entirely by neutral beams at a rate of 4.67 MW, with no pellet injection involved. The resulting magnetic geometry and plasma profiles give snapshots of the DIII-D experiment.

This work utilizes the resulting magnetic geometry and profile data as input for the gyrokinetic solver GENE³⁹ to simulate the inter-ELM pedestal micro-instabilities. To account for the stiffness of the transport mechanisms, sensitivity tests are performed with changes to local equilibrium gradients that fall within the tolerance of these reconstructions. Overall, these equilibria are accurate representations of the inter-ELM steady state for a given shot due to systematic reconstruction via CAKE/EFIT, and thus, the resulting gyrokinetic simulations are performed with experimental inputs that are as accurate and comprehensive as possible.

Figure 2 depicts the density and temperature profiles across the pedestal region. Ion temperature is calculated using charge exchange recombination (CER) to find carbon temperature, carbon is treated as the dominant impurity species in simulations, and T_i = T_C. The radial electric field, $Er/RBθ$⁠, is shown in Fig. 3. The magnetic geometry, pressure profile, and q-profile are shown in Fig. 4.

Advances in gyrokinetic codes have enabled accurate simulations of turbulence in fusion plasmas, matching experimental fluxes to an unprecedented degree. This is particularly true in the core plasma, where gyrokinetic orderings are more strictly satisfied and profile gradients are less extreme than in the pedestal.^40,44–53 Recently, simulations in the pedestal also demonstrate increasing correspondence with experimental trends and observations of transport and fluctuations.^{5–7,18–20,22,24,54}

The focus of this study is in the region of strongest electron temperature and density gradients near the plasma edge (⁠$ρ=0.945–0.99$⁠) in order to identify the mechanisms that mediate pedestal development during the inter-ELM cycle and, thus, contribute to the ultimate pre-ELM pedestal structure.

To gain insight into the driving mechanisms of observed turbulent fluctuations, a series of simulations using the gyrokinetic code GENE^3,39 was run based on the experimental scenario described above, scanning across a wide range of toroidal mode numbers. We consider both local and global simulations. Local linear simulations are performed at $ρtor=0.965$ where $ω*$ peaks, as this corresponds most closely to the frequency observed in global simulations and has been shown elsewhere to be the location at which MTMs peak.^22,31,56 For typical local simulations, a resolution of (⁠$kx,z,v∥,μ$⁠) = (12, 96, 45, 24) is used (where k_x is the radial wavenumber, z is the real coordinate along the field line, and $v∥$ and $v⊥$ are the velocities relative to the field line), while global calculations require (⁠$ρtor,z,v∥,μ$⁠) = (128, 96, 32, 16). Global simulations made use of the block structured grids feature in GENE,^57,58 allowing for lower resolutions in velocity space by resizing the velocity space domain to match the background temperature.

In Refs. 22, 31, 59, and 62, the alignment of mode-rational surfaces with the peak of the $ω*e$ profile was shown to be crucial for low-n MTM instability. This is an intrinsically global effect. However, in the present scenario, this phenomenon does not appear to be decisive, since many rational surfaces find close proximity to the peak of $ω*$ even at low toroidal mode numbers. Stated differently, the critical toroidal mode number for which rational surface alignment is important (see Ref. 62 for details) is very low for this discharge. The critical mode number is

where n_crit is the critical toroidal mode number, ρ_tor is the normalized radial flux surface coordinate, q is the safety factor $dϕ/dψ, ŝ$ is the magnetic shear $(1/q)(dq/dr)$⁠, and μ_crit is the distance in ρ between the nearest rational surface and peak in $ω*e$⁠. For toroidal mode numbers that exceed n_crit, multiple rational surfaces are found within the range of $δρ=2μcrit$⁠, and the modes are not subject to an offset stabilization effect.⁵⁹ The value of n_crit for this discharge is $≈4$⁠. Therefore, at toroidal mode numbers above n = 4, instabilities are not suppressed by misalignment between the rational surfaces and the $ω*$ profiles.

Growth rates and frequencies are shown in Fig. 5 for both local and global linear simulations. As can be seen, there are multiple branches (peaks). We first demonstrate that the modes are MTMs and subsequently probe the distinction between the branches.

To establish these modes as MTM, we first note that the experimental frequencies (⁠$≈100–400 kHz$⁠) closely match the MTM expectation for the given profiles: $ω=ω*e=kyρscs(1LTe+1Ln)$⁠, which results in frequencies in the range of $ω*e=120–500 kHz$ for toroidal mode numbers $n=3–14$ (we will describe detailed comparisons between GENE simulations and the spectrogram below).

GENE calculates heat flux using $Q=∫d3v12mv2f1vD$⁠, where $vE$ is the generalized $E×B$ velocity, which can be separated into electrostatic (⁠$vE,ES∝δϕ$⁠) and electromagnetic (⁠$vE,EM∝δv∥A∥$⁠) components. Figure 6 depicts the ratio $QEM/QES$ for the global linear simulations, for which a high value is a hallmark of the MTM. An additional point in favor of the MTM is the tearing parity of the $A||$ eigenfunction shown in Fig. 7. Both the real and imaginary parts are roughly symmetric about z = 0, indicating even parity for the parallel component of the magnetic vector potential.

Finally, the transport ratios shown in Fig. 8 indicate that the electron channel dominates the observed heat flux (⁠$χi/χe≪1$⁠), and the particle flux for each species is small in comparison with the electron heat diffusivity (⁠$Ds/χe≪1$⁠). Although this is also a signature of ETG modes, they can be excluded on the basis of the strong electromagnetic component of the flux and the large scales at which these modes are unstable.

Reference 31 demonstrates that two branches of MTM coexist in the DIII-D pedestal: a slab-like MTM at low n and a toroidal mode at higher n. These branches have distinct physical characteristics: the slab branch is insensitive to changes in ballooning angle and cannot exist in collisionless plasmas. By contrast, the toroidal mode is dependent on both collisionality and ballooning angle and can exist in collisionless systems. The coexistence of these two branches may offer insight into the virulence of microtearing modes across discharges.

As we show here, we identify the same phenomenon in this discharge, suggesting that it may be quite common in the pedestal. In contrast with Ref. 31, the toroidal branch in this discharge is unstable at lower toroidal mode numbers and in closer proximity (in n) to the slab branch.

We first examine the ballooning-angle dependence of two modes representative of the two branches, which is shown in Fig. 9 (this can only be examined for local linear simulations). The low-n mode exhibits very weak θ₀ dependence in contrast with the strong dependence of the high-n mode consistent with the characterization of each as slab or toroidal, respectively. The lower frequency electromagnetic fluctuation is associated with a lower toroidal mode number, due to the linear dependence of $ω*e$ on $kyρi$⁠. Sharp changes in the frequency of the n = 10 simulations are likely due to transitions between similar MTM mode branches.

We also investigate the collisionality dependence of the modes. As shown in Fig. 10, the high-n branch asymptotes to a constant, non-negligible, growth rate corresponding to a mode that retains MTM characteristics in the low collisionality limit. In contrast, the low-n branch transitions to an ion diamagnetic-directed instability with very low growth rate as collisionality approaches zero, indicating that this low-n MTM is strongly stabilized and becomes subdominant to another mode in this limit. Once again, this collisionality dependence is consistent with the modes' characterization as slab like and curvature driven. Several recent papers have investigated (analytically and numerically) collisionless microtearing instabilities.^31,60,61

While the anomalous transport in the edge region is of considerable interest, a significant portion of ion heat flux can be attributed to neoclassical transport. Using the code NEO,^63,64 the neoclassical transport levels for ions, electrons, and impurities are calculated. Figure 11 shows the total neoclassical heat loss from all three species, primarily in the main ion channel, indicating that neoclassical transport is significant, particularly toward the pedestal top. Pedestal top (⁠$ρ≈0.94$⁠) neoclassical heat losses are significant and decrease in the steep gradient region. For reference, we expect a total of $4.67MW$ of power losses in the pedestal, so these transport levels are significant. Figure 12 shows the total particle loss due to neoclassical transport. Although neoclassical transport is generally weak in the particle channel, these particle transport levels are close to those predicted by SOLPS modeling:³⁰ $∼1021$ particles/s. The code SOLPS predicts D and χ by iterating their value until modeled profiles (⁠$ne,Te,ni,Ti,Ptot$⁠) match experimental predictions.¹¹ In summary, these results suggest that neoclassical transport is important in the ion heat channel and in the particle channel.

To obtain estimates of anomalous power losses, fluctuation amplitudes, and transport coefficients, nonlinear simulations are required. Simulations of local nonlinear ETGs are performed at multiple radial locations in the pedestal by setting $ky,minρi=5$ and $ky,max=240$⁠, with sensitivity tests in the gradient drive to ensure that variations in the equilibrium profiles do not result in major changes to transport coefficients at that location, increasing $∇Te/Te$ by 20% and decreasing $∇ne/ne$ by 20% to keep $∇Pe$ fixed. Simulations of neoclassical heat losses are performed at the same radial locations as the ETG, with their value indicated by the blue-shaded region. The nominal profiles resulted in insignificant electron heat flux (P_ETG = 0.015 MW at its highest), and the modified profiles gave much higher losses (P_ETG = 1.26 MW at its highest), but the resulting increase in energy loss is still not enough to account for the experimentally observed losses as shown by the purple region in Fig. 13.

Nonlinear simulations (⁠$ky,minρi=0.02–0.24$⁠) of microtearing modes were carried out at the location of highest $∇Te$⁠. The simulations extend from $kyρi=0.02–0.94$ and employ kinetic electrons and adiabatic ions. The result is a high level of magnetic flutter (Q_em = 1.14 MW). An additional simulation with the same increase in $∇Te/Te$ by 20% and decrease in $∇ne/ne$ by 20% results in $Qem=1.32MW$⁠. Even with the modified gradients, there is still some heat flux deficit (1.5 out of 4.7 MW), particularly at the outermost point. This may be an indication of some deficiency in modeling tools, neglect of additional relevant transport mechanisms, and/or a consequence of the intrinsically uncertain input data. Clearly, however, the transport mechanisms considered are at experimentally relevant levels.

The snapshot of magnetic vector potential depicted in Fig. 14 shows a contour plot of the fluctuations caused by the microtearing modes. The corresponding heat flux spectrum is shown in Fig. 15.

Sensitivity studies of nonlinear microtearing saturation are under way, with variation of multiple driving mechanisms including β, $∇Te$⁠, and ν_ei with the target of formulating reduced models for pedestal MTM transport. The effect of E × B shear was tested and found to be negligible for this parameter point.

In this section, we compare predicted electromagnetic transport levels with a simple model relating heat diffusivity as directly proportional to the fluctuation amplitude through a Rechester–Rosenbluth-like relation:⁶⁵ 

where the parallel diffusivity is given by

The motivation is to characterize the extent to which the transport can be connected to the magnetic fluctuation level. This will inform, for example, interpretation of interior magnetic fluctuation diagnostics.²⁸ 

To test the validity of this model, we compare this estimate with the results found in GENE. In the nonlinear GENE simulation of this discharge, the electromagnetic electron heat diffusivity is found to be $χeem=0.16m2/s$ and the $B̃x/B0=0.055$⁠; this magnetic fluctuation amplitude is then used in a series of analytic predictions for electromagnetic heat flux. The calculation using the $k∥$⁠, calculated as an eigenfunction average of the parallel derivative at the peak linear growth rate (justified by the common observation that linear modes persist in the nonlinear turbulence) and the magnetic fluctuation amplitude from the nonlinear simulation, gives $χeem=0.68m2/s$⁠, which is reasonably close to the nonlinear simulation (i.e., same order of magnitude, $∼4$ times larger).

The use of the Rechester–Rosenbluth formula to model $χeem$ by using the k_z extracted from the nonlinear structure and $B̃x/B0$ from the nonlinear simulation fares quite well, $χeem=0.355$ (a factor of $∼2$ discrepancy). Estimating the diffusivity using the standard approximation $1/k∥=q0R$ fares much more poorly: $χeem=2.61m2/s$⁠. Figure 16 compares these diffusivities and the method of $k∥$ or $χe,∥$ used in their calculation.

Ultimately, this exercise shows that heat flux from magnetic flutter may be estimated (to within an order of magnitude) from experimental data, provided accurate observations of magnetic fluctuation amplitudes and parallel wavenumbers.

Motivated by the low-frequency activity in the magnetic spectrogram, we investigate low-frequency MHD instabilities by studying modified scenarios that produce such modes. Unstable local MHD modes are quite easy to produce. However, global MHD modes are much more challenging to produce for this scenario.

These results are obtained by increasing the pedestal β to levels surpassing the nominal experimental value (⁠$β=0.003 125$ at the center of global MTM simulations) without modifying the equilibrium profiles. By increasing the value of the pedestal beta by a factor of 2.4 for global (⁠$β=0.0075$⁠) and 1.8 (⁠$βnom=0.001$ and $βmod=0.001 88$⁠) for flux-tube simulations, keeping the profile gradients fixed, we identify unstable MHD modes. These values of β are clearly beyond experimental uncertainties. However, two considerations suggest that MHD modes may be unstable at more realistic values of β: (1) the mode is likely unstable at much lower values of β, but it is subdominant to MTM, and (2) for global simulations, the mode is likely more unstable with a more realistic radial boundary condition. The radial boundary condition for GENE simulations is a Dirichlet boundary conditions just inside the separatrix (MHD stability is well known to depend sensitively on the vacuum region). Moreover, there are additional uncertainties in density and temperature gradients as well as magnetic geometry (notably magnetic shear) that could be probed.

For the lowest toroidal mode numbers with unrealistically large β values, the resulting MHD-like instability becomes the most unstable mode. The real frequencies and growth rates for both local and global MHD-like simulations are shown in Fig. 17. Figure 18 shows transport ratios for a set of high-β simulations, which are consistent with the expectations from Ref. 2 for MHD-like modes. Note that the global simulations include the Doppler shift from flow shear, but the local simulations do not, which should be considered in the context of the mode frequencies shown in Fig. 17.

The characteristics of these modes are consistent with expectations of MHD-like modes, namely, significant particle diffusivity relative to heat diffusivity, real frequencies in the ion diamagnetic direction, similar heat transport in the ion and electron channels (⁠$χi/χe≈1$⁠), and heat transport dominated by electrostatic fluctuations (⁠$Qem/Qes<1$⁠). The global instability appears to be an Alfvénic mode, as it satisfies the relation $ω≈k∥vA$ rather than the KBM dispersion expectation of $ω≈ωi*/2$⁠. The parity appears to be ballooning, rather than tearing, as shown in Fig. 19. Further investigation into the nature of this instability is left for future work.

Another “fingerprint” that distinguishes microtearing instabilities from MHD-like modes comes in the form of parallel electric field cancelation. In kinetic MHD modes, $E∥$ is near zero while microtearing modes have a finite electric field linked to Ohm's law and the current that is driven around the rational surfaces. Figure 20 shows the normalized electric field cancelation factor as calculated by

differs significantly between MTM and MHD instabilities.

We note that this $E||$ criterion for MHD modes is more robust the closer the mode is to ideal MHD, i.e., $E∥≈0$ for ideal MHD modes, while a residual $E∥$ exists for their kinetic analogue. Extended MHD codes such as NIMROD⁶⁶ have the capacity to model considerable non-ideal behavior. Consequently, such MHD codes are clearly not constrained to model $E||∼0$ modes and the MHD-like transport ratios that stem from this constraint.² For highly electrostatic modes, i.e., ETGs, this ratio nears unity. MTMs are found to have significant $A∥$ and $ϕ$⁠, leading to $E∥≈0.5$⁠.

In this section, we will compare MTMs with MHD modes with the goal of drawing conclusions about their respective roles in pedestal transport. To that end, we will examine the sets of gyrokinetic simulations described above: (1) global and local linear MTMs, (2) local nonlinear MTMs, and (3) global and local linear MHD modes with modified β. Each set of simulations has its inherent strengths and weaknesses. The global linear MTM simulations capture background profile variation, which has been shown to be important in the past, and include kinetic ions (allowing for particle flux information) but do not predict fluctuation levels. The local nonlinear MTM simulations neglect background profile variation and kinetic ions but do predict a nonlinear saturation amplitude and heat flux. Nonlinear MHD simulations using gyrokinetics are extremely challenging, and so we limit the investigation to linear. The global linear MHD simulations with $2.4βnomglobal$ include background profile variation and information regarding the width of the drive region,⁶⁷ but extremely high β is required to produce an MHD mode.^68,69 In contrast, the local linear MHD simulations with $1.8βnom,local$ neglect the radially nonlocal effects but do not require such extreme values of β to destabilize as their global counterpart.

These simulations will be examined in the context of the following experimental (or edge modeling) data:

• Magnetic spectrograms.

• The ratio of density fluctuation amplitude to heat flux.

• The ratio of magnetic fluctuation amplitude to density fluctuation amplitude.

• The ratio of particle flux to heat flux (calculated from SOLPS simulations).

For some of these analyses, it is important to keep in mind that experimental diagnostics often target the outboard midplane. For instabilities such as the ITG (in the core) or KBM, this coincides with the peak fluctuation amplitude. However, microtearing instabilities in the pedestal peak off-axis, leading to a minimum in fluctuations at the outboard midplane. Figure 21 depicts the ratio of line-averaged magnetic to density fluctuations as a function of height (above or below the outboard midplane). Consequently, although measures of outboard midplane fluctuations are highly useful, they are not able to diagnose important aspects of our MTM predictions.

Figure 22 shows that the fastest-growing linear MTM is approximately at the same real frequency (⁠$∼400 kHz$⁠) as the experimentally observed signal in Fig. 1. The lower band in the spectrogram $f∼100–200 kHz$ is also quite close to the frequency of the low k_y MTM peak (slightly over 100 kHz). The experimentally observed frequency range is the broadest at the time from which the equilibrium data were acquired (⁠$t=3000 ms$⁠), which is perhaps consistent with the unstable MTMs in the simulations spanning the whole frequency range.

There is also a low-frequency band in the spectrogram (circled in black in Fig. 1), which we will compare with the MHD frequencies. The mode frequencies of the (global, which accounts for the Doppler shift) simulated MHD instabilities range from $35 to 55 kHz$ (recall Fig. 17), consistent with this band.

In summary, the higher frequency bands correspond quite well with the peaks in the global MTM growth rate spectrum, while the lower band corresponds quite well with the simulated MHD modes. We find it plausible that both are active simultaneously in the pedestal and affect different transport channels. The MTM instabilities would mediate the electron heat flux and temperature profiles, while the MHD modes would be limited primarily to particle transport.² We further explore the plausibility of this picture in Secs. IV B–IV D.

The ratio of magnetic fluctuations to density fluctuations is also a quantity that could differ between modes and can be measured by the recently developed Faraday effect radial interferometer–polarimeter (RIP) diagnostic^28,29 (although such fluctuation measurements are not available for this discharge).

MHD-like modes are expected to exhibit higher $δn¯/nδB/B¯$ in comparison with MTM modes at the outboard midplane. However, simulations of the increased β equilibria show that the MHD-like modes exhibit a $δB¯/Bδn¯/n$ ratio that is similar to (and, in fact, generally larger than) their microtearing counterpart, as shown in Fig. 23. Experimental expectations for KBMs predict that this ratio would be an order of magnitude lower than the value from MTMs. These MHD modes producing a higher value than their MTM counterpart is further evidence that this instability is unlikely to be a KBM. Consequently, this metric alone cannot cleanly discriminate between MTMs and all MHD-like modes, and thus, the following other quantities must also be compared.

Here, we examine another prospective experimentally accessible metric, the ratio of density fluctuations to heat flux $(δn/n0)/Q$⁠. In this discharge, beam emission spectroscopy (BES) is used to place an upper bound on the possible density fluctuation level. Shot 174082 is distinctive because the density fluctuation amplitude falls below the noise level of BES measurements. This places an upper bound on the possible fluctuation levels (⁠$δn/n≈0.1%$⁠), which we will use in comparison with simulated fluctuation amplitudes.

Figure 24 shows this ratio for the series of global MTM (blue solid line) and MHD (gold solid line) simulations, as well as for the threshold experimental fluctuation level $δn/n≈0.1%$ described above (gray solid line) and the local nonlinear microtearing simulation (gold dashed line). As shown in Fig. 24, the nonlinear MTM simulation produces a ratio quite close to this experimental estimate. However, the estimate from linear MHD is somewhat larger. Since this density fluctuation level is an upper bound, this would imply that the MHD mode is unlikely to account for the bulk of the heat flux. This does not eliminate the possibility of MHD fluctuations since (1) the value of $(δn/n0)/Q$ for the MHD modes is not too much larger than the experimental upper bound, and (2) even at a low fluctuation level, the MHD modes could still strongly impact the particle transport channel.

No conclusions can be drawn from the diagnostic signals regarding the significance of the density fluctuations. However, this information can be used to constrain our numerical experiments. The diagnostics show that the density fluctuations cannot exceed a certain value. That information gives insight from the set of linear simulations. The linear simulations do not provide transport levels (i.e., fluxes) or fluctuation levels, but they do provide ratios between these quantities. The lack of a density fluctuation signal places an upper bound on the density fluctuations for a given mode, which then, in turn, constrains the possible contribution of that mode to the various transport channels. In summary, this analysis is consistent with the nonlinear MTM simulations described above but cannot rule out the possibility of MHD fluctuations as well. The combined MTM and MHD scenario proposed above remains a possibility.

As a final comparison between MHD and MTM, we compare the ratio of particle to heat flux. A robust feature of MHD modes is that they produce roughly equal diffusivity in the particle and heat channels: $D∼χ$ (this criterion is increasingly stringent with proximity to the ideal MHD limit). In contrast, MTMs produce much higher heat transport: $D/χ≪1$⁠. The particle diffusivity is difficult to diagnose experimentally, but edge modeling can provide an estimate. SOLPS was used to this end for this discharge as described in Ref. 30.

Figure 25 depicts the values of $Γtot/Qtot$ for the linear simulations of MTMs (global) and MHD-like modes (global and local) along with the SOLPS¹¹ iterative model (recall that these nonlinear MTM simulations cannot be used here because of the adiabatic ion approximation). The unstable microtearing modes agree with the value found in SOLPS, whereas the MHD modes exceed the SOLPS prediction. Local values of $Γtot/Qtot$ in kinetic MHD modes are comparable with their global counterparts. If the MHD modes are active, their impact would be largely limited to the particle channel as shown by the significant particle to heat flux ratios in both local and global simulations. Our conclusion from this analysis is similar to that of Sec. IV C: MHD modes may be active but only at a level that can impact the particle transport rather than the heat transport.

Global GENE simulations of DIII-D shot 174082 have shown that the most unstable mode for a series of toroidal wavenumbers, with real frequencies corresponding to a broad band of observed magnetic fluctuations, is a microtearing mode. Simulations also indicate that ETG turbulence and neoclassical effects cannot account for the total heat transport observed in this shot. A significant cause of energy loss in the pedestal is found to be the microtearing mode, as evidenced by a combination of nonlinear local, linear local, and linear global simulations of the pedestal region. The MTM is the fastest-growing linear instability, with gyrokinetic fingerprints that correspond well with the expectations of analytic theory, the observations of experimental diagnostics, and the resulting gyrokinetic simulations of carefully reconstructed equilibria. Notably, the MTM frequencies are in good agreement with bands in the magnetic spectrogram.

Modifying the equilibrium β to $2.4×$ the nominal experimental value shows that an unidentified MHD mode, possibly a toroidal Alfvén eigenmode or ballooning mode, becomes unstable, with fluctuation frequencies consistent with a band of fluctuations observed in the experimental magnetic spectrogram. This low-frequency mode may be responsible for inter-ELM particle transport. Further characterization of this instability is left for a future project.

This paper has investigated transport and magnetic fluctuations in the pedestal using gyrokinetic (using the GENE code) simulations for DIII-D discharge 174082. The main findings are as follows: Simulated MTM frequencies are in good agreement with corresponding frequency bands in the magnetic spectrogram, and nonlinear MTMs produce experimentally relevant transport levels. We conclude that it is very likely that MTMs are responsible for the observed magnetic fluctuations and that it is likely that they also limit the electron temperature pedestal. We also speculate that MHD modes are responsible for low-frequency fluctuations found in the magnetic spectrogram. Although GENE does not identify unstable MHD modes as the most unstable mode for the nominal equilibrium reconstruction, high beta (⁠$β=2.4βnom$⁠) simulations produce MHD modes with frequencies in the right range. Although this β value is clearly beyond experimental levels, other factors may suggest that MHD modes may be unstable at more realistic values of β, notably limitations of the GENE radial boundary condition and the possibility of subdominant instability.

We investigate several metrics that may inform or constrain the possible roles of MHD and MTMs. Two metrics are, perhaps surprisingly, ineffective at discriminating between the two classes of fluctuations. The ratio of density fluctuation amplitudes to magnetic fluctuation amplitudes is comparable for MHD and MTM based on linear GENE simulations. Likewise, the ratio of density fluctuation amplitudes to heat flux is somewhat larger for MHD modes, but not so large as to preclude their activity in this discharge. One metric, however, clearly distinguishes between MHD and MTM: the ratio of particle to heat diffusivity is much smaller for MTMs in agreement with SOLPS predictions. While this does not preclude the possible activity of MHD modes, it does limit their role to the particle transport channel.

We also investigate neoclassical and ETG as prospective transport mechanisms. Neoclassical transport (as calculated with NEO) produces substantial ion heat flux and particle flux roughly in agreement with SOLPS interpretive modeling. ETG transport is negligible for the nominal gradients. For simulations with moderately higher η (within experimental uncertainties), ETG transport becomes relevant although still somewhat below experimental expectations.

Two possible transport pictures emerge, (1) one without MHD fluctuations and (2) one including MHD fluctuations. In scenario (1) (negligible MHD fluctuations), neoclassical transport accounts for the particle channel and ion heat channel, while ETG and MTM supply the electron heat transport. Simulated values for these (neoclassical, ETG, MTM) transport mechanisms are consistent with this picture. In scenario (2), MHD combines with neoclassical to account for the particle transport, and other channels are the same as in scenario (1). Although we are unable to probe this possibility with nonlinear MHD simulations, there is a low-frequency fluctuation band in the spectrogram that may signal such MHD activity.

This work is supported by U.S. DOE under Grant Nos. DE-FC02-04ER54698, DE-FG02-97ER54415, DE-AC02-09CH11466, and DE-SC0022164. A portion of this study's data analysis was performed using the OMFIT integrated modeling framework.⁴⁰ This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy (DOE). The publisher acknowledges the U.S. Government License to provide public access under the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). This report was prepared as an account of work sponsored by an agency of the U.S. government. Neither the U.S. government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. government or any agency thereof.

The authors have no conflicts to disclose.

Michael R. Halfmoon: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Ehab Hassan: Data curation (equal); Resources (equal); Software (equal); Writing – review & editing (equal). Emily Ann Belli: Formal analysis (equal); Resources (equal); Software (equal). Jeff Candy: Formal analysis (equal); Resources (equal); Software (equal). David R. Hatch: Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Michael Kotschenreuther: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal). Swadesh Mitter Mahajan: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal). Andrew Oakleigh Nelson: Data curation (equal); Investigation (equal); Resources (equal); Software (equal). Egemen Kolemen: Investigation (supporting); Resources (supporting); Supervision (equal). Max Tian Curie: Formal analysis (equal); Investigation (supporting); Resources (supporting); Visualization (equal). Ahmed Diallo: Formal analysis (equal); Investigation (equal); Validation (supporting). Richard Groebner: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.