# Gyrokinetic Simulations Compared with Magnetic Fluctuations Diagnosed with a
Faraday-Effect Radial Interferometer-Polarimeter in the DIII-D pedestal

M. T. Curie  Institute for Fusion Studies, University of Texas at Austin,
Austin, TX, 78705  General Atomics, San Diego, CA, 85608  Princeton Plasma
Physics Laboratory, Princeton, NJ 08540<EMAIL_ADDRESS>D. R. Hatch
Institute for Fusion Studies, University of Texas at Austin, Austin, TX, 78705
M. Halfmoon  Institute for Fusion Studies, University of Texas at Austin,
Austin, TX, 78705  J. Chen  Department of Physics and Astronomy, University of
California, Los Angeles, Los Angeles, CA, 90095  D.L. Brower  Department of
Physics and Astronomy, University of California, Los Angeles, Los Angeles, CA,
90095  E. Hassan  Oak Ridge National Laboratory, Oak Ridge, TN, 37830
Physics, Faculty of Science, Ain Shams University, EG, 11566  M.
Kotschenreuther  Institute for Fusion Studies, University of Texas at Austin,
Austin, TX, 78705  S. M. Mahajan  Institute for Fusion Studies, University of
Texas at Austin, Austin, TX, 78705  R. J. Groebner  General Atomics, San
Diego, CA, 85608  DIII-D team  General Atomics, San Diego, CA, 85608  See the
author list of ’DIII-D research advancing the physics basis for optimizing the
tokamak approach to fusion energy’ by M. Fenstermacher et al. to be published
in Nuclear Fusion (Accepted on 14 October 2021)

###### Abstract

Experimental data on electromagnetic fluctuations in DIII-D, made available by
the Faraday-effect Radial Interferometer-Polarimeter (RIP) diagnostic13, is
examined in comparison with detailed gyrokinetic simulations using Gyrokinetic
Electromagnetic Numerical Experiment (GENE). The diagnostic has the unique
capability of making internal measurements of fluctuating magnetic fields
$\frac{\int n_{e}\delta B_{r}dR}{\int n_{e}dR}$. Local linear simulations
identify microtearing modes (MTMs) over a substantial range of toroidal mode
numbers (peaking at $n=15$) with frequencies in good agreement with the
experimental data. Local nonlinear simulations reinforce this result by
producing a magnetic frequency spectrum in good agreement with that diagnosed
by RIP. Simulated heat fluxes are in the range of experimental expectations.
However, magnetic fluctuation amplitudes are substantially lower than the
experimental expectations. Possible sources of this discrepancy are discussed,
notably the fact that the diagnostics are localized at the mid-plane—the
poloidal location where the simulations predict the fluctuation amplitudes to
be smallest. Despite some discrepancies, several connections between
simulations and experiments, combined with general criteria discriminating
between potential pedestal instabilities, strongly point to MTMs as the source
of the observed magnetic fluctuations.

††preprint: AIP/123-QED

##  I Introduction

This paper presents comparisons between gyrokinetic simulations and novel
measurements of internal magnetic fluctuations in the pedestal of a DIII-D
discharge.

The pedestal forms when virulent fluctuations are suppressed at the edge of
the Tokamak. Due to the large impact of the pedestal on confinement, the
origin and role of the residual fluctuations is a key question for developing
a comprehensive understanding of the pedestal and projecting its behavior to
foreign parameter regimes like those envisioned for burning plasmas.

Gyrokinetic simulations have demonstrated ever greater fidelity in comparison
with experimental observations in the core plasma. 10; 33; 32; 29; 28

Notably, multi-channel transport, fluctuation levels, cross phases, and
turbulent spectra have all been favorably compared between simulation and
experiment. Although gyrokinetic pedestal simulations have been advancing 8;
27; 26; 25; 19; 24; 18; 23; 7 , detailed comparisons with fluctuation
amplitudes are rare. This paper reports on some of the first such comparisons
in the pedestal. Such a comparison is enabled by novel measurements of
internal magnetic fluctuations from the recently-developed Faraday-effect
Radial Interferometer-Polarimeter (RIP)13. This diagnostic directly measures
the radial magnetic fluctuation levels across the mid-plane. In contrast,
earlier fluctuation data from magnetic pick-up coils (external to the plasma)
were capable of determining fluctuation frequencies but not fluctuation
amplitudes.

In this paper, we describe gyrokinetic simulations 30; 31 of a DIII-D
pedestal. The simulations identify microtearing modes (MTMs) as the main ion-
scale instability. Nonlinear simulations produce frequency spectra, transport
levels, and collisionality dependence in good agreement with these
experimental measurements. However, the simulations predict magnetic
fluctuation levels lower than the experimental expectation. Possible sources
for this discrepancy are discussed. We also consider the ratio of magnetic
fluctuations to density fluctuations. Although there is also some discrepancy
between experiment and simulation for these measurements, the fluctuation’s
features are generally consistent with MTM and inconsistent with other
instabilities.

The structure of the paper is outlined as follows:

  * •

Background of the theory and experiment. The significance of MTM will be
discussed. And the brief description of the Faraday-effect Radial
Interferometer-Polarimeter (RIP) will be provided.

  * •

Linear Investigation We study the linear eigenmode spectrum, identifying the
salient modes and making preliminary comparisons with the RIP frequency
spectrum.

  * •

Nonlinear Investigation A set of direct comparisons between experiment and
local nonlinear MTM simulations will be performed including scans of the
frequency spectrum and collisional dependence. Local nonlinear ETG (electron
temperature gradient) simulations are conducted as a contrast with local
nonlinear MTM simulations to study spatial dependence and fluctuation ratio. A
comparison of electron heat transport between the experiments and simulations
is performed by adding the electron heat transport of ETG and MTM from
simulations.

##  II Theory and Experimental Background

Recent work has elucidated the potential roles of various transport mechanisms
in the pedestal. Historically, Kinetic Ballooning Mode (KBM) has been proposed
as the salient transport mechanism in the pedestal due to its utility in the
EPED model and simulation results suggesting that pedestal profiles lie near
the KBM limit 22; 21; 26; 20. Recent work, however, has determined that a
variety of mechanisms is required to account for pedestal transport in all
channels 17; 49; 15; 12. In particular, edge modeling in simulations 49
typically predicts that the effective electron heat diffusivity far exceeds
the electron particle diffusivity: $D_{e}/\chi_{e}\ll 1$. In contrast
Magnetohydrodynamic (MHD) modes, like KBM, are characterized by
$D_{e}\sim\chi_{e}$. Moreover, KBM has been found to be in a second stability
regime in some scenarios. While these observations do not eliminate KBM as a
prospective transport mechanism, it does demand investigation of additional
transport mechanisms and suggests that other mechanisms are necessary to
describe pedestal transport in all transport channels. Table 1 shows the
distinctive ‘fingerprints’ of various prospective transport mechanisms,
notably MTM, which is the main focus of this paper. There is growing evidence
that in the electron heat channel, electron temperature gradient (ETG)
turbulence and MTMs fill this role 26; 16; 9; 17; 46, likely often in tandem
18; 45.

Mode Type | $\chi_{i}/\chi_{e}$ | $D_{e}/\chi_{e}$ | $D_{z}/\chi_{e}$ | $Q_{em}/Q_{es}$ | Shear-
---|---|---|---|---|---
|  |  |  |  | suppressed?
MTM | $\sim 0$ | $\sim 0$ | $\sim 0$ | $>1$ | Sometimes
ETG | $\sim 0$ | $\sim 0$ | $\sim 0$ | $<1$ | No
MHD-like | $\sim 1$ | $\sim 2/3$ | $\sim 2/3$ | $>1$ | No
ITG/TEM | $\geq 1$ | $-0.2-1$ | $\sim 1$ | $<1$ | Usually
Table 1: Theoretical estimates of transport ratio for different instabilities,
$D_{s}$ is particle diffusion coefficient for species ’s’, $\chi_{s}$ is heat
diffusion coefficient for species ’s’, $Q_{em}$ and $Q_{em}$ are
electromagnetic and electrostatic heat flux respectively.

High-frequency magnetic fluctuations originating in the pedestal have long
been observed with no clear explanation for the underlying physical mechanism
35; 36; 36; 43; 42; 37; 38; 39; 41; 13; 14; 40. Recently, gyrokinetic
simulations have, with increasing certainty, established MTMs as the
underlying mechanism for at least a major class of these fluctuations. Ref.26
demonstrated that gyrokinetic simulations of MTMs can produce experimentally
relevant transport levels for a JET discharge, and Refs.17; 18; 16
demonstrated close agreement with the distinctive quasi-coherent frequency
bands measured by magnetic pick-up coils. Refs. 18; 16 provided an explanation
for the distinctive band structure: MTMs arise at the peak of the electron
diamagnetic frequency $\omega_{*,e}=k_{y}\rho_{s}c_{s}(1/L_{Te}+1/L_{ne})$
profile only for mode numbers whose rational surfaces align with this peak.
This numerical observation has been further elucidated by basic theory, which
demonstrates that this behavior is due to the profile variation in the
pedestal 15; 9.

###  II.1 Internal Measurements of Magnetic Fluctuations on DIII-D

Recent experimental work based on the RIP diagnostic has further established
MTM as the mechanism for magnetic fluctuations by demonstrating several
parameter dependencies that are consistent with MTM and inconsistent with
other candidates 14.

Radial-interferometer-polarimeter (RIP) is a newly developed Faraday-effect
polarimeter in DIII-D, which measures the electron density-weighted magnetic
fluctuation along horizontal chords at and around the mid-plane as the Fig. 1
shows. The measured line-averaged radial magnetic fluctuation amplitude from
the RIP can be expressed as

$\delta\bar{B}_{r}\approx\delta\bar{B}_{R}\equiv\frac{\sqrt{\overline{\left(\int
n_{e}\delta B_{R}dR\right)^{2}}}}{\int n_{e}dR}$ (1)

where $\sqrt{\overline{\left(\int n_{e}\delta B_{R}dR\right)^{2}}}$ is the
root mean square (RMS) of the electron density-weighted internal magnetic
fluctuation. 46 $n_{e}$ is electron density, $\delta B_{R}$ is the magnetic
fluctuation along horizontal chords. $dR$ is differential element of radius.

Figure 1: This figure shows the schematic of relevant diagnostic: 3 horizontal
lines show 3 horizontal chords of RIP diagnostic at the mid-plane (0 cm), 13.5
cm above the mid-plane, and 13.5 cm below the mid-plane. The green box is Beam
Emission Spectroscopy (BES) located at the pedestal. This figure is reused
from Chen (2021) 14

The detailed experiment setup can be found in Chen (2021)14

This diagnostic provides a direct measurement of the absolute magnetic
fluctuation amplitude inside the plasma. Although this diagnostic is not
localized radially, magnetic fluctuations in the core generally have lower
frequencies ($\sim 1kHz$) so that frequency filtering (in the range $150\sim
500kHz$) can effectively extract the relevant pedestal fluctuations. And the
fact that the RIP observed frequency band responds to ELMs (edge-localized
mode) and correlates to local measurements (BES) of electron density at the
pedestal further proves that the $150\sim 500kHz$ frequency band is localized
at the pedestal.14

For more experimental aspects of this research, the reader is referred to the
following publications: 13; 14; 46

###  II.2 DIII-D Discharge 175823

All the simulations conducted in the article are based on DIII-D H-mode shot
175823. This discharge exhibits type-I ELMs with ELM frequency $\sim 100Hz$.
The plasma is heated via neutral beam injection. Its magnetic geometry has an
upper single-null. Its line averaged density is $5.5\times 10^{19}/m^{3}$ and
$H_{98}\sim 1.1$. The further detail of this discharge can be found in Chen
(2021) 14. The profiles of density, temperature, electron diamagnetic
frequency, and safety factor $q$ are shown in Fig. 2, where $\omega_{*e}$ is
the electron diamagnetic frequency, $q$ is the safety factor
$q\equiv\frac{d\Psi_{T}}{d\Psi_{P}}$, $\hat{s}$ is the magnetic shear
$\hat{s}\equiv\frac{r}{q}\frac{dq}{dr}$, $L_{T_{e}}$ is the length scale of
the electron temperature gradient, $L_{n_{e}}$ is the length scale of the
electron density gradient. $a$ is the minor radius, $r$ is the minor radial
length for the given location, $\rho_{s}$ is the ion Larmor radius, $c_{s}$ is
the speed of sound in plasma.

Figure 2: The experimental equilibrium profile: electron temperature $T_{e}$
(top left), electron diamagnetic frequency $\omega_{*e}$ (bottom left),
electron density $n_{e}$ (top right), safety factor $q$ (bottom right). The
orange data points are experimental data, and the blue curves are the fitted
profile based on the experimental data (orange dots). And the red vertical
lines mark $rho_{tor}=0.98$, where the local simulations are conducted in the
article.

The RIP diagnostic provides line integrated radial magnetic fluctuation
amplitudes 13 $\frac{\int n_{e}\delta BdR}{\int n_{e}dR}$ across the mid-
plane. This provides us with a unique insight into the internal measure of
magnetic fluctuations.

The magnetic frequency spectrum is one of the key experimental signatures that
we wish to compare with simulation results. Consequently, we need an estimate
of the Doppler shift of the fluctuations. This is provided by CXRS (Charge
eXchange Recombination Spectroscopy), which determines the radial electric
field from the radial force balance of the carbon species. The resulting
Doppler shift for instability with toroidal mode number $n$ 18:

$\Omega_{Doppler}=\frac{nE_{r}}{RB_{\theta}}$ (2)

where $E_{r}$ is the radial electric field, $B_{\theta}$ is the poloidal
magnetic field, $R$ is the major radius, n is the toroidal mode number. For
this discharge, taking $n=1$, the pedestal rotation frequency is calculated to
be $E_{r}/RB_{\theta}\sim 10krad/s$. There are significant error bars on this
estimate, which are difficult to quantify. We take a rough estimate of the
error bars by examining uncertainties for the pressure gradient, which makes
the main contribution to $E_{r}$ in the radial force balance equation. This
exercise suggests that the contribution of the pressure gradient to the
rotation is $\sim 25\pm 10krad/s$. Consequently, we test two values for the
rotation in this work: $\Omega_{Doppler}=10krad/s$ and
$\Omega_{Doppler}=20krad/s$.

##  III Gyrokinetic Simulations

The profile and equilibrium for DIII-D discharge 175823 is constructed by
kinetic EFIT2. We use the reconstructed profiles and equilibrium for DIII-D
discharge 175823 for all simulations. In this paper, we focus on local flux
tube simulations. Previous work has shown that global simulations are required
to capture the sensitive selection of frequency bands and toroidal mode
numbers that are often observed in magnetic fluctuation data 18; 15. However,
this discharge has rather high magnetic shear ($\hat{s}=5$), and the
fluctuations are characterized by single broadband representing multiple
toroidal mode numbers ranging from 10 to 20. Consequently, a local
approximation is reasonable (with caveats for nonlinear simulations, which
will be discussed below). Previous results indicate that the global modes
localize at the peak of the $\omega_{*e}$ profile, providing a clear radial
location at which to perform the local analysis.

Numerical details as well as a discussion of convergence tests can be found in
the Appendix.

###  III.1 Nonlinear Simulations

We first describe investigations into the linear spectrum of unstable modes in
the pedestal of DIII-D discharge 175823. As will be seen, this is already
sufficient to make contact with experimental observations.

Figure 3: Local linear simulation results: Growth rate of unstable MTM over
radial location.

Fig. 3 shows local linear simulations scanning the radial domain of the
pedestal (from $\rho_{tor}=0.92\sim 0.99$), showing the growth rate of the
most unstable MTM at each location. As expected, the instability is most
unstable at the mid-pedestal $\rho_{tor}=0.98$, where
$\omega_{*e}=k_{y}\rho_{s}c_{s}(1/L_{T_{e}}+1/L_{n_{e}})$ peaks. Refs. 18; 15;
16 demonstrate that (1) pedestal MTMs peak in the potential well created by
the $\omega_{*}$ profile, and (2) at low toroidal mode number and/or low
magnetic shear, MTM stability is extremely sensitive to the alignment of this
potential well with mode rational surfaces. For this scenario, the combination
of high magnetic shear ($\hat{s}\sim 5$ and relatively high toroidal mode
numbers ($n>10$) produce MTMs that are not sensitive to these considerations.
Consequently, consistent with the results in Fig. 3 we proceed with a detailed
local analysis at a single location – $\rho_{tor}=0.98$. Relevant parameters
at mid-pedestal are shown in Table 2

Pulse | $q$ |  $T_{e}$ (keV) | $n_{e}(10^{19}/m^{3})$ | $\nu_{ei}(c_{s}/a)$
---|---|---|---|---
175823 | 5.59 | 0.197 | 3.57 | 7.38
$\rho_{tor}$ | $\hat{s}$ | $a/L_{T_{e}}$ | $a/L_{n_{e}}$ | $\beta(10^{-4})$
0.98 | 4.94 | 37.0 | 17.1 | 7.24
R (m) | $a(m)$ |  $B_{0}$ (T) |  $\rho_{s}$ (mm) |  $c_{s}$ (km/s)
1.73 | 0.739 | 1.977 | 1.023 | 69.1

$\Omega_{g}$ ($c_{s}$/a)
---
9362

Table 2: Basic parameters at the mid-pedestal: where q is the safety factor
$q\equiv\frac{d\Psi_{T}}{d\Psi_{P}}$, $\nu_{ei}$ is the electron-ion collision
frequency, $\beta$ is ratio of the plasma pressure to the magnetic pressure,
$\hat{s}$ is the magnetic shear, $L_{T_{e}}$ is the length scale of the
electron temperature gradient, $L_{n_{e}}$ is the length scale of the electron
density gradient, $R$ is the major radius, $a$ is the minor radius, $\rho_{s}$
is the ion Lamar radius, $c_{s}$ is the speed of sound in plasma, $\Omega_{g}$
is the ion gyro-frequency, $B_{0}$ is the magnetic field strength Figure 4:
Local linear simulation results: plot (A) on the left hand side shows the
growth rate over mode number ($k_{y}\rho_{s}=0.00776\times n$, where $n$ is
toroidal mode number). Red dots are the MTMs, blue dots are ESDWs, mode
numbers with no unstable modes are marked with grey dots. plot (B) on the left
hand side shows

Fig. 4 (A) shows growth rates scanned versus $k_{y}\rho_{s}$ and
$k_{x}\rho_{s}$ at $\rho_{tor}=0.98$ (the translation from $k_{x}$ to
ballooning angle $\theta_{0}$ is as follows:
$\theta_{0}=\frac{k_{x}}{\hat{s}k_{y}}$). The definition of $k_{y}$ is
$k_{y}=\frac{n_{tor}q}{a}$, where $n_{tor}$ is toroidal mode number. As in
earlier work 26; 16, we find that the modes peak at finite ballooning angle
$\theta_{0}$ (finite $k_{x}$). The MTMs coexist with an electrostatic drift
wave (ESDW, denoted with blue symbols) with strong electron
signatures—predominantly electron heat flux and frequencies in the electron
direction. The main distinction between ESDW and MTM is the very weak
component of electromagnetic heat flux in the former.

The MTM is subdominant at $\theta_{0}=0$ and becomes dominant only as
ballooning angle increases. Fig. LABEL:fig:LL_k_f shows frequencies in the
plasma frame in $kHz$. Although these local linear simulations are in the
plasma frame, one can already note the high frequencies ($>100kHz$)
characteristic of the MTMs.

For reference, the linear spectrum for high $k_{y}$ electron scales is shown
in Fig. 5. The corresponding single-scale nonlinear ETG simulations (described
briefly below) produce low transport levels, suggesting that the focus on MTM
is well-justified.

Figure 5: Local linear simulation results for high $k_{y}$ electron scales:
frequency in plasma frame and growth rate in electron scale. All instabilities
shown in these plots are ETG

As the first point of experimental comparison, we show the frequency spectrum
as calculated from the RIP diagnostic in comparison with the frequencies of
the linear MTMs in Fig. 6. In this figure, the experimental frequency spectrum
represents the magnetic fluctuation amplitude, while the linear growth rates
are used for the linear modes. The frequencies are translated into the lab
frame using $\frac{E_{r}}{RB_{pol}}=1.5kHz$, as described above. The
frequencies that correspond to the peak growth rates are in good agreement
with the peak of the experimental spectrum. Note from Fig. 4 (A) that, as
$k_{y}$ increases, the electrostatic modes dominate just beyond the peak in
MTM growth rates (growth rates are shown in Fig. 4 [B] ). This suggests that
higher frequency MTMs corresponding to the upper tail of the experimental
spectrum are likely still unstable but subdominant to the electrostatic modes.

Figure 6: Comparison of local linear simulation of MTM with experimental
observation: the red dots are lab frame frequency VS growth rate of the local
linear simulations, the blue dots are the experimental observation of magnetic
fluctuation amplitude

Figure 7 and Figure 8 show the $A_{\|}$ mode structures of representative MTM
and ESDW modes, respectively, from the local linear simulations. The mode
structure of MTM and ESDW are distinct from one another: the MTM has (roughly)
an even parity of $A_{\|}$ (imaginary part or real part) around ($\theta=0$),
while ESDW has (roughly) odd parity of $A_{\|}$ (imaginary part or real part)
5. Aside from that, the modes are not the simple structures at the outboard
mid-plane familiar from core-like scenarios. For example, they extend further
in $\theta$ and generally exhibit more structure.

Figure 7: $A_{\|}$ Mode structure of MTM with $k_{x}\rho_{s}=0.6$,
$k_{y}\rho_{s}=0.14$. The y axis has arbitrary unit that is proportional to
$A_{\|}$ Figure 8: $A_{\|}$ Mode structure of ESDW with $k_{x}\rho_{s}=0.6$,
$k_{y}\rho_{s}=0.2$. The y axis has arbitrary unit that is proportional to
$A_{\|}$

As a final investigation into the linear properties of the mode, we
investigate the collisionality dependence of the growth rates as shown in Fig.
9. The MTM growth rates are plotted in addition to those of the electrostatic
mode using results from the GENE eigenvalue solver. The MTMs exhibit the
expected non-monotonic collisionality dependence 47; 48; 15, while the other
modes exhibit much milder collisionality dependence. The simulations are
conducted at $\rho_{tor}=0.98$, $k_{y}\rho_{s}=0.14$, $k_{x}\rho_{s}=0.6$
where the MTM growth rate peaks. The experimental magnetic fluctuation levels
also exhibit non-monotonic behavior with respect to collisionality. We will
return to this point in the next section in the context of comparing a
nonlinear collisionality scan with experimental data.

Figure 9: Local linear collisionality scan of at the $\rho_{tor}=0.98$,
$k_{y}\rho_{s}=0.14$, $k_{x}\rho_{s}=0.6$. $\nu_{ei}$ is the electron-ion
collision frequency, $\omega$ is the mode frequency

###  III.2 Nonlinear Simulations

A set of local nonlinear simulations has been conducted at the location where
the growth rate of MTM maximized — $\rho_{tor}=0.98$. Nonlinear simulations in
the pedestal are very challenging due to strongly shaped flux surfaces,
proximity to MHD limits, and extreme gradients. As will be seen, these
simulations do make some clear connections with experimental observations.
However, they also have certain limitations. Most notably, following an
intermediate time period with saturated heat fluxes, the simulations
transition to a period of runaway growth toward unrealistic transport levels.
Despite these limitations, we have decided to present these simulations in
order to (1) inform the community regarding numerical issues in this
challenging parameter regime, and (2) analyze the intermediate saturation
phase in comparison with experimental observations. The long-time runaway is
the consequence of low-$k_{y}$ modes that grow without bound. It is our
tentative conclusion that the intermediate saturation phase is at least
qualitatively meaningful and that global effects are critical for stabilizing
these low-n modes—i.e. global simulations would constrain the low-$k_{y}$
fluctuations to radial scales consistent with the width of the pedestal and
the width of the driving range of the modes. Global simulations exploring
these effects are currently under investigation and will be published
elsewhere.

Figure 10: Local nonlinear simulations of MTM: the top plot is simulations
using the nominal profile with saturated total electron heat transport of
about 2.3MW; the bottom plot is simulations using the $120\%$ nominal
$a/L_{T_{e}}$ with saturated total electron heat transport of about 3.99MW.
The red vertical lines show the start of the saturated period, and the blue
vertical lines mark the end of the saturated period. The electromagnetic
portions of the electron heat transports are represented by the orange lines,
and their averages are represented by green horizontal lines. The
electrostatic portions are represented by the blue lines, and their averages
are represented by orange horizontal lines.

The simulations have $k_{y}\rho_{s}$ ranging from 0 to 0.72. Simulations
employ the three dynamic particle species (electron, deuterium, and carbon).
They are based on the nominal equilibrium profile, and a profile with 120% of
the electron temperature gradient at the pedestal. The increase of the
electron temperature gradient is intended to increase the transport level of
MTM using profile variation within the experimental uncertainty, as MTM is
driven by electron temperature gradient. Fig. 10 shows time traces of heat
fluxes for two simulations: one with the nominal electron temperature gradient
and another with a 20% increase in an electron temperature gradient. The
simulation with the nominal profile produces electromagnetic electron heat
transport $Q_{EM}=1.67\pm 0.21MW$, electrostatic electron heat transport
$Q_{ES}=0.48\pm 0.11MW$, in total produces $Q_{e}=2.15\pm 0.32MW$. The
simulation with the 20% increase in electron temperature gradient produces
electromagnetic electron heat transport $Q_{EM}=2.34\pm 0.27MW$, electrostatic
electron heat transport $Q_{ES}=1.65\pm 0.35MW$, in total produces
$Q_{e}=3.99\pm 0.63MW$. As can be seen in Fig. 10, the electromagnetic
component of the transport saturates for a considerable length of time before
the electrostatic component grows uncontrollably with the nonphysical
transport level (just beyond the plot range).

####  III.2.1 Construction of synthetic diagnostics

Local nonlinear GENE simulations produce fluctuation data (electrostatic and
magnetic vector potentials, density, temperature fluctuations, etc.) in a
three-dimensional flux tube (a domain that follows the magnetic field lines,
is extended in the parallel direction, and is limited in the perpendicular
directions). This data is processed in various ways for comparison with
experimental fluctuation data, as illustrated in the flowchart in Fig. 11.

Fig. 11 shows the processes of the simulation diagnostic tools being
constructed to directly compare with the experimental results. The diagnostic
takes $A_{\|}(k_{x},k_{y},z,t)$ from GENE simulations, then converts to
$B_{r}(k_{x},k_{y},z,t)$ in Gaussian units. The magnetic field is then summed
over the $k_{x}$: $B_{r}(k_{y},z,t)=\sum_{k_{x}}B_{r}(k_{x},k_{y},z,t)$. To
mimic the RIP diagnostic, the synthetic diagnostic tool does a Jacobian-
weighted sum over a height of 7cm around the height of interest:
$B_{r}(k_{y},t)=\frac{\int B_{r}(k_{y},z,t)J(z)dz}{\int J(z)dz}$, where $J(z)$
is the Jacobian that is a function of $z$. The coordinate of the simulation is
non-Cartesian. In order to have a normalized sum over its Cartesian area, we
employ the Jacobian-weight sum. The auto-correlated Fourier transform was
performed to translate the information from time to frequency space:
$B_{r}\left(k_{y},f_{plasma}\right)^{2}=\int_{t_{i}}^{t_{f}}d\tau\int_{t_{i}}^{t_{f}}dt\overline{B_{r}}\left(k_{y},t-\tau\right)B_{r}\left(k_{y},t\right)e^{-i2\pi
ft}$. In this analysis, $t_{i}$ is the starting time, $t_{f}$ is the ending
time, $B_{r}\left(k_{y},f_{plasma}\right)$ is in units of $Gauss/\sqrt{kHz}$
as calculated spectral density. The Doppler shift is also added to boost the
frequency from the plasma frame to the lab frame:
$f_{lab}=f_{plasma}+f_{Doppler}$, where the $f_{Doppler}=1.5kHz$ for toroidal
mode number n=1 (for more details about the Doppler shift, refer to appendix
section). The contour plot in terms of $B_{r}$ across $k_{y}\rho_{s}$ and
$f_{lab}$ then can be plotted from $B_{r}(k_{y},f_{lab})$. Interpolation will
be conducted to unify the frequency space in preparation for the summation
over $k_{y}$ to arrive at the final spectrogram $B_{r}(f_{lab})$. To sum over
a bandwidth of frequency the formula is the following:
$B_{r,sum}=\sqrt{\int_{f_{min}}^{f_{max}}|B_{r}(f_{lab})|^{2}df}$ The
following calculation of $B_{r}$ as a function of height will be summed around
the mid-plane, and integrated over the frequency of 150kHz to 500kHz to match
the integration range from the experiment14 The collisional dependence
calculation is integrated over the full frequency range at the mid-plane to
match with the method of data processing in the experiment46; 14. All analysis
is performed during the intermediate time period when the transport is in a
saturated state (the starting time and ending time are marked as red and blue
vertical lines respectively in Fig. 10).

Figure 11: Processes of the synthetic diagnostic tools being constructed

####  III.2.2 Frequency Spectra and Fluctuation Amplitudes

Building upon the agreement between linear growth rates/frequencies and the
RIP frequency spectrum in Fig. 6, here we analyze frequency spectra from the
nonlinear simulations. The contour plot of $B_{r}$ as a function of mode
numbers and frequency is shown in Fig. 12. The fluctuations are maximum at
n=12, in good agreement with the experimental expectation of toroidal mode
numbers in the range ($10-20$) as noted in Ref. 14. For reference, a line
representing $\omega_{*e}=k_{y}\rho_{s}c_{s}(1/L_{T_{e}}+1/L_{n_{e}})$ is also
shown in the figure along with the frequencies of the linear eigenmodes. All
quantities—from basic theory, linear simulation, and nonlinear
simulation—align nicely.

Fig. 13 shows the spectral density of magnetic fluctuations as diagnosed by
both RIP measurement (blue) and the GENE local nonlinear simulations (red).
The experiment has $25\%$ of the uncertainty of the magnetic fluctuation
measurement which is represented as the vertical bar. In this figure, we show
the frequency spectra for different simulation scenarios representing various
uncertainties in the electron temperature gradient and the Doppler shift:
nominal gradients and Doppler shift (top left); nominal gradients with the
increased Doppler shift (top right—recall appendix section for a discussion of
the Doppler shift); electron temperature gradient increased by 20% with
nominal Doppler shift (bottom left); and the increased temperature gradient
and increased Doppler shift (bottom right). The simulation with the increased
temperature gradient agrees very well with the peak and the general shape of
the frequency spectrum. This substantial shift to higher frequencies for the
increased gradient case is attributable to two factors: (1) a slight increase
in $\omega_{*e}$, and (2) a shift of the nonlinear fluctuation spectrum to
higher $k_{y}$ (and higher frequency). In order to be consistent with the
experimental calculation for the spectrum, both simulations and experiments
use Welch’s method11 with ’hann’ window.

The apparent good agreement between simulation and experiment in Fig. 13 is,
unfortunately, a coincidence, since the experimental data is averaged over the
entire radius of the device (including regions in the core with, presumably,
much lower magnetic fluctuation levels). In contrast, the GENE data is
averaged over only the small flux-tube radial domain localized where the MTM
fluctuations are most potent.

We may speculate on various reasons for the discrepancy in fluctuation
amplitude. First, the recent RIP diagnostic upgrade 41 provides lower noise
floor measurement. Interestingly, the measured broadband magnetic fluctuation
amplitude in discharges similar to 175823 has been reduced since then. An
example can be seen in Chen (2022) 1. In shot 183225, a shot not identical but
similar to 175823 (shape, q95, density, etc.), measured line-averaged
broadband magnetic fluctuation amplitude (150-500kHz) is only 3-4 Gauss,
compared to 15 Gauss measured in 175823. It appears that a higher noise floor
in FY18 might have introduced an uncertainty leading to an overestimate of
absolute magnetic fluctuation amplitude. This potentially could explain part
of the discrepancy. Second, as noted above, these local nonlinear simulations
have limitations that may produce discrepancies with the experimental
observations. Third, it may point to the need to construct more sophisticated
synthetic diagnostics. Another possible source of the discrepancy can be
identified in the poloidal dependence of the fluctuations. This can be seen in
Fig. 7, which shows $A_{||}$ as a function of the parallel direction
(parameterized by the poloidal angle, $\theta$). Note that the amplitude peaks
strongly away from the outboard mid-plane where the diagnostics are centered.
In fact, the flux and fluctuation amplitudes are very low at this location.
Consequently, the simulated amplitudes at this location are perhaps a less-
robust reflection of the quality of the simulation in comparison with the
experiment. Stated differently, the peak amplitudes are the clearest
reflection of the thermodynamic drives of the system and would thus be
expected to be the most reliable signals of the simulation. This is reinforced
by the fact that the simulations produce realistic transport levels—2.1MW to
4.2MW of electromagnetic heat flux, which is closely connected to magnetic
fluctuation levels 34 (the best estimates of the experimental electron heat
losses range from $2.3-4.3$ MW as discussed below). Consequently, we view this
hypothesis to be very plausible.

Similar considerations should be kept in mind when comparing other features of
the simulations. The RIP diagnostic was used to measure magnetic fluctuations
at multiple vertical locations (recall Fig. 1). The magnetic fluctuation
amplitude strongly decreases in the vertical direction with no apparent
symmetry around the outboard mid-plane as shown in Fig. 14 for discharge
179451. In order to compare the spatial dependence between experiments and
simulations, we integrate over the peak frequencies (150kHz to 500kHz) and
extract the fluctuation amplitude ($\delta
B_{r}=\sqrt{\int^{500kHz}_{150kHz}B_{r}(f)^{2}df}$) at various vertical
locations. The simulated fluctuation amplitudes are also shown in Fig. 14. Due
to the fact that the simulation is based on inputs from a different discharge
(175823), only qualitative comparisons should be made. The nonlinear MTM
simulation (red) exhibits a relatively constant amplitude in the vertical
direction. Interestingly, the linear eigenmode (shown with grey symbols)
exhibits a vertical dependence closely matching that of the experimental
observation.

For reference, we also plot the magnetic fluctuation amplitude extracted from
a nonlinear ETG simulation (green) performed at the same radial location. This
fluctuation amplitude is negligible: $\delta B_{r}=10^{-3}Gauss$ despite the
fact that the ETG fluctuations do peak at the outboard midplane. This is a
clear observation distinguishing MTM and ETG fluctuations, which otherwise
have many ‘fingerprints’ in common (i.e., they are driven by electron
temperature gradients, have negative frequencies, and produce almost
exclusively electron heat transport).

Figure 12: This plot shows the comparison between local nonlinear simulations
(contour plot), local linear simulations (orange dots), and theory (blue
line). The contour plot shows $B_{r}$ as a function of frequency and
$k_{y}\rho_{s}$ calculated from local nonlinear simulations, where $B_{r}$ has
the unit of $Gauss/\sqrt{Hz}$. The orange dots are the results of the local
linear simulations with different $k_{x}\rho_{s}$ and $k_{y}\rho_{s}$. The
size of the orange dots represents the growth rate of each mode. The blue line
shows the MTM frequency expected from theory17: electron diamagnetic frequency
in the lab frame. The red horizontal lines correspond to the toroidal mode
numbers, starting from n=0, the darker solid red lines have an increment of 5
for toroidal mode number n.  Figure 13: The comparison of frequency of the
experiment 175823 spectrogram (blue) and the local nonlinear simulations (red)
where the top two plots show the simulation with nominal profiles and the
bottom two plots show the simulation with $120\%$ of $a/L_{T_{e}}$. The left
two plots are using $1.5kHz$ of Doppler shift for toroidal mode number
$n_{tor}=1$, while the right two plots are using $3.0kHz$ Figure 14:
Comparison between the experiment discharge 179451 and simulations of $B_{r}$
over height around the mid-plane. The blue dots are experiment observation,
the red dots are the local nonlinear simulation for MTM, the green dots are
the local nonlinear simulation for ETG, the grey dots are the local linear
simulation for MTM. The local linear $B_{r}$ has been normalized in order to
be compared with other quantities.

####  III.2.3 Collisionality Dependence

Non-monotonic collisionality dependence is a hallmark of microtearing modes.
The collision frequency at which the MTM peaks is sensitively dependent on a
multitude of variables—$\eta\equiv L_{n}/L_{T}$, magnetic shear, $k_{y}$ 15;
47. In Ref. 14, a set of discharges (175816, 175821, 175823, 176303, 176462,
176463, 176464) was analyzed in order to survey the collisionality dependence
of pedestal magnetic fluctuations. This set of discharges varies
collisionality while minimizing variation in other relevant parameters. The
fluctuation amplitudes from the RIP diagnostic and local nonlinear simulations
with different collisionality are shown in Fig. 15, qualitatively reproducing
the non-monotonic behavior expected from MTMs. The amplitudes are normalized
to their respective peak values.

Notably, the simulations and experiment find good agreement in the value of
collisionality at which the fluctuation amplitudes peak. They also exhibit
reasonable agreement in the breadth of the distribution, although the
experimental results are somewhat narrower in collision frequency. A similar
(preliminary) comparison—between the experimental fluctuation amplitudes and
simulated linear growth rates—was shown in Ref. 46. The comparison with
nonlinear simulations shown here is much more compelling—both the peak
collisionality and the breadth of the distribution are in much better
agreement, pointing to key nonlinear modifications to the linear picture. The
sharper falloff of the nonlinear simulations than linear as collisionality
increases may be attributed to the ESDW becoming increasingly dominant,
consistent with the dependencies shown in Fig. 9.

Figure 15: Collision dependent comparison between experiments (blue dots with
blue axis) and local nonlinear simulations (red dots with red y axis), the
star is denoted as the simulation with the nominal value taken from discharge
175823. $\nu_{ei}$ is the electron-ion collision frequency, $\omega$ is the
mode frequency. It is important to note that the experiment and simulations
$B_{r}$ is plotted on a different scale. This plot is intended to show the
qualitative comparison between experiments and simulations.

####  III.2.4 Fluctuation Ratio

In addition to the fluctuation amplitudes, fluctuation ratios are highly
relevant since they can (1) be compared to linear eigenmodes, and (2) often
clearly distinguish between different classes of modes.

BES 44 can measure $\delta n_{e}/n_{e}$ at the outboard mid-plane, radially
localized in the pedestal. The calculation of $\delta n_{e}$ from local
nonlinear simulations is done in the same way as $\delta B_{r}$ calculations
except the quantities being used – one uses $\delta n_{e}$ the other used
$\delta B_{r}$. With this, one can calculate the fluctuation ratio
$\frac{\delta B_{r}/B_{0}}{\delta n_{e}/n_{e}}$. For the discharge of interest
(175823) at the mid-plane, this is measured to be $\frac{\delta
B/B_{0}}{\delta n_{e}/n_{e}}=0.08\pm 0.03$ (150kHz $\sim$ 500kHz) 46.

This can be directly compared with simulation results. For MTM, $\delta
B_{r}=12Gauss$, $\delta n_{e}=6.16\times 10^{16}/m^{3}$; for ETG $\delta
B_{r}=0.067Gauss$, $\delta n_{e}=3.6\times 10^{16}/m^{3}$. Recall from the
Table 5, $B_{0}=19773Gauss$, $n_{e}=3.57\times 10^{19}/m^{3}$. Combining these
calculations, we arrive at estimates from the local nonlinear MTM simulations
of $\frac{\delta B/B_{0}}{\delta n_{e}/n_{e}}=0.36$. On the other hand, ETG
has the fluctuation ratio of $\frac{\delta B/B_{0}}{\delta
n_{e}/n_{e}}=0.0038$. Therefore, the ratios calculated from MTM, ETG,
experiment have the following relation.

$\text{ratio}_{ETG}\ll\text{ratio}_{experiment}\ll\text{ratio}_{MTM}$ (3)

The ratio from the MTM simulation is over a factor of 4 greater than the
experiment. There are a few possibilities for such a discrepancy. It is
possible that other electrostatic instabilities are also present but not
captured in the scale ranges of our simulations. Another possibility is that
KBM fluctuations are present in the discharge but not captured in the
simulations (Ref. 3 investigates such possibilities for another DIII-D
discharge). Such fluctuations would produce relatively stronger density
fluctuations while the MTM is the primary contributor to the magnetic
fluctuations. We also note once again the fact that the outboard mid-plane is
a region of low simulated fluctuation levels, and so maybe a challenging
location for comparisons.

One clear conclusion, however, is that ETG fluctuations are very unlikely to
be responsible for the observed magnetic fluctuations, since
$\frac{\text{ratio}_{ETG}}{\text{ratio}_{experiment}}=0.0475$. Such ratios can
be used broadly for experiments and simulations to determine the types of
instabilities17.

####  III.2.5 Transport

The experimental estimate of the total electron heat transport for discharge
175823 is $2.3\sim 4.3MW$. The ONETWO analysis uses carbon ion temperature.
However, the recent study 8 shows that using carbon ion temperature may
underestimate the power loss in the ion channel and therefore overestimate the
electron power loss. The real power loss due to electron could be roughly 2MW
lower which is a grey box marked for ONETWO uncertainty in Fig. 16 and Fig.
17.

Fig. 16 shows the contributions from the nonlinear ETG and MTM simulations.
For the nominal scenario, these two components of the transport reach the
lower band of the ONETWO uncertainty estimate. If we consider also the MTM
simulation with a higher temperature gradient, the simulations lie more
clearly in the expected range. Fig. 17 shows the transport with MTM
simulations with $120\%$ of the electron temperature gradient. In light of the
fact that magnetic fluctuation levels are closely connected to transport
levels, this may suggest that the peak magnetic fluctuation amplitudes
accurately reflect those in the experiment while the simulated levels at the
outboard midplane are not as accurate.

Despite the aforementioned long-time runaway of the simulations, the fact that
the intermediate saturation period produces reasonable transport levels adds
some credence to the simulations.

Figure 16: Power balance: the electron power loss calculated from the ONETWO
is $4.3MW$ (red horizontal line). The grey transparent rectangle represents
the uncertainty of ONETWO estimation . The simulation calculated transport
from MTM (blue bar) is $2.15\pm 0.32MW$, ETG (orange bar) is $0.17\pm 0.01MW$.
The total simulated transport added up to $2.32\pm 0.32MW$ which is marked as
green bar Figure 17: Power balance: the electron power loss calculated from
the ONETWO is $4.3MW$ (red horizontal line). The grey transparent rectangle
represents the uncertainty of ONETWO estimation . The simulation (with $120\%$
of the nominal electron temperature gradient) calculated transport from MTM
(blue bar) is $3.99\pm 0.63MW$, ETG (orange bar) is $0.17\pm 0.01MW$. The
total simulated transport added up to $4.16\pm 0.63MW$ which is marked as
green bar

It is also worth mentioning that local nonlinear simulation yield total ion
transport of $Q_{i}=0.15\pm 0.13MW$ with electrostatic ion heat flux is
$Q_{es,i}=0.15\pm 0.13MW$ , and electromagnetic ion heat flux
$Q_{es,i}=0.005\pm 0.003MW$. The transport coefficients ratio can be used as a
fingerprint 17 to identify the modes of instabilities. Table 3 shows such
ratios for the ion scale local nonlinear simulations. And they point toward
MTM.

$Q_{em,e}/Q_{es,e}$ | $Q_{i}/Q_{e}$ | $D_{e}/chi_{e}$
---|---|---
2.56 | 0.13 | 0.043
Table 3: Transport coefficients ratios for the ion scale local nonlinear
simulations, where $Q_{em,e}$ is electron electromagnetic heat flux,
$Q_{em,e}$ is electron electrostatic heat flux, $Q_{e}$ is electron heat flux,
$Q_{i}$ is ion heat flux, $D_{e}$ is electron particle transport coefficient,
$D_{e}$ is electron heat transport coefficient.

##  IV Conclusions and Discussion

Local linear and nonlinear gyrokinetic simulations of DIII-D discharge 175823
identify MTMs as the most likely source of the magnetic fluctuations measured
by the RIP diagnostic. Linear MTMs have growth rates that peak at frequencies
closely matching the peak of the experimentally-diagnosed frequency spectrum.
Despite some numerical issues with nonlinear simulations, there exists an
intermediate time range in which the simulations are well-behaved and produce
turbulence that finds clear connections with the experimental observations.
Notably, nonlinear simulations produce frequency spectra of $\delta B_{r}$
that are in good agreement with the peak and shape of the diagnosed spectrum
of the magnetic fluctuations. Nonlinear simulations also reproduce the
variation of magnetic fluctuation amplitudes observed in an experimental scan
of collision frequency—both experiment and simulations agree on the value of
collisionality at which the fluctuations peak. The simulations also produce
realistic levels of electron heat transport (2-4 MW within reasonable
uncertainty bounds, which is precisely in the range of experimental
expectation).

The simulations do not find close agreement with the absolute magnetic
fluctuation levels. This may be attributable to deficiencies in the
simulations (e.g., the local approximation produces artificial saturation
levels), or the need for more sophisticated synthetic diagnostics. However, we
posit that the main cause of the discrepancy is the fact that the RIP
diagnostic targets the midplane, where simulation identifies the fluctuation
levels to be at a minimum. We expect the peak fluctuation amplitudes further
away from the midplane to be a much more robust quantity for comparison. This
hypothesis is reinforced by the observation that the simulations produce
electromagnetic transport levels (which are closely connected to magnetic
fluctuation levels) in good agreement with experimentally-expected electron
heat losses.

Comparisons between the ratio of magnetic to density fluctuations also do not
find close agreement. This may be due to fluctuations from other instabilities
(KBM or some electrostatic mode) that are not captured in the scale ranges or
parameter points we simulate. However, the experimentally diagnosed
fluctuation ratios are in such a range that typical electrostatic fluctuations
can be eliminated as candidates.

These results strongly support the conclusion that MTMs are very likely to be
responsible for the observed magnetic fluctuations. Figs. 12 and 13 are
especially relevant. Fig. 12 demonstrates close agreement between basic
theory, linear simulation results, and nonlinear simulation results, and Fig.
13 extends this agreement to the experimentally measured frequency spectrum
for the magnetic fluctuations.

Quantity | Experiment | MTM | ETG
---|---|---|---
Frequency | 250kHz | 250kHz |  $>$500kHz
Direction | Electron | Electron | Electron
$\frac{\delta B_{r}/B_{0}}{\delta n_{e}/n_{e}}$ | $0.08\pm 0.03$ | 0.36 | 0.0038
| Non-monotonic
---
collisional
dependence
Yes | Yes | No
Transport |  $2.3\sim 4.3$MW |  $3.99$MW |  $0.17$MW
Table 4: Comparisons of MTM, ETG simulations, and experiment observations

As shown in Table 4, the results show a high level of agreement between GENE
simulations for MTM and RIP observations. All the evidence gathered supports
the idea that MTM is an important electron heat transport mechanism.

Figure 18: Flowchart for distinguishing the MTM, ETG and KBM.The top diamond
is to judge if the instability is largely magnetic, which distinguish MTM from
KBM and ETG. The second diamond is to judge if the frequency is in the
electron direction which can be used to differ KBM and ETG.

Inspired from the fingerprint method in Kotschenreuther (2019)17, Fig. 18
shows, experimentally, one can exercise the following flowchart to separate
MTM, ETG, and KBM using observation of frequency in plasma frame, magnetic
fluctuations, electron density fluctuations: If the $\frac{\delta
B_{r}n_{e}}{B_{0}\delta n_{e}}>0.01$, then the instabilities is likely to be
MTM 6. Otherwise, it could be KBM or ETG. If the frequency of the instability
is in the electron direction in the plasma frame then the instability is
likely to be ETG; otherwise, it will be KBM. If one applies such a flowchart
to the results listed in the Table 4, it is rather easy to find that what the
experiment observed is MTM.

##  V Acknowledgements

Thanks to the fruitful discussions with Tao Xie, and Joel Larakers.

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Fusion Energy Sciences, using the DIII-D National
Fusion Facility, a DOE Office of Science user facility, under Award(s): DE-
FC02-04ER54698, DE-SC0022164, DE-AC02-05CH11231, DE-SC0019004,

This work was supported by U.S. DOE Contract No. DE-FG02-04ER54742 at the
Instituted for Fusion Studies (IFS) at the University of Texas at Austin.

This research used resources of the National Energy Research Scientific
Computing Center, a DOE Office of Science User Facility. We acknowledge the
CINECA award under the ISCRA initiative, for the availability of high
performance computing resources and support.

This research was supported at Oak Ridge National Laboratory supported by the
Office of Science of the U.S. Department of Energy under Contract No. DE-
AC05-00OR22725.

This research used resources of the National Energy Research Scientific
Computing Center, a DOE Office of Science User Facility, and the Texas
Advanced Computing Center (TACC) at The University of Texas at Austin.

##  VI Disclaimer

This report was prepared as an account of work sponsored by an agency of the
United States Government. Neither the United States Government nor any agency
thereof, nor any of their employees, makes any warranty, express or implied,
or assumes any legal liability or responsibility for the accuracy,
completeness, or usefulness of any information, apparatus, product, or process
disclosed or represents that its use would not infringe privately owned
rights. Reference herein to any specific commercial product, process, or
service by trade name, trademark, manufacturer, or otherwise does not
necessarily constitute or imply its endorsement, recommendation, or favoring
by the United States Government or any agency thereof. The views and opinions
of authors expressed herein do not necessarily state or reflect those of the
United States Government or any agency thereof.

##  VII Notice of copyright

This manuscript has been authored in part by UT-Battelle, LLC, under contract
DE-AC05-00OR22725 with the US Department of Energy (DOE). The publisher
acknowledges the US government license to provide public access under the DOE
Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

##  VIII Appendix

###  VIII.1 Simulation parameters

For local linear simulations for

| Simulation type | nx | nz | nv | nw | hypz | Shear?
---|---|---|---|---|---|---|---
| Local linear | 9 | 90 | 32 | 16 | -2 | Off
| Local nonlinear | 128 | 180 | 40 | 8 | -2 | Off
Table 5: Important parameters used in the local linear and nonlinear
simulations

Hyperdiffusion in the z-direction is non-zero, and hyperdiffusion is zero for
the rest of the directions

For ETG nonlinear simulations, ion and impurity are adiabatic, $nky=48,\
kymin=5$. For MTM nonlinear simulation, all 3 species are non-adiabatic,
$nky=24,\ kymin=0.03$.

A convergence test has been performed for both linear and nonlinear
simulations by doubling the numerical grid points. The growth rate (for
linear) and saturation level (for nonlinear) stay the same.

Local nonlinear simulations for MTM have the following resolution:
$k_{x}\rho_{s}$ is ranging from 0 to 4 with the definition of $\frac{1}{32}$,
$k_{y}\rho_{s}$ is ranging from 0 to 0.72 with the definition of 0.03
($k_{y}\rho_{s}=n_{tor}\times 0.00776$). The simulations have $E\times B$
shear turned off since $E\times B$ shear stabilizes the turbulence. The
frequency has been translated into the lab frame during the post-processing
procedure. Figure 10 shows the result from local nonlinear simulation. Hyper-
diffusivity in $k_{x}$ and $k_{y}$ are 0. Hyper-diffusivity along the magnetic
field is 2. After $t=55(a/c_{s})$, the low k electrostatic mode starts to run
away to the nonphysical transport level. However, from the discussion in
Section III.2.2, the possibility of the theory, local linear simulation, local
nonlinear simulation, and experiment results tightly match at the same time by
accident is little to none. Therefore, one could argue that the results from
the early saturation stage of the MTM local nonlinear simulations are still
valid.

Simulating the local nonlinear MTM is a nontrivial task. The saturation level
is near zero if the range of ky is too small. And one species simulation seems
to produce a much lower transport level for MTM compared with the simulation
with three species.

###  VIII.2 Doppler shift

The Doppler shift must be estimated in order to make quantitative comparisons
between simulation frequencies and fluctuation data.

There is no ion temperature measurement beyond $r/a\sim 0.97$. The profile is
fitted so that $E_{r}\sim 0$ at the last closed flux surface (LCFS). The
Doppler shift calculated for translation from plasma frame to lab frame is
using the fitted profile near the edge and varies around such value within
experimental uncertainty ($\pm 20\%$).

Based on the nominal value of calculation from 14, the $v_{E\times B}\sim
10km/s$ in electron direction. Since $f_{Doppler}=\frac{v_{E\times
B}B_{tor}}{2\pi RB_{pol}}$, The Doppler shift then can be calculated from the
Table 2. $R=1.73m$, $B_{tor}\sim 1.55T$, $B_{pol}\sim 0.3T$, $f_{Doppler}\sim
4.75kHz$.

On the other hand, from the neoclassical analysis, the Doppler shift for n=1
can be up to 16kHz. Therefore the Doppler shift for n=1 can range from 5kHz to
16kHz.

###  VIII.3 Normalization

The collisionality is calculated as $\nu_{ei}=5\times
10^{-11}Z_{eff}n_{e}/T_{e}^{1.5}$ in experiment14 assuming the $ln\Lambda=17$.
In GENE, the collision frequency can be expressed as $\left\langle
v_{\mathrm{ei}}\right\rangle=\frac{2^{1/2}n_{\mathrm{i}}Z^{2}e^{4}\ln\Lambda}{12\pi^{3/2}\epsilon_{0}^{2}m^{1/2}T_{\mathrm{e}}^{3/2}}$.
4 The data and plots in this article have been normalized to experimental
expression.

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