<figure><img src="image_47.png" /><figcaption>**Figure 44.** Measured $\delta T_e$ spatial correlation function from a CECE diagnostic, compared to a GENE nonlinear simulation synthetic diagnostic, based on an AUG L-mode scenario. The simulation $T_e$ logarithmic gradient is reduced by 0.87 compared to the nominal value. Reprinted from [1003], with the permission of AIP Publishing.</figcaption></figure>

In summary, a sufficient body of evidence has emerged that supports the adoption of nonlinear gyrokinetic simulations as sufficient ground truth for validating reduced order models. However, some open issues remain, such as systematic deviations in $\delta T_e/T_e$ spectra in some cases, necessitating further investigation into both higher fidelity simulations and diagnostic methodology. In addition, further investigation is necessary in wider experimental parameter ranges, particularly those more relevant for burning plasmas. New capabilities in nonlinear gyrokinetic simulation on transport timescales should be further explored for validation of nonlinear gyrokinetics against radial profiles of kinetic profiles [1016]. As discussed in sections 3 and 5, open questions remain on the validation of nonlinear gyrokinetics against intrinsic rotation and impurity transport.

### 9.3. Reduced quasilinear models of gyrokinetic turbulent transport

The coupling of fast reduced turbulent models to integrated modeling frameworks enables the evaluation of tokamak core plasma stationary state or temporal scenario evolution in reasonable computational time, ranging from minutes to days (on a single compute node) depending on the application. This is key for interpretation of physics mechanisms, which are often emergent through the nonlinear coupling of various physical processes such as sources, sinks, transport, and equilibrium on confinement time scales. Fast simulation further enables scenario optimization, experimental design, and machine design. Accurate 'predict-first' simulation is a required component of ITER pulse design. Surrogate models of turbulent transport through machine learning techniques (see section 9.3.9) provide further computational acceleration.

The quasilinear approximation is an accurate method for construction of tokamak reduced transport models [1017]. It is a weak turbulence approximation, which assumes that linear cross-phases between the fluctuating electromagnetic

fields and moments of the perturbed distribution function of the dominant linear instabilities are preserved in the nonlinear turbulent state, at least for the wavenumbers responsible for the majority of the transport fluxes. The assumption that the linear cross-phases dominate in the nonlinear regime for transport-driving scale-lengths was demonstrated to be accurate to within 5% for energy fluxes and 18% for the particle fluxes for a set of 64 nonlinear gyrokinetic turbulence simulations scanning plasma parameters about a standard mid-radius reference case [288]. Besides the preservation of the linear cross-phases, other linear instability characteristics are observed in turbulence simulations as discussed in the next section. The computational burden of quasilinear transport models is limited to the solution of the instability spectrum from the linear dispersion relation. Quasilinear fluxes are calculated for each of these instabilities, and summed while weighting over the amplitude spectrum of the electromagnetic fields. The saturated fluctuation amplitudes are not calculated by linear theory; they are modeled through a 'saturatio n rule', essentially dimensionally constrained functions of the wavenumber and linear growth rate spectrum calibrated by nonlinear gyrokinetic simulations. Recently a 3D model of the saturated electric potential fluctuation spectrum from gyrokinetic turbulence was developed [347]. When using the same gyrokinetic code (CGYRO [309]), applying this 3D saturation model on the linear instability spectrum achieves unprecedented accuracy in the plasma core for a quasilinear model when comparing to the analogous nonlinear simulations (10% for energy flux and 19% for particle flux [288]). The accuracy of quasilinear models in the core is often sufficient to be used to validate the gyrokinetic theory with experimental measurements for much larger data sets than is possible with direct turbulence simulation. An efficient validation hierarchy is thus established, with quasilinear models used for large experimental surveys and gyrokinetic simulations used to verify the quasilinear results.

Discrepancies however are sometimes found between experiment or higher fidelity gyrokinetics and the quasilinear models. They trigger higher fidelity validation against gyrokinetic simulations, leading to reduced model improvements, or if the higher fidelity validation fails, to challenging the approximations higher up the fidelity chain. In particular, discrepancies between quasilinear approaches and nonlinear flux-driven gyrokinetics have been reported in the core [1018]. They point towards a possible breakdown of the assumption of a scale separation between mean and fluctuations due to the emergence of long-lived mesoscale structures near marginal stability (further remarks can be found in section 9.4). Discrepancies have also been reported in the edge [1019] when Kubo numbers approach unity. In such cases, edge quasilinear models accurately predict edge electron energy fluxes but seem to under-predict edge fluxes of ion energy and particle (by factors of 3–4). These questions are still somewhat preliminary and remain a matter of active investigation.

In the following subsections we further justify the quasilinear approximation, provide updates on transport model development, and show examples of validation against experiments within integrated modeling.
<figure><img src="image_48.png" /><figcaption>**Figure 45.** Comparisons of $\delta\phi$ and $\delta T_{\perp}$ cross-phases from nonlinear simulations (contours) and linear calculations (markers) for various cases corresponding to AUG L-mode discharge parameters. The dominant linear modes are the red squares; secondary modes are green circles. Reprinted from [1022], with the permission of AIP Publishing.</figcaption></figure>

### 9.3.1. Verification of the quasilinear approximation with gyrokinetic turbulence simulations.
Fundamental justification of the quasilinear approximation is based on comparisons to nonlinear simulation. Comparisons of nonlinear and linear cross-phases, e.g. the phase difference between the fluctuating electrostatic potential and fluctuating density (among the possible comparisons), was carried out in multiple settings for ITG, TEM, and resistive modes [830, 1020–1022], showing good correspondence for transport-relevant wavenumbers, see figure 45.

Similar results were observed in a related comparison between nonlinear and quasilinear transport weights [1023], defined as the transport flux divided by the sum of the electromagnetic potentials (or just by $|\phi|^2$ for electrostatic transport). Nonlinear and quasilinear transport weights are observed to have a non-unity, but constant ratio across wide parameter space. The non-equality of the transport weights may be due to the existence of stable and on-average non-transport-inducing modes in the nonlinear turbulence spectrum [1024]. A constant non-unity ratio can be absorbed by nonlinear saturation rules. However, care must be taken if the nonlinear saturation

physics changes across parameter space, potentially modifying the transport weight ratio and necessitating refinements to the saturation rule [1025].

Agreement between mean nonlinear frequency (per wavenumber) and linear eigenvalue frequencies was also observed in multiple settings [830, 1020, 1022, 1025]. Agreement between the magnitude of nonlinear frequency broadening and the linear growth rate was also observed in multiple academic and experimentally derived parameter regimes [1009, 1025]. This signifies the balance between nonlinear dissipation and the linear drive at those wavenumbers, pointing to the maintenance of the linear mode characteristics, as well as justifying the use of the full complex eigenvalue in the linear response when evaluating the quasilinear transport fluxes [1026].

Another important verification of the quasilinear approximation is the evaluation of the Kubo number [1027], which is the ratio between a test particle autocorrelation time in the turbulent field, and the transit time of the particle around a turbulent eddy. The Kubo number is a tentative measure of nonlinearity strength based on dimensional arguments. To justify the
implementation of predicted quasilinear transport model coefficients into a convection-diffusion transport equation, this number must be lower than unity. This was verified in [1020], as well as in [1025] in multiple cases when directly calculating the Eulerian autocorelation time (in the field reference frame) from nonlinear simulations. More recent comparisons when calculating a Lagrangian autocorrelation time (in the particle reference frame) showed Kubo numbers larger than, but still of order, unity both in the core [1018] and in the edge [1019]. A Kubo number lower than unity approximately states that even though the underlying transport processes are nonlinear, the strength of the nonlinearity is weak or moderate. In practice however, Kubo numbers only partially inform about validity of the quasilinear approximation. Kubo numbers are indeed not a prognostic quantity and are only computed *a posteriori*. Interestingly, even in cases where Kubo numbers are of order unity and where quasilinear fluxes have been reported to be significantly under-predicted with respect to the nonlinear calculation [1018], linear and nonlinear cross-phases still seem in reasonable agreement.

In all these studies, the best agreements were observed for the lower wavenumber (higher wavelength) portion of the spectrum, responsible for driving the majority of the turbulent flux.

### 9.3.2. Specific model updates and saturation rules

#### 9.3.2.1. TGLF
The TGLF linear equations [1028] are velocity space moments of the linear gyrokinetic equations that have been closed in such a way to fit the local plasma response functions including the imaginary residue from kinetic resonances. This system of equations built on a progression of theoretical developments starting with the Landau Fluid equations [1029], which accurately reproduce the Landau damping of the plasma response function. The curvature drift resonance was then added [1030] making a Gyro-Landau Fluid system (GLF23). The GLF23 system of equations were used for nonlinear turbulence simulations that were the first demonstration of the quenching of ITG turbulence by equilibrium ExB velocity shear [1031]. The linear GLF23 equations were also employed in a quasilinear model of the transport fluxes [1032]. Finding the linear eigenmodes of a fluid system of linear equations is an efficient matrix eigenvalue problem. The GLF23 quasilinear transport model used a spectrum of poloidal wavenumbers and a simple saturation model that was calibrated to the available state of the art nonlinear gyro-fluid simulations. The GLF system was extended to include more velocity space moments in [1033, 1034] and bounce averaged trapped electron fluid equations were also developed [1035].

The objective of the TGLF project was to build a more accurate quasilinear model than GLF23. The TGLF system incorporates bounce averaged trapped particles in a more integrated way by taking velocity moments over trapped and passing regions of velocity space. An adaptive wavefunction representation in the ballooning angle space was developed for TGLF that is more accurate than the trial wavefunctions used in GLF23. This enabled the use of the Miller model [1036] for finite aspect ratio, shaped flux surface, geometry.

The closure of the TGLF system uses a similar method as the Beer-Hammett 6-moment system [1034], but the closure coefficients depend upon the fraction of trapped particles. Since the original model was published, electromagnetic fluctuations were added [287], the electron pitch angle scattering collision model in TGLF was greatly improved [1037] and equilibrium toroidal velocity and velocity shear were added [1038]. A new eigensolver for TGLF that can change velocity space resolution and has a much simpler closure is soon to be released [1039]. These gyro-fluid eigensolvers are much faster than initial value solutions using the gyrokinetic simulation codes.

The original saturation model used in TGLF was 1D. It has a spectrum of poloidal wavenumber ($k_y$) like GLF23. The fluctuation intensity for each ky was independent of all the others and only depended on the linear growth rate and wavenumber at that ky. The first 2D (poloidal $k_y$ and radial $k_x$ wavenumber) model for the flux surface averaged saturated electric potential fluctuation intensity was motivated by the observed radial wavenumber shift due to equilibrium $E \times B$ velocity shear [307] in the nonlinear spectrum. This 'spectral shift model' enabled TGLF to compute the Reynolds stress due to the sheared Doppler shift. A 2D model of the coupling of electron and ion scale modes across wavenumbers was later developed [915] that included mixing between poloidal wavenumbers for the first time. This model introduced the concept of ZF mixing as the primary saturation mechanism. This new paradigm was needed because the ZF shearing paradigm failed for electron scale turbulence simulations. The recent 3D model of the saturated electric potential fluctuation spectrum (including poloidal angle dependence) [288] improves the accuracy of the model for shaped magnetic flux surfaces. A new saturation model that includes a change in wavenumber dependence for electron modes vs ion modes is able to capture the isotope dependence observed in the gyrokinetic simulations for the first time [52].

#### 9.3.2.2. QuaLiKiz
The QuaLiKiz model solves an approximated form of the linear gyrokinetic dispersion relation [244, 246]. For a detailed review of the model derivation and approximations see [1040]. ITG, TEM, and ETG linear modes are calculated within approximately 1 CPU second per instability. The main approximation is that the eigenmode shape is not self-consistently calculated, but prescribed using a shifted Gaussian ansatz. The Gaussian width and shift is obtained in an auxiliary calculation based on a high-frequency expansion of the dispersion relation. The shift, emerging from the presence of plasma rotation (both parallel and perpendicular) and its shear, is responsible for the symmetry breaking that leads to momentum transport [308]. Further approximations include pitch-angle averaging, strongly-passing and strongly-trapped particle trajectories, low inverse aspect ratio, shifted circle geometry, and the electrostatic limit. An ad-hoc model for incorporating fast-ion-enhanced EM-stabilization has been developed for use in integrated modeling [195]. The saturation rule has a model to determine the spectral form of the saturated electrostatic potential $|\phi|^2$, based on the poloidal wavenumber $k_\theta$ corresponding to the maximum of $\gamma_k/k_\perp^2$, and
spectral falloffs on either side. There are additional corrections on the saturation level at low magnetic shear ($\hat{s}$) due to enhanced ZF coupling, as well as nonlinear contributions to radial wavenumbers ($k_x$) [1025]. A simple multi-scale rule is in place, based on relative magnitudes of maximum $\gamma/k_\theta$ on ion and electron scales [384]. Recent modifications to the collision operator has shown significant improvement in regimes where TEM drive is important [1041].

#### 9.3.2.3. Multi-mode-model (MMM).
The MMM takes the approach of cumulatively adding the transport predicted by disparate models, each developed to describe specific branches of microinstabilities. See [1042] and references therein, including for the following models: the Weiland model for ITG, TEM, and KBMs, a drift-resistive-inertial ballooning mode (DRIBM) model which may be important for the L-mode edge, and an ETG model. An improved version (EDWM) of the Weiland model, now including calibration against nonlinear gyrokinetic simulations of the saturated potential amplitude and spectral shape, has recently been developed [1043].

#### 9.3.3. Progress in modeling the saturated fluctuation spectrum.
There are several ways in which a turbulent system can achieve saturation. The concept of a 'mixing length' saturation follows from the linear growth of an instability being balanced by the transport relaxation of the driving gradients. For example, consider a linear instability of the density $n$ with growth rate $\gamma$, and a diffusive model for the turbulent particle transport:

$$ \Gamma = -D\partial n/\partial x. \qquad (26) $$

The time evolution of the density then satisfies the 1D equation

$$ \frac{\partial n}{\partial t} = \gamma n - D \frac{\partial^2 n}{\partial x^2}. \qquad (27) $$

Saturation is achieved when the $D = \gamma L_x^2$ where $L_x$ is the density gradient scale length. This is sometimes called quasilinear flattening since the plasma profiles flatten to achieve a saturated steady state. Local flux tube gyrokinetic simulations evolve radially periodic fluctuations that do not allow the equilibrium gradients to flatten, so this type of saturation cannot occur. The flux tube gyrokinetic simulations saturate through mode coupling by the nonlinear advection since profile gradients are held fixed. In symbolic form, the evolution of the fluctuation in the distribution function $\delta f$ with linear drive $\gamma$ and nonlinear advection is given by

$$ \frac{\partial \delta f}{\partial t} = \gamma \delta f - \delta \vec{V}_{E \times B} \cdot \vec{\nabla} \delta f. \qquad (28) $$

The electric potential part of the nonlinear advection has the form

$$ \delta \vec{V}_{E \times B} = \vec{B} \times \vec{\nabla} \delta \phi / B^2. \qquad (29) $$

The early models for the saturated fluctuation spectrum of the electric potential did not include any mode coupling. They were 0D single mode models (Weiland) or 1D in $k_y$ models like in GLF23, QualiKiz or TGLF (SAT0) where each poloidal wavenumber saturated independent of the others. The nonlinear advection in the gyrokinetic equation mixes the turbulence in both directions $(x,y)$ perpendicular to the equilibrium magnetic field. In the Fourier transform space, in which flux tube gyrokinetic simulations are typically evolved, this means there is mode coupling between both poloidal $k_y$ and radial $k_x$ wavenumbers. The most unstable linear instabilities in a tokamak geometry (torus) usually have $k_x = 0$ for finite $k_y$. The $E \times B$ flow (eddies) of these instabilities is in the radial direction with alternating sign. These radial 'streamers' can be closed into finite radial extent eddies by coupling to the poloidal $E \times B$ flows of the zonal potential. The zonal ($k_y = 0$) part of the mixing velocity $\delta \vec{V}_{E \times B}$ couples modes with the same poloidal wavenumber $k_y$, but different radial wavenumbers $k_x$. A 2D saturation model was developed for TGLF (SAT1) that includes mixing between the poloidal wavenumbers and mixing of the radial wavenumbers mediated by the fluctuating ZFs. Fluctuating ZF mixing causes saturation by facilitating a chain of radial wavenumber coupling leading to higher $k_x$ modes that are damped. The width of the $k_x$ electric potential fluctuation spectrum at each $k_y$ is a measure of the rate of transfer of fluctuation intensity from the peak instability at $k_x = 0$. This ZF mixing rate (mixing in $k_x$) must balance the growth rate of the peak instability. The ZF mixing rate can be approximated by $\gamma_{ZF} = V_{ZF}k_y$ where $V_{ZF}$ is an average ZF poloidal velocity of the nonlinear advection. The radial wavenumber average intensity of the fluctuating zonal $E \times B$ velocity can be computed from the nonlinear spectrum as

$$ V_{ZF} = 0.5 \sqrt{\sum_{k_x} k_x^2 |\delta \phi_{k_x, 0}|^2}. \qquad (30) $$

If the ZF mixing of radial wavenumbers is the dominant saturation mechanism, the ZF mixing rate $\gamma_{ZF}$ would be comparable to the linear growth rate $\gamma_{k_y}$ for this same $k_y$ at $k_x = 0$. Since $V_{ZF}$ does not depend on the poloidal wavenumber the matching of the mixing rate to the growth rate cannot be satisfied at all $k_y$. This suggests a ZF saturation rule given by

$$ V_{ZF} = \text{MAX}_{k_y < 1} \left[ \frac{\gamma_{k_y}}{k_y} \right]. \qquad (31) $$

This ZF saturation rule has been verified with a large number of ion-scale gyrokinetic turbulence simulations and found to be quite accurate [288]. The ZF shearing rate (which is not well defined) could match the ion scale linear growth at some particular $k_y$ but since the shearing rate does not depend on the poloidal wavenumber it is very much exceeded by the electron scale linear growth rates. The ZF mixing rate $k_y V_{ZF}$ can and does exceed the electron scale linear growth rates and was shown [33, 381] to predict the nonlinear threshold of electron scale temperature gradient modes (ETG) observed in
multi-scale turbulence simulations [563]. The nonlinear ETG threshold condition

$$ \text{MAX}_{k_y>1} \left[ \frac{\gamma_{k_y}}{k_y} \right] > \text{MAX}_{k_y<1} \left[ \frac{\gamma_{k_y}}{k_y} \right] \quad (32) $$

has been successfully validated experimentally [385], but may require modification for intermediate scale TEMs.

The ZF saturation rule equation (31) identifies the poloidal wavenumber $k_{ymax}$ where the ion-scale maximum occurs, and the corresponding linear growth rate at this peak $\gamma_{max} = V_{ZF}k_{ymax}$, as the physical reference scales of the poloidal wavenumber and linear growth rate that adapt to the plasma conditions. The saturation model for the electric potential fluctuation spectrum is $\gamma_{max}/k_{ymax}^2$ times a dimensionless function of the poloidal wavenumber and linear growth rates normalized to these reference values. This property is a very powerful reduction of the problem of finding a model of the saturated fluctuation spectrum. The high accuracy 3D saturation model (SAT2), fit to CGYRO simulations using CGYRO linear eigenmodes [288, 347], was found to require geometry factors in order to model the poloidal angle dependence of the 3D saturated fluctuation spectrum. An overall factor of $L_p/R$, where $L_p$ is the total pressure gradient scale length from the MHD force balance and $R$ is the major radius at the magnetic axis, was also needed for SAT2 in order to better fit the gradient scans. This factor may be due to a change in the balance between ZF mixing and poloidal mode mixing in the saturated state. A recent study of the isotope dependence of the gyrokinetic fluxes showed that there was a change in the saturation for TEM dominant cases compared to ITG dominant cases and that this was responsible for the observed isotope dependence [52], leading to an improved saturation model (SAT3) [52], better fitting the isotope scaling, among other refinements. It will be included as a new option in TGLF. Since the newest models (SAT2 & SAT3) are fit using linear gyrokinetic eigenvalues from the same nonlinear simulation with CGYRO they are independent of the error of the linear eigenmodes due to the reduced gyrofluid equations used in TGLF. This separates the task of modeling the saturation from the task of developing a fast reduced linear eigensolver.

### 9.3.4. Validation against experiments within integrated modeling.
In this section we summarize validation exercises against experimental measurements, applying within integrated modeling the reduced transport models introduced in the previous section. We focus on TGLF and QuaLiKiz, which have incorporated comparison against nonlinear gyrokinetics in their model verification. Validated models are key for interpretation of physics transport mechanisms, scenario optimization, and extrapolation to future devices. For ITER the main goal of the present validation exercises is to demonstrate predictive capabilities of core turbulent transport for regimes foreseen in the ITER Research Plan, with respect to relevant transport mechanisms and absolute performance predictions.

The results summarized here were produced with an array of integrated modeling suites, specifically JINTRAC [242,

1044], ASTRA [351, 1045], and the stationary-state solver TGYRO [240].

Multiple aspects of reduced model validation within integrated modeling as well as standalone analysis including comparison with higher fidelity modeling have already been reviewed in previous sections of the TC review. We refer the reader to section 2 for particle transport including the relative role of inward convection versus source for density peaking, multiple-isotope transport, and integrated modeling with multiple pellet cycles; to section 3 for impurity transport; and to section 4 for multi-scale ETG transport, reduced ion thermal transport due to EM-stabilization of ITG, and isotope effects. To avoid repetition we review below specific aspects of reduced model performance. Predictive capabilities in the L-mode near-edge is covered in section 9.3.5, transport predictions across the LOC/SOC transition is reviewed in section 9.3.6. In section 9.3.7, we focus on Predict First modeling (model predictions ahead of and potentially guiding specific experiments) and extrapolations to future reactors, including ITER scenarios. Open questions regarding modeling fidelity are discussed in section 9.3.8.

#### 9.3.5. L-mode edge (pedestal forming region).
The transport fluxes in the gyrokinetic formulation are second order in the expansion parameter $\rho_* = \rho_i/L_p$ where $\rho_i = v_{thi}/\Omega_i$, with $v_{thi} = \sqrt{T_i/m_i}$ the ion thermal velocity, $\Omega_i = Z_i eB/m_i c$ the ion gyro-frequency for magnetic field strength $B$, and $1/L_p = -d\ln(P)/dr$ is the equilibrium pressure gradient scale length. Because the first order gyrokinetic equations do not explicitly depend on $\rho_*$ the reference magnetic field and length scale can be chosen to suit the code normalizations. However, the physical gradient length scale is from the total pressure $L_p$ since this is the only plasma profile gradient in the zeroth order force balance $\vec{J} \times \vec{B} = \vec{\nabla}P$. Saturation by the nonlinear mode coupling advection requires that the saturated fluctuations have an amplitude relative to the background of order $\rho_*$ [1046]. The fluxes driven by the turbulence are then of order $\rho_*^2$. This scaling is referred to as gyro-Bohm scaling, which is short for gyro-reduced Bohm scaling since the effective diffusion scales like $\rho_*$ times a Bohm diffusion coefficient $D_B = cT/(16eB)$. From the early days of gyrokinetic theory is was recognized that the strong temperature scaling of the gyro-Bohm energy fluxes may lead to steep gradients at the low temperature boundary of the tokamak in the low confinement regimes (L-mode) [1032]. The quasilinear transport models have a built-in gyro-Bohm scaling and had previously not been successful in matching the edge temperature profiles of a tokamak in L-mode or Ohmic heating regimes [1047]. This failure has hampered the prediction of the power required to transition to the high confinement regime H-mode. In addition, accurate L-mode modeling up to the LCFS is vital for $I_p$ ramp-up and ramp-down modeling and scenario optimization studies. While the precise mechanism of the L-H transition itself is still under investigation [1048], see discussion in section 7, significant improvements have been made in the predictive capability of L-mode turbulence in the pedestal-forming-region, as outlined below.
One way around the low transport implied by the gyro-
Bohm scaling for the L-mode edge is if the turbulence is strong
so that the gyrokinetic ordering breaks down. This is the case
in the open field line region known as the SOL that connects
to the walls in a tokamak. Braginski fluid turbulence simula-
tions found a regime of very strong resistive interchange turbu-
lence that does not preserve its linear phase [1049]. The plasma
parameter domain where the SOL is in this strong turbulence
state aligns well with L-mode conditions [1050]. However, this
explanation does not hold for the closed flux surfaces inside
the separatrix, since the closed flux surfaces suppresses inter-
change modes and the increasing temperature stabilizes resist-
ive ballooning instabilities [1051].

Another mechanism for matching the observed flux-
gradient relationship, in spite of the small gyro-Bohm flux, is
with sufficient linear drive and saturated nonlinear flux. Recent
local gyrokinetic simulations for JET and AUG parameters
have shown that the L-mode edge transport is characterized by
electron drift-wave instabilities with a non-monotonic depend-
ence on collisionality, and is consistent with the quasilin-
ear approximations [829, 830, 891]. At low collisionality, the
dominant modes are more TEM in nature, and are stabilized
by collisionality. At higher collisionality, the dominant modes
are of a drift-resistive type, and are destabilized by collision-
ality. Nonlinear local gyrokinetic simulations were able to
match observed power balance. A β threshold was predicted
in the nonlinear simulations, at values below the linear KBM
threshold, for transition to virulent turbulence beyond power
balance levels due to electromagnetic effects. However, for the
cases studies this threshold β value was above the observed
value. The isotope mass dependence of the parallel dynam-
ics leads to predicted trends in the virulence of the turbulence,
consistent with observed trends of L-mode confinement scal-
ings [830, 1052]. In the absence of significant linear drive,
non-local mechanisms such as turbulence spreading from the
SOL (see section 9.4), may also contribute to the observed flux
levels [1053].

Regarding incorporation of these effects into the quasilin-
ear models, the MMM model incorporates a reduced model
of the drift-resistive interchange ballooning mode (DRIBM)
transport, and was able to produce sufficient transport for
full radius temperature predictions in high-density DIII-D L-
mode cases [1042]. In TGLF, two recent developments have
improved the quasilinear modeling fidelity in the L-mode
edge. The electron temperature gradient mode was shown
to have a sufficiently high electron energy transport contri-
bution in L-mode edge conditions to improve the prediction
of electron temperatures in some cases [916]. Furthermore,
improving the dependence of the saturation model on the flux
surface shape increased the overall transport in TGLF [288,
347], sufficient for the predicted turbulent transport to simu-
late temperature and density profiles with the ASTRA code in
agreement with AUG L-mode observations not only for low-
density, low-collisionality ECRH heated cases [1054, 1055],
but also for more general variations of engineering paramet-
ers such as plasma current, plasma density [1055] and mag-
netic field [1056], up to the LCFS, see figure 46. This change
to the TGLF saturation model does not make it violate the

gyrokinetic ordering in the natural ρ* parameter. This is con-
sistent with earlier findings that turbulence simulation with
the GENE gyrokinetic code in a flux tube were able to match
the power balance fluxes in L-mode near the separatrix [829,
1057]. QuaLiKiz TEM predicted transport is also able to pre-
dict JET temperature profiles in a low-density regime, up to the
LCFS [1058]. Validation of TGLF and QuaLiKiz predictions
at higher density, where drift-resistive modes may dominate,
is ongoing.

The progress in validating both local gyrokinetic and
reduced model prediction of L-mode near-LCFS heat trans-
port, provides confidence in proceeding towards L-H trans-
ition predictive simulations and experimental validation (at
least in low-density regimes), including projections of the H-
mode power threshold. A challenge remains with fully self-
consistent particle transport predictions, which are complic-
ated by uncertainties in calculating the particle source arising
from gas puffing and recycling.

9.3.6. LOC/SOC transition with theory-based transport models. In Ohmic L-modes, the energy confinement time first increases linearly with increasing electron density, which is known as the LOC regime, and then saturates or slightly decreases. This second phase is the SOC regime. More than 40 years after its first observation, the LOC/SOC transition has been thoroughly investigated experimentally and has driven numerous theoretical studies [565]. The progress of theory-based transport models embedded in an integrated modeling framework has proven particularly useful in the understanding of the LOC/SOC transition. Dedicated modeling of density ramps in ASDEX Upgrade Ohmic L-modes [1059] was performed within ASTRA, using TGLF-SAT1 [381] for turbulent transport and an analytical model for neoclassical transport. The current profile, magnetic equilibrium, electron and ion temperature, electron density were all predicted. The impurity content, temperature and density at the separatrix were prescribed according to a parametric fit of experimental data. The simulations reproduced accurately the evolution of the energy confinement time with the electron density, except at low current and high density, where the ion heat transport is somewhat underestimated. This study unveiled the respective role of key plasma parameters in the dynamic evolution of the plasma during the density ramp-up. In the beginning of the LOC phase, at low density, the transition from TEM turbulence to ITG turbulence was shown to be responsible from an increase of the density peaking, which was decorrelated from an abrupt transition to the SOC regime. When the density is further increased, the SOC phase occurs. The saturation is due to an increase of the ion heat flux driven by ITG turbulence. Here, the decrease of the impurity content ($Z_{eff}$) during the density ramp plays a key role, since impurity content dilutes the main ions and reduces the ITG drive. The LOC phase can be understood as being due to increasing electron-ion temperature coupling in the TEM dominant phase, whereby the electron Ohmic heating is progressively transferred to the ions, relatively increasing $T_i$ while ITG is still stabilized due to high $Z_{eff}$.
<figure><img src="image_49.png" /><figcaption>**Figure 46.** Comparison of TGLF-SAT2 profile predictions up to the LCFS (solid triangles) vs. measurements (open circles) for an AUG L-mode power scan. $T_e$ (upper left), $T_i$ (upper right) and $n_e$ (bottom left) comparisons are shown at various radial locations. The total thermal energy comparison is shown on the bottom left. Reproduced from [1055]. © EURATOM 2022. CC BY 4.0</figcaption></figure>

Finally, the importance of subdominant modes has been emphasized in a quasilinear approximation model [568, 1060].

9.3.7 Predict first modeling for ITER and other machines. A recent success of predictive transport modeling was the resolution of the long-standing puzzle of core temperature perturbation inversion. In specific regimes typically at low density, triggering an outer core electron temperature reduction due to impurity injection and associated energy losses, leads to a subsequent core electron temperature increase on timescales faster than the energy confinement time. See [1061] for a review. The rapid change in transport properties was hypothesized as evidence for non-local phenomena. However, recent simulations with ASTRA-STRAHL-TGLF were able to explain the fast temperature inversion from within a local

paradigm. Simulations of C-Mod with a prescribed fast density pulse induced by the impurity injection showed that density flattening stabilized TEM (which dominates the transport in the low-density regime) leading to a temperature inversion [1062]. This led to Predict First simulations for DIII-D with subsequent experiments that reproduced the phenomenon as predicted [978]. Subsequent simulations with predicted particle transport as well as heat transport on AUG cold pulse experiments also reproduced the magnitude and timescale of the inversion as measured, with the fast density transport timescale a consequence of the TEM regime [117].

Both QuaLiKiz and TGLF were recently found to be valid for AUG scenarios in a variety of regimes [1063].

Within the framework of steady-state scenario projections to ITER, high $\beta_{pol}$ scenarios with high safety-factor and dominant non-inductive bootstrap current have been developed
at DIII-D [1064]. TGLF modeling successfully reproduced these experiments [1065], as well as similar experiments at EAST [1066]. A complex regime is predicted, with bifurcations between edge and internal transport barriers through the interplay between Shafranov shift, bootstrap current, and magnetic shear, impacting KBM turbulence thresholds [330, 976]. Projections to ITER predict a viable long-pulse $Q \sim 5$ scenario in this regime [1067].

The recent JET DTE2 experimental campaign was accompanied by a significant modeling effort aiming to extrapolate D scenarios to DT fusion power predictions, both for baseline and hybrid scenarios. For theory-based core turbulence modeling, QuaLiKiz and TGLF were applied. Pedestal extrapolations included both empirical and theory-based scalings. The JET experimental campaigns that preceded the DT campaign incorporated theory-based DT scenario extrapolation activities, using as a reference the high performance scenarios achieved. These generally led to successful predictions of the reference scenarios, and extrapolations to DT power in the range 10–16 MW, depending on various assumptions and heating power [1068–1071]. The eventual record fusion energy DT discharges themselves produced fusion power in line with the theory-based predictions, for the input power available [1072]. This result provides significant confidence in extrapolations to ITER performance.

DIII-D has developed an ITER-like scenario designed to mimic as closely as possible the dimensionless parameters corresponding to the high-current ITER baseline scenario. This scenario has been the subject of extensive high-fidelity and reduced transport model validation and extrapolation to ITER performance. Dedicated validation exercises using TGLF stationary state modeling with TGYRO reproduced this scenario (among others) [1073]. The same workflow predicted ITER performance within ITER goals, as long as a pedestal height in line with EPED predictions is achieved. Successful validation of TGLF predictions for the DIII-D hybrid scenario, including for the ITER-like-shape, was also achieved [1074]. More recent TGYRO-TGLF predictions also reproduced the DIII-D scenario, and further shed light on the turbulent pinch mechanism leading to density peaking, and predicted $Q \sim 10$ for the baseline ITER scenario using the same modeling workflow [1075].

Study and optimization of ITER scenarios with the theory-based reduced transport models has generally shown that core confinement is compatible with ITER goals [1073, 1076]. Optimization of $Z_{eff}$ (implied through core impurity seeding) was carried out in TGLF integrated modeling [1077]. In ITER long-pulse and hybrid scenario modeling with GLF23, heating and current drive optimization studies was carried out, through q-profile tailoring which impacts ITG critical thresholds [1078–1080]. Simulations with MMM, GLF23, and TGLF in various ITER scenarios has predicted a potentially significant impact on confinement of toroidal flow shear, arising from turbulent momentum pinch, depending on assumptions on rotation at the pedestal top [1081–1083].

Recent ITER baseline simulations comparing TGYRO-TGLF and JINTRAC-QuaLiKiz show agreement between the models, both predicting the achievement of $Q \sim 10$, as long

<figure><img src="image_50.png" /><figcaption>**Figure 47.** ITER baseline modeling with TGYRO-TGLF-SAT0 (dotted curves), TGYRO-TGLF-SAT1 (dashed curves) and JINTRAC-QuaLiKiz (solid curves), predicting $T_e$ (upper panel), $T_i$ (middle panel) and $n_e$ (lower panel). The pedestal boundary condition at $\rho_{tor} = 0.93$ is consistent with EPED modeling. The average fusion gain predicted is $Q \sim 9.5$. Reproduced from [272]. © IOP Publishing Ltd All rights reserved.</figcaption></figure>

as pedestal confinement is consistent with theory-based EPED extrapolations [272]. See figure 47, where the pedestal is simulated with a continuous ELM model consistent with theory predictions of height and width.

For modeling initial ITER Pre-Fusion Power Operation (PFPO) phase plasmas, it is encouraging that validated theory-based models in this specific low density high electron heating regime are available, as shown in section 9.3.5. Recent modeling with ASTRA-TGLF of 1/3 field PFPO-1 ITER plasmas has suggested that additional ECRH heating is necessary for H-mode attainment [1084].

We briefly mention the use of reduced turbulent transport models in design and performance predictions of future machines aside from ITER. QuaLiKiz and TGLF have both been applied for supporting the design of the Divertor Test Tokamak (DTT) [1085]. TGLF has been extensively used for illustrating the physics basis of SPARC performance predictions [1086]. CFETR baseline and hybrid scenario prediction and optimization was studied with TGLF [1087, 1088]. EU-DEMO studies have incorporated TGLF for core transport predictions to guide design choices [19, 1089, 1090].
9.3.8. **Challenges for reduced models and integrated modeling.** Multiple challenges remain for validation of reduced transport models in various regimes. The Dimits shift, while often small in simulations with kinetic electrons [1091], has parameter dependencies, e.g. collisionality [1092] and parallel scale lengths [1093], meaning that extrapolation to future devices with low collisionality is uncertain. A fully predictive Dimits shift reduced model is not yet available and fully validated, although efforts are ongoing [1094, 1095]. The validity of the local $\delta f$ in near-marginality is in general a topic of active debate, as summarized in section 9.4. KBM turbulence, which may dominate in specific domains and scenarios such as high $\beta_p$ scenarios [976, 1096], or the inner-core in MHD-free hybrid scenarios [998], may have a different saturation mechanism compared to ITG/TEM, thus necessitating a specific saturation rule in reduced models [1097]. This holds similarly for turbulence from MTMs, which may dominate in pedestals or spherical tokamaks, necessitating further understanding of saturation mechanisms and dedicated saturation rules [1098, 1099]. In general, reduced models of inter-ELM pedestal turbulence are necessary for full-device integrated modeling at high fidelity; this includes models for pedestal ETG and MTM turbulence, for which dedicated studies are ongoing [1100, 1101]. The importance of drift-resistive modes, predicted prevalent in high density L-mode edge conditions, also must be determined for accurate predictive modeling, particularly for ramp-up studies and predictive L-H transition modeling [830]. A further challenge for reduced models, discussed in detail in section (TODO), is the incorporation of fast-ion-enhanced EM-stabilization. While efforts have been made to construct effective models [1102] or modify saturation rules based on finite-$\beta$ modified ZF coupling [353, 1103], these have not yet been incorporated or tested within existing reduced transport model frameworks.

Routine application of advanced methodologies for validation and UQ is also key. For meaningful comparison between transport model predictions within integrated modeling and experimental measurements in various regimes, it is vital to provide validation metrics and UQ of both model and measurements such that statistical mismatch can be ascertained [1000, 1001]. This is vital for determining where model improvement is needed, and for trusting extrapolations to future scenarios and devices in specific regimes. Incorporation of this methodology has improved in recent years. More routine incorporation of statistical methods experimental profile fitting is taking place, such as Gaussian Process Regression [1104, 1105] and Bayesian analysis [1106]. UQ of integrated modeling output should include propagation of model uncertainties or prescribed experimental variables such as boundary condition values or impurity content. This is not routinely done due to computational expense, but has been demonstrated with various methods [1105, 1107]. Fast surrogates of transport models (see section 9.3.9) provide an opportunity for extended UQ analysis, facilitated by automated workflows. In stand-alone analysis of transport models, a method for statistical mismatch analysis through a surrogate-based strategy and UQ of local input uncertainties has been

developed [386]. Further efforts in incorporating and extending the above methods for more routine verification, validation and UQ applications is highly desired.

9.3.9. **Surrogate modeling with NNs.** Machine learning techniques such as NNs are powerful tools for regression. By fitting a pre-generated database of transport model runs, they can accurately reproduce the static nonlinear input-output mapping defined by the model. The inference time of the fitted NN model is on the order of 1 ms, orders of magnitude faster than the original model, leading to a surrogate transport model with near-realtime capability. This opens up a plethora of many-query applications, such as extensive UQ, large-scale model validation, inter-shot scenario optimization, and controller development.

An early proof-of-principle was a regression of QuaLiKiz with adiabatic electrons, 4D inputs and a single ion heat flux output [1108], providing faster-than-real-time simulation of ITER scenarios within the RAPTOR [1109] control-oriented simulator. This work was later extended to kinetic electrons and particle transport modeling [1110]. An extension to a lattice input set, with ten dimensionless input parameters, bounded by the typical ranges encountered by experiments, was then carried out [1111]. The training cost function and regressor variables incorporate physics-informed elements to ensure consistency with known physical structure such as sharp instability thresholds with a common threshold to all transport channels. The NN model, QLKNN-hyper-10D, is not bounded by specific experiments and could thus be used for study of future devices, and has been applied for DTT [1085] and ITER [1112] scenario optimization.

The incorporation of more input dimensions demands restricting the input space to the sub-space defined by experimental data, modeled extrapolations to future devices, and variations thereof. This approach was taken by the TGLF-NN, first developed based on training set inputs derived from DIII-D data [1077], and later extended to a multi-machine database including ITER simulations from the TRANSP database [239]. Physics-informed instance weighting is applied for increased fitting accuracy near critical thresholds. TGLF-NN has been demonstrated in coupled core-pedestal simulations together with a NN surrogate of the EPED pedestal model, including for ITER scenario optimization [239]. A version of the QuaLiKiz NN with training set inputs also constrained to experimental data was recently developed, based on a large database of JET discharges [1113]. This QLKNN version, QLKNN-jetexp-15D, has 15 input dimensions compared to the more general, but dimensionally-constrained QLKNN-hyper-10D. QLKNN-jetexp-15D has recently been successfully applied for JET hybrid scenario ramp-up optimization for the tritium campaign [1114]. A NN regression of the MMM model has also been developed, based on a DIII-D input database, and applied for control-oriented simulations [1115].

The surrogate modeling avenue has proven extremely promising for delivering fast and accurate tokamak simulation
necessary for flight simulator modeling. Multiple avenues of further improvement are possible. Surrogates can be generated based on higher-fidelity modeling, such as quasilinear gyrokinetic modeling of full GK codes with fewer approximations, and can be multi-fidelity by incorporating a limited numbers of nonlinear simulations. This approach would result in both faster and more accurate transport models and has been pioneered through a quasilinear GKW model in JT-60U parameter space [1116]. For such higher-fidelity surrogates, the computational burden of the training set generation necessitates data efficient approaches; QLKNN and TGLF-NN training sets are over-sampled (since it is cheap and does not harm the final output), but for higher-fidelity modeling only truly informative data-points should be generated. An active learning pipeline aiming to achieve this goal has been recently developed [1117]. Progress in open data sets from fusion devices and simulation databases would help further accelerate surrogate model development. With regard to physics-informed approaches, the NN topology can be structured to better capture the salient features of the input-output structure, e.g. through a parameterized critical-gradient-model [1118]. Finally, surrogate modeling techniques can be directly applied to accelerate simulations pipelines involving high-fidelity nonlinear simulations, as recently demonstrated for a stationary-state transport solver with turbulent fluxes from CGYRO nonlinear simulations [1086]. Such techniques enable more routine ultra-high-fidelity validation of reduced transport models.

## 9.4. New frontiers in gyrokinetic simulation

Model reduction, as has been emphasized above, is critical to explore parameter space and to interpret measurements. The accuracy of model reductions especially hinges on the ability to understand the fundamentals of plasma organization. The primitive models for turbulence are based on the gyrokinetic formalism. The prevailing paradigm for gyrokinetic computations assumes separations of scales between rapidly evolving micro-scale fluctuations, typically at electron $\rho_e$ or ion $\rho_i$ Larmor scales atop temporally fixed and smooth macro-scale background gradients, typically at the scale $L$ of the device. Modeling strategies which enforce this scale separation are ‘gradient-driven’: fluctuation-fluctuation feedback on the mean is discarded. Mean gradients are fixed in time and small-amplitude perturbations are computed about this state [30, 142, 302, 311, 1034, 1119–1121]. The scale separation allows for computational efficiency. Such frameworks may either be local (flux-tube, radial periodicity is enforced) or global (no radial periodicity enforced). They have proven fruitful in the plasma core yet are questioned where scale separations become debatable, e.g. near marginal stability or in the plasma edge. Near marginality is to be understood in a non-linear sense, as a regime of minimal available free energy (or driving source) which allows sustainment of a finite level of turbulence activity. Strong stiffness (clamping of profiles near turbulence onset) or reduced ratios (as expected in ITER) of

external drive (heating or fueling power) over plasma volume with respect to current-day machines contribute to the relevance of near marginal regimes for fusion operation.

A more physically consistent approach has been proposed in which known, imposed fluxes drive the system and both mean gradients and fluctuations dynamically evolve on the same timescale, driving and back-reacting on one another. This ‘flux-driven’ framework [255, 278, 495, 502, 710, 1122–1128] relaxes assumptions of scale separation, keeps the fluctuation-fluctuation feedback on the mean (mean gradients become an output of the computation) and results in a tenfold or more increase in computational demands. Integrated modeling frameworks, while flux-driven in the sense that a stationary state is reached at power-balance, still implicitly make the scale-separation assumption, since the transport PDEs (macro model) call the reduced turbulence model (micro model) for an instantaneous update following a macro timestep, typically on order of 1 ms.

The validity of gradient-driven approaches is debated, especially at the plasma edge, where scale separations are questionable, or near marginal stability, i.e. between linear and nonlinear thresholds, in the so-called ‘Dimits shift’ region [730, 1129]. Flux-driven approaches indeed show near marginality, i.e. that (i) distinct turbulent states and flux levels can be obtained from almost identical underlying mean plasma gradients. This fact underlies a multi-valued (mostly bistable), non-monotonic flux-gradient landscape [257, 496, 1128, 1130, 1131], with competition [906] between mean (equilibrium) flow shear and zonal (fluctuating) flow shear.

In addition, (ii) turbulence spreading [1053, 1132–1136] is ubiquitous in flux-driven regimes, its more visible manifestations being the observation of fronts or avalanches [255, 1122, 1123, 1126, 1137–1141]. This fact underlies departure from a local or Fickian transport paradigm [1139, 1142, 1143]. The connection between spreading and departure from local, single-valued flux-gradient landscapes is still being debated. Aside from the breakdown of Fickian transport at microscales, flux-driven frameworks also report (iii) mesoscale organization near marginal stability with emergence of a tertiary structure, the ‘E × B’ staircase' [256, 710, 1139, 1141, 1144–1158]. Staircases are a sequence of zonal mean flow layers co-located with profile corrugations which statistically enclose regions of avalanching (strong transport). See figure 48 for an illustration. The zonal mean flow interspacing scale is about 40 to $50\rho_i$, with a fat tail [257, 1141, 1151, 1156]. This emergent mesoscale near marginality (iv) is consistent with a transition from Bohm-like (avalanche-dominated) transport for small plasma sizes to gyro-Bohm like (staircase-regulated avalanching) transport for large plasma sizes $\rho_* = \rho_i/L \lesssim 1/300$ [257]. Above marginal stability, as staircase organization fades, avalanching activity is less regulated and Bohm-like confinement can be observed [496]. Interplay between layering and spreading, permeability of shear layers (staircase-related or otherwise) to extended transport events are seldom explored though interesting angles to pursue a deeper understanding of transport [257, 1141, 1151,
<figure><img src="image_51.png" /><figcaption>**Figure 48.** GYSELA full f global simulation based on Tore Supra parameters. Long lived structures form where narrow regions of $E \times B$ shear locally reduce heat fluxes and lead to temperature corrugations (staircase steps), with higher heat flux (temperature avalanching) in between. Reprinted figure with permission from [256], Copyright (2015) by the American Physical Society.</figcaption></figure>

1159, 1160]. Regarding robustness, (v) staircase organization is numerically observed with adiabatic or kinetic [1154, 1155] electron responses, in electrostatic or electromagnetic [1158] frameworks. Gradient-driven approaches hinder emergence of such structures [257, 1150]. Experimentally, density fluctuations measured with fast swept reflectometry on TORE SUPRA [256, 1151] have identified regularly-spaced shear flow layers consistent with the staircase phenomenology. Electron imaging diagnostics in KSTAR have emphasized avalanche regulation by such shear flows [1153, 1155]. More experiments are required to fully document staircase regimes. Near-marginal regimes are relevant to large-scale devices such as ITER. Observation of (vi) significant discrepancies in heat transport predictions between flux-driven approaches and gradient-driven or quasilinear approaches [1018] has been reported near nonlinear threshold. These observations may

have important consequences yet are still in their infancy and require further investigation.

Gradients are steep in the peripheral plasma (Scrape-Off Layer (SOL), pedestal and edge), with intrinsic temporal variability. Precise experimental measurements are challenging. Fluxes may vary by factors in gradient-driven frameworks upon scanning imposed mean gradients within experimental error bars. Predictive assessment of transport levels in the peripheral plasma is therefore especially difficult. Through the possibility of ion-scale and equilibrium coupling, flux-driven approaches have emphasized core-edge-SOL turbulence interplay (and hence turbulence spreading) as a potentially important player for edge dynamics [920, 1053, 1128, 1131, 1161]. Indications of such interplay have been experimentally reported [1162]. Recent modeling [1053] from core to material boundaries in flux-driven gyrokinetics shows that combination of plasma-boundary interaction and spreading of boundary-borne, poloidally-localized fluctuations provides a pathway to avoid transport shortfalls of computed fluctuation levels near the plasma separatrix, even when no local linear drive is present. The former deeply modifies convective stability close to the magnetic separatrix. The latter emphasizes edge turbulence as the result of a globally organized state, ‘nonlocally’ controlled by fluxes of turbulence activity. This spreading dynamics spontaneously leads to improved confinement through onset of a stable transport barrier at the separatrix. Given the central importance for fusion of the edge region, a significant effort is currently under way to improve [1163–1165] or expand [1053, 1166–1172] major kinetic approaches towards consistent interplay between confined plasma and SOL, incorporating aspects of sheath and neutral physics. The latter have been shown to play a key role in fluid modeling of the peripheral plasma dynamics [1173–1177].

Synergies between neoclassical (collisional) and turbulence dynamics have been hinted at experimentally [1178–1180] and emphasised numerically [258, 259, 503, 1181–1183] which, given the importance of edge rotation or high-Z impurity transport are matters of current interest.

Last and by no means least, the perspective of burning plasmas is an exciting new frontier for fusion. Transport and turbulence may be significantly modified from current-day devices as additional feedback loops must be considered in which fast particles, thermal ions and slow modes can interplay [285, 1184, 1185]. This new physics, relevant to reactor operation is also challenging. It is flux-driven in nature: fast particle production depends on profile dynamics and plasma history; it also likely requires global integration of core, edge and possibly SOL to account for the interplay between thermal modes and the large orbits of suprathermal particles.

An important caveat throughout these results is that the majority of flux-driven simulations were carried out with the adiabatic electron assumption. This strong assumption is known to enhance the relative impact of ZFs and increase the Dimits shift [1091]. In future work, it is imperative to leverage
increased HPC capacity for fully realistic physics simula-
tions with kinetic electrons, global, full-f and flux driven.
Through comparisons of these simulations with their δf flux-
tube counterparts, the true impact of relaxing the scale separa-
tion approximation in various regimes can be assessed, with
a crucial contribution to the validation and development of
reduced models.

Acknowledgments

Authors would like to extend our thanks and enormous appre-
ciation to all colleagues who have contributed to reduced
model development and validation within Transport and
Confinement ITPA activities.

10. Summary

M. Yoshida, R.M. McDermott and J. Snipes

Over the last decades, great progress has been made
on improving our understanding and modeling capability
of plasma TC via TC ITPA joint activities and interna-
tional experiments. Physics mechanisms important to the
transport in all channels, particle, impurity, thermal, and
momentum transport, were identified thanks to develop-
ments in experimental techniques, theoretical models, and
operation capabilities in multiple machines. Predictive cap-
abilities for ITER and burning plasmas were significantly
improved with the developed/validated theoretical models,
integrated codes, and global confinement scaling approach.
On the other hand, there are still important issues remain-
ing for future burning plasmas. The TC Chapter will be
closed by describing the important remaining issues of each
section.

In the particle transport studies, improved boundary con-
ditions that mimic plasma-wall interaction and experimental
quantification of neutral particle sources and sinks in the edge
region are some of the challenging, but essential issues to
develop density profile modeling from the separatrix to the
core region. The measurements of inward particle pinch con-
firmed the crucial role collisionality is playing in contrib-
uting to density peaking. Experimental observations of the
isotope scaling of particle transport are limited compared to
those of thermal energy transport and are not always explained
by gyrokinetic simulations. Further detailed studies will be
needed with additional experimental data including the latest
JET hydrogen, deuterium and tritium experiments to predict
density profiles in ITER DT phases. As the fusion power
depends critically on the density, it is necessary to develop
a fundamental understanding of both density limits and how
RMPs will impact the density profile. In addition, the out-
ward particle flux in the outer pellet deposition zone under
ELM suppression conditions should be evaluated with both
experiment and theory. Although compatibility of pellet fuel-
ing and ELM mitigation with RMP was demonstrated, pel-
let triggered transient ELM free H-mode remains a potential
issue.

In the impurity transport studies, more complete character-
ization of helium transport coefficients and more stringent val-
idation of the present predictive capabilities remain import-
ant research items. Model validation of the measured out-
ward convection and hollow density profiles of light impurit-
ies are also challenging tests. For quantitative comparison, full
nonlinear gyrokinetic simulations, which entail a considerable
level of effort and computational resources, will be required.
Characterization of heavy impurity transport behavior in ITER
relevant conditions would certainly also be of interest. Another
important research area is to investigate impurity transport
mechanisms in the plasma edge and SOL, which determines
the overall impurity content in the plasma core. Joint works
with the DIVSOL and pedestal TGs will be beneficial.

In the thermal transport studies, the validity of fast ion
effects should be investigated for the MeV range. Global flux-
driven simulations coupling turbulent and equilibrium scales
will be necessary to assess the MeV-class fast-ion effects on
ITG stabilization in burning plasmas as well as to quantify
SOL-edge-core coupling and barrier onset mechanisms. In
parallel, an upgrade of quasi-linear models by including non-
linear electromagnetic stabilization effects is important for
hybrid and advanced scenario plasma modeling. Incorporating
near marginal turbulence dynamics in quasi-linear saturation
rules could also prove important. The ETG contributions are
still not sufficient to explain the electron flux even with multi-
scale simulations. The impact of impurities or hidden variables
would be worth investigation. Quasi-linear models for elec-
trons also need to be updated and further validated against
gyrokinetic simulations and experiments. Measurements of
electron-scale fluctuations would add precious information.

In the momentum transport studies, quantitative agree-
ment with gyrokinetic predictions of diffusion and convec-
tion coefficients has only been obtained under ITG domin-
ated conditions. Validation of theoretical predictions of the
turbulent momentum fluxes under TEM dominant conditions
or in mixed ITG-TEM conditions are important for reliable
predictions in plasmas with dominant electron heating. The
construction of a momentum database is required to con-
duct such a systematic model validation. Additional work
remains to identify the impact of core residual stress on rota-
tion profile formation. However, the largest uncertainty for
predicting the core rotation profile in ITER, is the pedestal
top rotation, which, through diffusion and especially convec-
tion, can strongly influence the core profile. Significant exper-
imental and theoretical work remains to understand the origin
and scaling of the residual stress momentum flux in this region.
The development of modeling capabilities for residual stress,
and other effects that influence the rotation, in the edge pedes-
tal and the connection of the core flows with those in the SOL
are indispensable to develop a predictive integrated modeling
framework.

For the effect of 3D fields on transport studies, one of
the most crucial issues for ITER is to achieve ELM sup-
pression while minimizing plasma confinement degradation
by the application of RMP. Comprehensive validated models
capable of predicting the 3D effects on edge transport will
be required to achieve quantitative prediction in ITER with
applied RMP field or a stellarator-based reactor. For example, effects of 3D fields require a synergistic approach for the various phenomena observed in the different machines. Effects of plasma boundary and plasma-facing components should be also investigated. The complex interplay between neoclassical and turbulent transport properties should be fully understood which requires synergistic research between tokamaks and stellarators.

In the studies of confinement transitions, the scaling of the L-H power threshold, $P_{th}$, should be refined in view of ITER and burning plasma relevant conditions. For instance, the dependence of $P_{th}$ on density in different isotope mixtures, impurities from different wall materials, and different boundary configurations need to be clarified consistently in multiple machines. Those parameters might be worth including in the scaling variables. The L-H transition power should be explored in dominant electron heating, small external torque, low collisionality, and low normalized ion gyro-radius conditions, although those conditions might be challenging in present machines. Modeling of L-H transitions closely matched to experimental parameters are key to improve the prediction of $P_{th}$ and understand the L-H transition physics.

In global confinement studies, there are missing variables such as the separatrix density, plasma rotation, and the electron-to-ion heating ratio, in the confinement scaling. There is actually a clearly decreasing trend with increasing separatrix density in the confinement enhancement factor. The physics behind the updated scaling dependencies, such as a reduced dependence on machine size, should be resolved. Inter-dependencies between predictor variables in the regression analysis remains an issue for a more unambiguous estimate of the individual dependencies. The confinement scaling prediction for ITER hybrid scenarios will be the next target. Expanding the database by adding data from additional devices and new operational areas in the existing devices, as well as by collecting other possible variables affecting the confinement should be continued to solve those questions. Physical constraints would be key to further regularize the scaling.

In the reduced transport modeling studies, multiple challenges remain for development and validation of reduced transport models in various regimes. Fast ion enhanced electromagnetic stabilization should be incorporated in the reduced transport model frameworks as the fast ion effects were clearly demonstrated in non-linear turbulence simulations and experiments. Saturation rules in KBM and MTMs were not examined yet and will be required for high fidelity modeling in the core region in the high $\beta_p$ regime and in the H-mode pedestal region, respectively. Development of reduced models of the H-mode pedestal and L-H transition are challenging, but essential issues for integrated modeling. This work is closely linked with the question of which plasma edge scenarios will be present in ITER and reactors. Joint work with multiple topical groups (PEP, DSOL, IOS) would be important. Further development of surrogate modeling with NNs is key to accelerate simulation pipelines involving high-fidelity nonlinear simulations.

The above-mentioned remaining issues are quite challenging. Observed complicated phenomena can likely not be explained by turbulent and neoclassical transport physics only. The phenomena closely link with other physics areas, such as energetic particles, MHD, pedestal stability, and boundary conditions as well as sources and sinks from the DIV/SOL regions. Strengthening coordination and cooperation with other topical groups will be key to providing answers to those issues for ITER. Assessing uncertainties of transport models and scalings for confinement and L-H power threshold are very important, as this can have a large impact on the predicted performance of ITER plasmas and design of future reactors. Consistent integrated modeling including rotation, impurities and 3D magnetic fields is the next essential target of the TC group with realistic error estimates. Consistency of the profile modeling with global confinement scaling is also key to minimize uncertainties. There are also important variables that are absent in present machines, but become important in burning plasma conditions, such as MeV class fast ions, small gyroradii, and high density with low collisionality. Model validations with new data from JET DT plasmas, JT-60SA high beta steady-state plasmas, and existing machines with machine enhancements, could help to close this gap.

The TC TG will tackle these challenging issues by exploring transport model development and validation based on the requirements for burning plasmas. The theoretical works will have two directions: higher-fidelity modeling to give more accurate predictions and for the design of future machines and reduced modeling to contribute to real-time plasma control. Experiments should focus more on ITER or reactor relevant conditions, such as dominant electron heating, small torque input, higher beta, and low collisionality. Both experiment and modeling will continue to work to achieve those targets.

## Acknowledgments

The author team would like to acknowledge the tremendous effort and help of the Guest Editors in charge of this chapter, Richard Hawryluk and Baonian Wan for this Nuclear Fusion Special Issue. In addition, we would like to thank the all of the experts of the ITPA Transport and Confinement (TC) topical group for their valuable comments and suggestions.

## Funding acknowledgements

Support for the authors of this Chapter as follows: The work by P. Mantica has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200 EUROfusion). Views and opinions expressed are however those of the authors only and donot necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the
European Commission can be held responsible for them. The work by G. Verdoolaege has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 and 2019–2020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission or the ITER Organization. The research was also partly supported by U.S. DOE Contract DE-AC02-09CH11466. The work by E. R. Solano is supported in part by Grant PID2021-127727OB-I00 of the Spanish Ministry of Science and Innovation, funded by MCIN/AEI/10.13039/501100011033 and by ERDF/EU. The work by G. Dif-Pradalier is supported by EPSRC Grant EP/R014604/1. This work is partially supported by a grant from the Simons Foundation. GDP would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme 'Anti-diffusive dynamics: from sub-cellular to astrophysical scales',

where work on this paper was undertaken. The work by B.A. Grierson 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 Awards DE-FC02-04ER54698. The work by S.M. Kaye is supported by the U.S. Department of Energy, Office of Science, under Contract DE-AC02-09CH11466. The work by J.W. Hughes is supported in part from the U.S. Department of Energy, under Awards DE-SC0021629 and DE-SC0014264. The work by S. Mordijck is funded by the U.S. Department of Energy, Fusion Energy Sciences, DE-SC0021306. The work by L. Schmitz is supported in part from the U.S. Department of Energy, under Awards DE-SC002-0287. The work by G. Staebler is supported by the U.S. Department of Energy, under DE-AC05-00OR22725 (ORNL). This work by J. Snipes is supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under Award DE-AC02-09CH11466 (PPPL).
# Appendix

<table><caption>**Table 3.** JEX/JA table.</caption><thead><tr><th>Number</th><th>Title</th><th>Coordinators</th><th>Period</th></tr></thead><tbody><tr><td>TC-1</td><td>Confinement scaling in ELMy H-modes: beta degradation</td><td>C.C. Petty</td><td>before 2009–2012</td></tr><tr><td>TC-2</td><td>Power ratio—Hysteresis and access to H mode with H 1</td><td>Y.R. Martin</td><td>before 2009–2012</td></tr><tr><td>TC-3</td><td>Scaling of the Low-Density Limit of the H-mode Threshold</td><td>J. Hughes</td><td>before 2009–2012</td></tr><tr><td>TC-4</td><td>H-mode transition and confinement dependence on ionic species</td><td>P. Gohil</td><td>before 2009–2012</td></tr><tr><td>TC-5</td><td>Determine transport dependence on Te/Ti in hybrids and steady-state</td><td>—</td><td>by 2009</td></tr><tr><td>TC-6</td><td>Effect of rotation on plasma performance</td><td>—</td><td>by 2009</td></tr><tr><td>TC-7</td><td>ITG/TEM transport dependence on Ti/Te, q and rotation in L-modes</td><td>P. Mantica</td><td>2009–2010</td></tr><tr><td>TC-8</td><td>QH/QDB plasmas</td><td>—</td><td>by 2009</td></tr><tr><td>TC-9</td><td>Intrinsic rotation and torque for ITER rotation prediction in L and H modes</td><td>J. Rice, T. Tala/B. Duval</td><td>before 2009–</td></tr><tr><td>TC-10</td><td>Turbulence and transport in the L-mode core-edge region</td><td>C. Angioni, C. Bourdelle/G. Staebler</td><td>before 2009–2023</td></tr><tr><td>TC-11</td><td>Impurity transport studies dedicated to ITER priorities</td><td>C. Angioni/M. Valisa</td><td>2009–</td></tr><tr><td>TC-12</td><td>H-mode transport and confinement at low aspect ratio</td><td>M. Valovič</td><td>2009–2013</td></tr><tr><td>TC-13</td><td>Ion and electron critical gradient and profile stiffness</td><td>P. Mantica</td><td>2009–2023</td></tr><tr><td>TC-14</td><td>RF Rotation Drive</td><td>J. Rice</td><td>2009–2014</td></tr><tr><td>TC-15</td><td>Electron and mixed-ion particle transport, isotope scaling and density peaking</td><td>T. Tala</td><td>2009–</td></tr><tr><td>TC-16</td><td>Dependence of momentum pinch and diffusivity on Beta</td><td>T. Tala</td><td>2011–2012</td></tr><tr><td>TC-17</td><td>rho* scaling of intrinsic torque</td><td>W. Solomon/T.Tala</td><td>2011–2019</td></tr><tr><td>TC-18</td><td>Dimensionless Identity Experiments in I-Mode</td><td>F. Ryter</td><td>2011–2013</td></tr><tr><td>TC-19</td><td>Characteristics of I-mode plasmas</td><td>J. Rice</td><td>2011–</td></tr><tr><td>TC-20</td><td>Validation of transport models in ITER-similar current-ramp plasmas</td><td>D. Mikkelsen</td><td>2012–2013</td></tr><tr><td>TC-21</td><td>Characteristics of the LOC/SOC transition</td><td>J. Rice</td><td>2012–2020</td></tr><tr><td>TC-22</td><td>Evolution of transport during and immediately after H-L transitions</td><td>D. McDonald</td><td>2012–2014</td></tr><tr><td>TC-23</td><td>Physics of nonlocality phenomena</td><td>H. Sun</td><td>2012–2014</td></tr><tr><td>TC-24</td><td>Impact of resonant magnetic perturbations on transport and confinement</td><td>M. Jakubowski/T. Evans/S. Youwen, D. Orlov</td><td>2013–</td></tr><tr><td>TC-25</td><td>Particle transport of pellet fuelled H-modes</td><td>M. Valovič</td><td>2015–</td></tr><tr><td>TC-26</td><td>L-H/H-L scaling in the presence of metallic walls</td><td>D. McDonald/E.Delabie</td><td>2015</td></tr><tr><td>TC-27</td><td>Fueling vs particle pinch in opaque edge Plasmas</td><td>S. Mordijck</td><td>2015–2019</td></tr><tr><td>TC-28</td><td>Update and reanalysis of International H-mode Database</td><td>S. Kaye</td><td>2016–2020</td></tr><tr><td>TC-29</td><td>Update of 1D profile database to support neural network training</td><td>J. Citrin</td><td>2017–2023</td></tr><tr><td>TC-30</td><td>Gyro-kinetic database</td><td>Y. Camenen</td><td>2018–2022</td></tr><tr><td>TC-31</td><td>PFPO-relevant L-modes to analyze H-mode accessibility</td><td>J. Citrin</td><td>2019–2023</td></tr><tr><td>TC-32</td><td>Negative Triangularity and Impact of Shaping</td><td>M.J. Pueschel</td><td>2022–</td></tr></tbody></table>

The titles of TC-9, 10, 11, 13, 15, 19, 24, 25, 28, 29, 30, 31 were reworded to reflect modification in focus on ITER HPIs in 2019. The numbers were redefined in 2009 (CDB-xx and TC-xx were changed to TC-xx) according to the merger between the Transport Physics Topical Group and the Confinement Database and Modeling Topical Group. TC-5,6, and 8 were closed at the transition from CDB/TP to TC.
<table><caption>**Table 4.** Major database of transport and confinement.</caption><thead><tr><th>Name</th><th>Responsible Officer</th><th>Citation</th><th>Section</th></tr></thead><tbody><tr><td>International Global H-Mode Confinement Database</td><td>G. Verdoolaege, S.M. Kay</td><td>[950]</td><td>8</td></tr><tr><td>International Threshold Global Database (IGDBTH)</td><td>Y.R. Martin</td><td>[1]</td><td>7</td></tr><tr><td>Multi-machine density peaking databases</td><td>C. Angioni</td><td>[9]</td><td>2</td></tr><tr><td>Multi-machine momentum database (IMDB)</td><td>M. Yoshida</td><td>[525]</td><td>5</td></tr><tr><td>Intrinsic Rotation Profile DB (RotDB)</td><td>B. Duval</td><td>[408]</td><td>5</td></tr><tr><td>Multi-machine I-mode database</td><td>A.E. Hubbard</td><td>[881]</td><td>7</td></tr><tr><td>Gyro-Kinetic DataBase (GKDB)</td><td>Y. Camenen</td><td>http://database.gkdb.org</td><td>9</td></tr></tbody></table>

<table><caption>**Table 5.** Major theoretical models.</caption><thead><tr><th>Transport channel</th><th>Models</th><th>Section</th></tr></thead><tbody><tr><td>Particle transport</td><td>GENE<sup>GK</sup>, GKW<sup>GK</sup>, NEO<sup>NC</sup>, QuaLiKiz<sup>GK</sup>, TGLF<sup>GF</sup>, EDWM<sup>GF</sup></td><td>2</td></tr><tr><td>Impurity transport</td><td>CGYRO<sup>GK</sup>, TGYRO<sup>GK</sup>, GENE<sup>GK</sup>, GKW<sup>GK</sup>, GYSELA<sup>GK</sup>, GT5D<sup>GK</sup>, QuaLiKiz<sup>GK</sup>, TGLF<sup>GF</sup>, NCLASS<sup>NC</sup>, NEO<sup>NC</sup></td><td>3</td></tr><tr><td>Thermal transport</td><td>CGYRO<sup>GK</sup>, GENE<sup>GK</sup>, GS2<sup>GK</sup>, GKV<sup>GK</sup>, GKW<sup>GK</sup>, GYSELA<sup>GK</sup>, GT5D<sup>GK</sup>, XGC<sup>GK</sup>, QuaLiKiz<sup>GK</sup>, TGLF<sup>GF</sup></td><td>4</td></tr><tr><td>Momentum transport</td><td>GS2<sup>GK</sup>, GKW<sup>GK</sup>, GTS<sup>GK</sup>, GYRO<sup>GK</sup>, GTC<sup>GK</sup>, NEO<sup>NC</sup></td><td>5</td></tr><tr><td>Reduced transport</td><td>QuaLiKiz<sup>GK</sup>, TGLF<sup>GF</sup>, MMM, EDWM<sup>GF</sup></td><td>9</td></tr></tbody></table>

GK: Gyrokinetic models, GF: Gyrofluid models, rGK: Quasilinear gyrokinetic models, rGF: Quasilinear gyrofluid models,

NC: Neoclassical transport models.

## ORCID iDs

M. Yoshida (Chair Transport and Confinement) https://orcid.org/0000-0002-2899-1323

R.M. McDermott (co-Chair Transport and Confinement) https://orcid.org/0000-0002-8958-8714

C. Angioni https://orcid.org/0000-0003-0270-9630

Y. Camenen https://orcid.org/0000-0003-1361-3722

J. Citrin https://orcid.org/0000-0001-8007-5501

M. Jakubowski https://orcid.org/0000-0002-6557-3497

J.W. Hughes https://orcid.org/0000-0003-4802-4944

Y. Idomura https://orcid.org/0000-0002-2829-0498

P. Mantica https://orcid.org/0000-0001-5939-5244

A. Mariani https://orcid.org/0000-0003-0476-3825

S. Mordijck https://orcid.org/0000-0001-8537-4383

E.J. Paul https://orcid.org/0000-0002-9355-5595

T. Tala https://orcid.org/0000-0002-6264-0797

G. Verdoolaege https://orcid.org/0000-0002-2640-4527

A. Zocco https://orcid.org/0000-0003-2617-3658

F.J. Casson https://orcid.org/0000-0001-5371-5876

G. Dif-Pradalier https://orcid.org/0000-0003-4869-7049

B. Duval https://orcid.org/0000-0001-7783-129X

B.A. Grierson https://orcid.org/0000-0001-5918-6506

S.M. Kaye https://orcid.org/0000-0002-2514-1163

M. Maslov https://orcid.org/0000-0001-8392-4644

T. Odstrcil https://orcid.org/0000-0003-4173-4230

J.E. Rice https://orcid.org/0000-0001-8319-5971

L. Schmitz https://orcid.org/0000-0003-1346-0914

F. Sciortino https://orcid.org/0000-0002-5159-1889

E.R. Solano https://orcid.org/0000-0002-4815-3407

G. Staebler https://orcid.org/0000-0002-1944-1733

M. Valovič https://orcid.org/0000-0002-0855-1056

E. Wolfrum https://orcid.org/0000-0002-6645-6882

J.A. Snipes (co-Chair Transport and Confinement) https://orcid.org/0000-0002-7077-5077

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