frequency is comparable to the resistive frequency, the change in topology is suppressed [645, 646]. Reduction of the resonant field by one to two orders of magnitude is observed when a resistive response is taken into account, for large enough plasma flows [647]. On the other hand, non-resonant components of the perturbation are not screened, can penetrate, and thus contribute to NTV, see section 6.2.1.

Shielding currents at rational surfaces induce a braking torque, known as an electromagnetic (EM) torque. This arises due to **J** caused by eddy currents near the rational surface and the magnetic field perturbation due to an RMP, providing a **J** × **B** torque [583]. The steady-state rotation frequency is determined by the balance between the viscous and EM braking forces. In contrast with the NTV torque, which typically scales as $T \sim V - V_{\text{offset}}$, EM torque decreases with increasing rotation. Furthermore, EM torque is typically localized near rational surfaces, but can sometimes become more global due to radial diffusion [647].

The basic physical description of the interaction between 3D rotating magnetic structures and external magnetic perturbations (or resistive walls) was put forth by Fitzpatrick [645]. Here it was understood that rotating magnetic islands are stabilized by rotation, to some extent, if they are small enough. However, when the island width exceeds a critical value, mode locking occurs prior to complete stabilization. Because the EM torque achieves a local minimum for a given rotation frequency, below a critical rotation frequency, a bifurcation to very small rotation occurs [645, 646], known as mode locking. Slow rotation causes the penetration of the RMP and rapid growth of an island on a time scale associated with the resistive decay of current sheets. Due to the nature of EM forces, mode unlocking due to increasing of rotation occurs at significantly larger critical rotation value than that of mode locking, indicative of hysteresis.

The original picture has been further refined by improving on the plasma model by considering viscous-resistive effects, thus identifying three main parameters that characterize the response to a static, resonant, error-field of a rotating tokamak plasma in the vicinity of the rational surface. These are indeed the plasma viscosity, rotation, and resistivity. In total, eleven distinguishable response regimes have been identified [646]. An improved version of the original model of Fitzpatrick [646] was then proposed to include diamagnetic and semicollisional effects [648] to the layer response, that is finite equilibrium pressure gradients and electron thermal conduction. From the point of view of the layer response, electron diamagnetism can completely suppress a magnetic perturbation [648] for large enough values of the control parameter $\hat{\beta} = 0.5\beta_e L_s^2/L_n^2$, via a screening of the perturbed current [649] (here $\beta_e$ is the ratio of the kinetic to the magnetic plasma pressure, for the electrons, and $L_s$ and $L_n$ are some characteristic lengths for global shear and equilibrium density gradients, respectively).

The theory of resonant field screening and rotation is further modified in the presence of stochastic layers [650, 651]. A parallel current is established by electron streaming along field lines, while a neoclassical ion radial current arises to maintain ambipolarity. The associated self-consistent $E_r$ modifies the

screening of resonant fields, and the ion radial current leads to a modification to the rotation profile.

The inclusion of NTV (section 6.2.1) in the rotation balance can modify the locked mode physics. In a regime where NTV torque exceeds perpendicular momentum diffusion, the rotation is maintained at $V_{\text{offset}}$ outside the resonant layer, and the threshold for locking typically increases with this effect [652–655]. Newer analytic models which include drift-MHD [656] and nonlinear [655] effects more accurately capture the linear scaling of the critical perturbation amplitude with density observed in JET, DIII-D, and Alcator C-Mod. Extrapolation of this theory to ITER indicates that avoidance of locked modes lies just within the capabilities of the error-field correction system [628].

Some aspects of the theory of the plasma response to magnetic perturbations at resonant locations have been investigated numerically [657], confirming the importance of semicollisional effects, but also pointing out the crucial role played by kinetic Alfvèn waves [649]. Other crucial phenomena, such as drift wave radiation [658], have been proposed to have an impact on the penetration of magnetic perturbations (either rotating or locked). A comprehensive theory [581] is yet not available, but high fidelity modeling [659] provides a general physical understanding in which screening effects increase for lower resistivity, stronger rotation, and smaller RMP amplitude. A further important effect is that of the nonlinear evolution of the radial electric field produced by RMPs. The $E \times B$ rotation due to such fields can be locally compensated by electron diamagnetic rotation. Then, RMPs can indeed penetrate and form magnetic islands. This is generally true for a large class of examples based on DIII-D, JET, and ITER type plasmas [659].

More direct experimental evidence of the interplay between magnetic perturbations, plasma rotation, and high fidelity layer modeling has been supported by extended MHD computations, which have been successful in reproducing the parametric space of ELM suppression via RMPs in the DIII-D and KSTAR tokamaks [660]. Careful modeling and dedicated experiments brought the important conclusion that ELM suppression by RMPs can occur over wide $q_{95}$ ranges in DIII-D ITER similar shape (ISS) plasmas with $n = 3$ RMPs [661, 662]. Because the field penetration threshold decreases with density, the range of $q_{95}$ for ELM suppression increases with decreasing electron density. The access to this regime at substantially higher pedestal pressures is facilitated by applying higher toroidal mode number ($n = 4$) RMPs.

In conventional (far from quasisymmetric) stellarators, the NTV torque plays an outsized role in establishing the flow profile. The presence of $1/\nu$ transport in a boundary layer outside an island allows the EM torque to be overcome, and healing of the island [663], analogous to mode unlocking in a tokamak. The theory has been validated on TJ-II and LHD, which have observed the healing of magnetic islands with an increase in poloidal flow [664]. If the NTV is in the $1/\nu$ regime, it is observed to compete with viscosity and dominate the rotation profile for ITER parameters [636]. This leads to penetration in the edge and shielding in the core. The NTV torque has
been computed via MHD-kinetic hybrid modeling of an ITER baseline scenario coupled with a drift-kinetic model. The results indicate that the EM torque dominates over the NTV torque for ITER parameters with $n = 3$ perturbations [665].

In practice, both the resonant and non-resonant torques play a role in determining the rotation profile in tokamaks with 3D perturbations. Plasmas respond to RMP fields by amplifying non-resonant harmonics that have the same sign of the pitch as resonant harmonics [647, 666] through a near-marginal kink-like perturbation. This leads to an effective increase in NTV that becomes more substantial near marginality. The effect has been observed in DIII-D [667], where the magnetic braking of an $n = 1$ perturbation was measured, showing, however, the importance of both resonant and non-resonant torques in determining the global plasma rotation profiles. Similar effects have been observed on KSTAR [668]. For non-resonant configurations, a “quiescent” behavior is observed in which no density pump out and loss of stored energy are detected. Resonant configurations feature stronger rotation damping along with confinement degradation and density pump out, in line with DIII-D results. The modeling of such phenomenon in ASDEX Upgrade (different $n$ and phasing) [669] is consistent with the general picture: the resonant EM torque, the Reynolds stress, and NTV all contribute, to different extents, to the flow damping. The relative importance of each factor depends on the specifics of the configuration. Specifically, the NTV and the EM torque tend to be largest in the pedestal, for smaller Reynolds numbers.

### 6.2.3. Influence of 3D fields on impurity transport.
The physics of impurity transport, neoclassical and turbulent, has been recently reviewed by Angioni [113] and is also discussed in section 3 of this chapter. One of the most important developments for neoclassical impurity transport studies is in the realization that poloidal asymmetries of the impurity density can develop [185, 199, 200], thus affecting their radial transport [188, 201, 202]. While this might be suggestive of specific features of nonaxisymetric devices, where the electrostatic potential also shows variations on flux surfaces [670–672], it cannot be considered a truly 3D effect, but rather a collisional and finite poloidal Larmor radius effect [113]. Truly 3D effects are induced by the presence of (saturated) MHD modes, which can be mimicked with stellarator equilibrium codes [673]. However, contradictory results in JET hybrid scenarios [673] (where impurities peaking is observed and simulated) and ASDEX Upgrade ECRH centrally heated ones [253] (where an impurity hole is observed) are limiting our ability to make conclusive statements. Nevertheless, this area of research shows a great potential for synergistic tokamak-stellarator studies. Indeed, the observation of an impurity hole in a plasma with an ion internal transport barrier on the Large Helical Device [674] has been a key result, and a long standing open problem [675, 676] in the stellarator community. The analogy seems to apply also to the ITER pedestal, for which numerical studies have been performed to quantify the effects of non-axisymmetric fields on the ambipolar electric field and radial particle fluxes [677]. Here the authors studied both the

toroidal field ripple and the externally applied fields for ELM control. We know that, in ITER, the toroidal field ripple effects on high-Z impurity transport are negligible [677]. However, ELM control fields cause a significant modification of the pedestal ambipolar electric field (becoming less negative), allowing for the appearance of multi-valued solutions for the electric field, analogous to core stellarator transport [607]. Thus, ELM control external fields significantly increase the outward neoclassical pedestal transport in ITER, for both the main plasma ions, and high-Z impurities.

### 6.2.4. 3D neoclassical effects on particle transport.
Non-axisymmetry has an important effect on collisional particle transport for both impurities and the bulk species. For this reason, the formal tools used to address the physics of impurities and their 3D properties are the traditional ones of neoclassical theory [113, 678]. Since even in axisymmetry, magnetic field inhomogeneities due to the poloidal variation of the magnetic field strength cause the presence of trapped particles and, therefore, of neoclassical effects, it is not so surprising that the neoclassical theories of particle and impurity transport in tokamaks (whether 3D or not) and stellerators (whether modular or helical) are mutually interrelated. A ubiquitous particle transport effect induced by 3D fields is the large density pump-out phenomena routinely observed in ELM control experiments using RMP fields. It is now understood that this behavior is triggered by the same physics that produces the toroidal NTV torque, mediated by the drift resonance between the 3D field perturbation and the toroidal precession of trapped thermal particles [679]. Pump out becomes stronger at low collisionality and near the offset of the plasma rotation, which are relevant regimes for ITER. ELM suppression mechanisms reduce the $E \times B$ drift frequency at the pedestal, thus allowing for the resonance to occur. High-fidelity modeling of neoclassical transport in the presence of broken magnetic surfaces with the NEO and M3D-C1 codes has reproduced the density pump-out observed on RMP-suppressed DIII-D plasmas [680]. To date, the modeled pump-out level does not always match the experimental value. One of the reasons could be the effect of island stochasticity, which also affect the values of $E_r$ [639]. Experimentally, a correlation between density pump-out, the increase of $E_r$ with RMPs, and a spin-up in the co-current direction, possibly due to loss of electrons in the stochastic layer, is observed [681].

## 6.3. Turbulent transport

Stellarators provide a unique 3D configuration in which turbulence can be studied and in recent years with the optimization of stellarator geometries to limit neoclassical transport, turbulence research has progressed rapidly, both theoretically [682–690] and experimentally [603, 691–697]. Multiple tokamaks have observed changes in turbulence characteristics when 3D fields are applied. As 3D fields have been beneficial at regulating confinement and stability, there have been a variety of observations, some of them at first glance contradicting results from other devices. Turbulence, even in axisymmetric fields,
depends strongly on the plasma characteristics, such as collis-
ionality, local gradients in both densities and temperatures,
im purity dilution, the normalized plasma pressure β<sub>N</sub>, plasma
species, plasma rotation, temperature ratios, just to provide an
incomplete list. As such, without taking into account the fun-
damental plasma parameters and turbulence conditions inde-
pendent of the eventual breaking of axisymmetry, it is challe-
ging to compare and contrast observations. An additional com-
plication is that, as section 6.1.2 describes, 3D magnetic field
configurations are not identical in mode structure and each will
have a different impact on turbulent transport. The situation is
similar to the case of stellarators, where the classification of
geometries is notoriously not straightforward. While all exper-
imental research provides information on the 3D actuators that
are employed, not all authors provide information on the actual
3D field configuration that leads to the observed changes in
turbulence. In the following sections, we limit ourselves to
those observations that can be linked to known and reproduc-
cible 3D field configurations.

Commonly, 3D effects in tokamaks are introduced by
applying RMPs. The presence of RMPs often leads to
enhanced particle transport which causes a so-called dens-
ity pump-out which alters the pedestal pressure gradient and
pushes the plasma into a stable peeling-ballooning regime, see
section 6.5. As a result, there has been an enhanced focus on
investigating how 3D effects alter turbulence and contribute to
the changes in particle transport. Early research did not always
have the tools yet to identify the underlying local 3D structure
and link this to the observed changes. Several devices such
as DIII-D [599, 601, 698], EAST [699–701], KSTAR [702],
HL-2A [228], MAST [703] and AUG [704] showed changes
in turbulence characteristics that would indicate an enhance-
ment of turbulent transport. This was in contrast with earlier
work on ergodic divertors on TEXTOR [596] and TORE-
SUPRA [705], where changes in turbulence were directly
linked to specific changes in magnetic geometry. Enhanced
understanding of changes to the magnetic topology in toka-
maks has allowed for joint tokamak-stellarator turbulence
studies [706]. These studies provide critical insights on the
ubiquitous properties of turbulence in 3D magnetically con-
fined fusion plasmas.

In the next sections, we first discuss observations where
the 3D-induced changes in turbulence correlate with changes
in profiles and turbulence drive as calculated with gyrokin-
etic codes. An expansion on this work, discussed in the next
section, investigates the direct impact of the altered radial
electric field by 3D fields on turbulence and turbulent trans-
port. Finally, we discuss how specific magnetic topologies can
affect turbulence, from 3D corrugation of flux surfaces, to cre-
ation of islands, stochastic fields, natural resonances, and the
impact of mixing open and closed field lines. These changes
in geometry can directly affect both turbulent flow generation
as well as changes in electrostatic and magnetic fluctuations.

### 6.3.1. Indirect 3D impact on turbulence and transport.
Several experiments have reported that after the application
of RMPs, the changes that have been observed at the edge

of both L- and H-mode plasmas can be reproduced using
quasi-linear gyrokinetic modeling, without including any 3D
deformations to the magnetic geometry. DIII-D H-mode plas-
mas showed that the density fluctuations in the steep part of the
pedestal were directly correlated with the linear growth rates
calculated by TGLF [600] and more recently similar results
were obtained for the changes in electron temperature fluctu-
ations [707]. Similarly, simulations with GTC showed that the
changes in 3D field structure in similar H-mode plasmas on
DIII-D were not sufficient to directly influence changes in tur-
bulence characteristics [708].

In DIII-D L-mode plasmas, the applications of RMPs did
lead to small changes in gradients at the plasma edge. Using
a perturbative gas puff to probe particle transport, a reversal
of the particle pinch from inward to outward was found, and
this switch in direction coincided with a change from ITG to
TEM driven turbulence according to TGLF simulations. This
predicted switch from ITG to TEM was also observed in the
reversal of the turbulent flow measured by the DBS [597].
None of these results can explain what caused the initial
change in confinement, but indicate that once a new equi-
librium is found that the 3D field geometry might not be
the dominant component to the changes in turbulent trans-
port in specific conditions. Similar observations on the role
of local gradient are found when studying the impact of mag-
netic islands and 3D corrugation of magnetic flux surfaces, see
sections 6.3.3.1 and 6.3.3.2.

### 6.3.2. Impact of $E_r$ well and $E \times B$ shear.
The majority of the research on RMPs is linked to the suppression of ELMs by reducing the pressure gradient in the pedestal region. In the pedestal region, the $E \times B$ shear plays an important role in regulating and reducing turbulent transport [549]. In the pedestal region, the $E \times B$ shear is dominated by the pressure gradient and as a consequence any changes to the electron density and temperature profiles, will directly affect the $E \times B$ shear. Additionally, the changes in toroidal rotation through neoclassical effects, resonant and non-resonant braking (see section 6.2), cannot be neglected as a contribution to the $E \times B$ shear, especially in L-mode where the pressure gradient has a much smaller contribution.

The application of RMPs showed a flattening of the radial
electric field in MAST L-mode consistent with changes in
both perpendicular and parallel flows and a broadening of
the ion saturation fluctuations [703]. These effects required
the applied RMP field to exceed a minimum value and the
effects were not toroidally localized. Similar, in AUG L-
mode plasmas, using various RMP configurations also showed
a threshold below which no changes to flow, $E_r$ and dens-
ity fluctuations were observed using the DBS [704]. Similar
observations were observed in DIII-D L-mode plasmas, where
below a threshold changes to the radial electric field and
density fluctuations were within error bars [597]. Above the
threshold, density fluctuations increased along a substantial
change in the toroidal rotation and a change in the tur-
bulent flow as measured by the DBS. Similar changes are
observed before the L to H-mode transition in DIII-D, which
will be discussed in section 6.4. Both the MAST and AUG experiments showed a differentiation between the turbulence response to RMPs inside the separatrix and in the SOL. This differentiation is attributed to difference in magnetic topology and the creation of a laminar and stochastic zone. This strong dependence of localized turbulence measurements on the underlying magnetic field structure highlights the need, when studying the impact of various RMP spectra in plasmas with different densities and $q_{95}$ values, to calculate the plasma response. A comparison between L- and H-mode plasmas revealed that in H-mode, even for very small levels of RMPs, a density pump-out was observed, counter to the need for a threshold in L-mode [709]. Aside from potential effects of the plasma response on the strength of the applied field, it was noted that H-mode plasmas are close to marginality and a small change in the $E \times B$ shearing rate (through for example neoclassical effects) could have a nonlinear effect on turbulence and transport, through a self-reinforcing cycle until a new transport equilibrium is achieved.

Further examination of how RMPs impact DIII-D H-mode plasmas shows that the $E \times B$ shearing rate at the top of the pedestal and inward $\Psi_N \sim 0.7-0.9$ decreases, along with an increase in density fluctuations measured both by the BES and DBS [600]. The reduction in the $E \times B$ shearing rate means that the linear growth rates of the dominant turbulent modes are now larger than the $E \times B$ shearing rate. This strongly limiting the $E \times B$ shearing rate ability to suppress turbulent transport. Gas puff modulations to measure the changes in particle transport indicated that the outward diffusion increases and the inward particle pinch decreases over the whole plasma radius. Identical results were obtained on EAST with $n=2$ RMP perturbations and using the supersonic molecular beam to extract the perturbative particle transport coefficients [710]. EAST reproduces that the diffusive coefficient increases at the plasma edge, while the inward pinch decreases. A comparison with neoclassical simulations shows that the experimental obtained values are much larger. Similarly on DIII-D, a reduction of the $E \times B$ shearing rate was observed, along with an increase in density fluctuations. Further research on EAST found that there was a toroidal dependence of the $E \times B$ shear and fluctuations linked to the corrugation of 3D field when RMPs are rotated around the device [711]. KSTAR found an increase in perpendicular velocity fluctuations when the $E \times B$ shear is altered through neoclassical effects [712]. These results were confirmed using gyrokinetic modeling with the GTC code, where they found that changes in the $E \times B$ shear were necessary to explain the increase in density fluctuations with 3D fields [713].

### 6.3.3. Direct 3D geometry link to turbulence & transport.
Early research on limiter tokamaks such as TEXTOR [596, 714, 715] and TORE-SUPRA [705], and recent stellarator results [697, 716] showed that the underlying 3D geometry is important in how the turbulence characteristics are affected. In diverted tokamak configurations the rotation of RMPs and visible imaging of the X-point region showed that the separatrix

is deformed and there is a corrugation [717, 718]. The 3D corrugation will affect long wavelength turbulence and turbulence flows directly. Moreover, the 3D homoclinic tangles [719]—topological structures created by splitting of the separatrix or any other resonant flux surface by resonant components of the magnetic perturbation—close to the X-point result in the creation of a region where closed and open field lines are intermixed. Experiments on TEXTOR studied the ergodic divertor concept, focused on how turbulence is affected by the mixing of chaotic field lines of different lengths. In the ergodic zone in TEXTOR, the turbulent particle flux reverses from outward to inward, which results in an improvement in particle confinement. Along with the reversal of the particle flux, the poloidal phase velocity of the fluctuations altered direction from the electron to the ion diamagnetic direction [596, 720]. The changes in turbulence in the laminar region are linked to a reduced rotational shear, and blobby transport is reduced in the SOL [715]. In the next sections, we will first address the impact of the simple 3D corrugation of magnetic flux surfaces, the impact of island structures, and finally the impact of chaotic field lines.

H-mode experiments in DIII-D in which the 3D-field perturbation was altered showed an immediate change in the density fluctuations as a function of the applied RMP field strength [601]. The increase in density fluctuations was observed over a wide span of $q_{95}$ values, from mid-radius outwards. This was independent of ELM suppression and the increase in density fluctuations was much larger than the local changes in density. The observed changes in density fluctuations were instantaneous and indicated that the 3D fields had a direct impact. GTC modeling using 3D fields obtained using plasma response model, were not able to reproduce these results [713]. To match the observations, the GTC model in addition to the 3D field required the altered $E \times B$ shear to be included. Regardless of the reliability of the physical model used to numerically describe turbulence at the L-H transition, a key aspect of a successful modeling is the study of the effect of the bootstrap current on the local value of the global shear, which has a great impact on the character of turbulence itself [721].

#### 6.3.3.1. 3D corrugation of magnetic flux surfaces.
DIII-D H-mode plasmas showed that turbulence, as well as the plasma profile modifications, are linked to the 3D corrugation of the plasma boundary [722]. While the changes in fluctuations followed the same trends as those observed in the local profiles, they indicate that changes in turbulence as a result of the 3D corrugation might not be uniform. Long wavelength turbulence and flows such as GAMs and ZFs benefit from toroidal symmetry in tokamaks. Experiments on MAST have shown that RMPs can damp GAMs [723]. In MAST, the RMPs result in a shift of the mode with respect to plasma rotation and eventual suppression of the mode. An increase in the power of turbulence fluctuations is observed after suppression, indicating a nonlinear coupling to higher wave number turbulence. Theoretical work has focused on how the breaking of the toroidal symmetry of the magnetic flux surfaces can affect flows. Initial theoretical work showed that in the presence of RMPs,
ZFs are damped and partially transfer their energy to a resonant vortex-flow pattern [724]. The resulting vortex-flow has a multi-scale nature that can lead to streamer-like transport properties. The theory also shows that RMPs can lead to transport bifurcation and alter energy-transfer from ZFs to turbulence [725], which in turn will affect changes in confinement as will be discussed in the LH-transition section 6.4. Finally, further development of this 1D theory indicated the importance of $E \times B$ flow in aiding the damping of the ZF.

Theoretical investigations using gyro-kinetic simulations of DIII-D plasmas show that a direct decrease in ZF levels is linked to an increase in turbulence fluctuations levels [726]. The nonlinear gyrokinetic simulations in GENE do not contain the 3D deformation, but GENE has an ad-hoc magnetic field component that can be scaled. A scan of this ad-hoc magnetic field perturbation in GENE, using experimental profiles as input, is able to reproduce the experimental observations. The turbulence increases at smaller scales and a reduction of the turbulence ZF are observed. Further theoretical studies looking at the impact of tearing modes on ZFs and electro-static microturbulence found the charge loss as a result of RMPs on a flux surface damps the ZF and leads to higher electron-scale fluxes while the microtearing saturation physics is unaffected [380].

The understanding of kinetic microinstabilities and turbulence in 3D is not yet established, but the field is advancing considerably. The effects of general 3D geometry on the toroidal ITG and KBMs where studied in [727, 728]. A general theory of the electrostatic TEM was proposed in [729], while its electromagnetic counterpart is still missing. Rosenbluth-Hinton residuals [730], in this context, undergo a new type of oscillation and collisionless damping [731–734], which has no equivalent in axisymmetry, nor in helical symmetric devices. The growing analogies between tokamak pedestal and stellarator core behaviors [735] are allowing us to draw a more reliable picture of 3D effects on microinstabilities.

### 6.3.3.2. Magnetic islands.
Magnetic islands in the plasma edge are either spontaneously present, due to the MHD activities affecting magnetic flux surfaces with low rational safety factor [736], or result from external perturbation fields interacting with the plasma equilibrium [593]. As a magnetic island breaks the nested magnetic flux surfaces, it modifies plasma transport both locally and globally. The enhanced radial heat diffusivity ($\chi_r$) due to the parallel transport along the field lines is increased over a region of about the island width $w$ [737]. The effects are significant if the island width is larger than critical island width [737, 738]

$$w_c = (\chi_{\perp}/\chi_{||})^{1/4} (\epsilon_s r_s / L_{q0s} n)$$

where $\epsilon_s = r_s/R$ is aspect ratio of the resonant flux surface with radius $r_s$ and $L_{q0s} = q_0/q'_0$. This section will provide a short summary of the impact of magnetic islands on turbulence and transport, and we refer the reader for a more extensive overview to Choi [739].

Earlier research had shown that heat pulses propagate at different speeds depending on whether they pass through the center of an island or the X-point in both stellarators and tokamaks [740, 741]. This geometric dependence on the heat flux resulted in a series of experiments on J-TEXT [742], DIII-D [743], HL-2A [744] and KSTAR [745] to better understand the origin and investigate the role of turbulence. All devices found a clear increase in density fluctuations using multiple diagnostics that measure density fluctuations at multiple wavelengths. Further research showed that the reduction in turbulence at the O-point is linked to the width of the island and the flattening of the temperature gradient [744, 746]. In general, most devices found good agreement between local gyro-kinetic models, based on the measured gradients, and the turbulence changes, which could include a change from ITG to TEM dominant, depending on the temperature ratio. As such, the magnetic O-point reduced gradients and thus transport, whereas no such reduction in gradients was observed at the X-point, nor a reductions in turbulence characteristics [742–753]. These results would suggest that flow shear and flow shear suppression are less relevant, but not all results fit within this framework. At the outer edge of the island, at the poloidal location of the O-point, turbulence is strongly suppressed, even with the reappearance of a temperature gradient. Experiments on KSTAR revealed a strong poloidal flow [754]. This agreed well with observations of radial electric field reversals in islands observed on LHD [755, 756] and TJ-II [757, 758]. Further experiments on KSTAR found that plasma turbulence becomes strongly inhomogeneous around a magnetic island due to the combined effect of the pressure gradient (driver) and flow shear (regulator) modifications by the island [739].

On LHD Jakubowski et al [759] showed that the natural $m/n = 1/1$ magnetic island enlarged with RMPs leads to an increased particle diffusion coefficient in the vicinity of the island. Temperature ($L_T$) and diffusion ($L_n$) scale lengths showed strong variation with the island width for the island width ($w_t$) larger than 5–6 cm. These findings are in line with gyrokinetic simulations of the static island by Bañón Navarro et al [690], showing a threshold in the island width significant for the transport changes. There are two distinct regimes depending on the island full-width, with a threshold occurring at $W_t = 33\rho_i$, where $\rho_i$ is the ion gyroradius. Both turbulent radial particle and heat fluxes are reduced at the O-point while only the total heat transport is increased across the island region. The hysteresis relation between density fluctuation ($\tilde{n}/n$) and temperature modulation during the heat pulse propagation into a magnetic island at DIII-D provided experimental evidence of the turbulence spreading from the X-point to the O-point of the magnetic island [755].

### 6.4. L-H transition

Transition from L-mode to H-mode, i.e. the rapid reduction of turbulent transport at the plasma boundary, which leads to the formation of the so-called pedestal in the plasma pressure profiles, was first observed in tokamaks and later also in stellarators. The current most broadly accepted mechanism for
the L-H transition is through self-organization of the turbu-
lence at the plasma edge, a strong $E \times B$ sheared velocity flow
is created which shears edge turbulent eddies, reducing their
scale length and amplitude. Once in H-mode, the strong $E \times B$
sheared flow is sustained through the pressure gradient. The L-
H transition is strongly influenced by initial conditions in the
L-mode discharge, such as collisionality, electron density, tor-
oidal and poloidal plasma flows and the ability for those con-
ditions to support ZFs and Geodesic Acoustic Mode (GAM)
generation. An overview of the current understanding on the
L-H transition can be found in section 7 of this paper. This
section first discusses the observed changes in power threshold
when 3D fields are applied, next we discuss how 3D fields
alter the initial conditions, such as macroscopic flows and tur-
bulence characteristics and how 3D fields alter ZF and GAM
behavior. Next we discuss how the 3D field nature of the stel-
larator affects L-H transitions.

6.4.1. *Impact on power threshold for L-H transition.*

Historically, a threshold in power has been observed to trigger
the plasma to transition from L- to H-mode. The threshold
is an important parameter which allows the identification of
power requirements for future fusion reactors as well as act-
ing as a simple metric to assess whether the L-H transition
is affected by a change in plasma conditions and provide a
scaling metric across multiple devices to aid in uncovering
common physics.

The application of RMPs in L-mode, before the L-H transition occurs can result in an increase of the power threshold, which has been observed on DIII-D [760, 761], MAST [762, 763], AUG [764–766], NSTX [767] and, KSTAR [768]. All tokamak experiments indicated that the power threshold increases with increasing resonant magnetic field strength as calculated using plasma response models. This would mean that the power threshold is less affected if the applied field is mostly non-resonant [760–763, 766] and on K-STAR they did find that the power threshold in general, due to smaller errorfields was below those from DIII-D [768]. DIII-D also found that even with similar momentum injection, RMPs applied in ECH heated plasmas increased the power threshold more than RMPs applied in NBI heated plasmas [761]. Moreover, the applied RMPs further amplified the increased power threshold by a factor 2 at low collisionality, with little or no change at high collisionality. MAST and AUG were able to show that the power threshold is directly linked to the magnetic field as calculated using a plasma response model, instead of a vacuum calculation [763, 766]. Devices that employ a plasma response model to calculate the radial magnetic field strength all show a minimum requirement in the applied 3D magnetic field amplitude to increase the L-H transition power threshold. To better understand how the RMP spectrum affects the power threshold for the L-H transition the focus shifted to

(i) investigating the impacts of RMPs on L-mode plasma conditions before the transition and comparing those to physical parameters that are known to affect the L-H transition without the application of RMPs,

(ii) investigating how RMPs could alter the self-organization
of turbulence through long-wave length turbulence flows
which can facilitate the L-H transition.

Both observations and theoretical understanding will be
discussed in the next sections with respect to both the impact
of altered plasma conditions before the L-H transition as well
as the changes during the transition.

6.4.1.1. *RMP impact on pre L-H transition plasma conditions.*

As indicated earlier, RMPs and 3D-fields in general can alter both neoclassical (see section 6.2) and turbulent transport (see section 6.3). The power threshold has been linked to a minimum required ion heat flux close to the plasma edge as well as requirements linked to pre-existing $E \times B$ flow shear and turbulence characteristics. The observations on the impact of RMPs in the L-mode phase prior to the transition are similar to those reported in previous sections and the discussion in this section focused on how these changes are linked to the changes in power threshold.

6.4.1.2. *Plasma edge flows.* As indicated in section 6.2, the application of 3D-fields will alter the plasma rotation through neoclassical effects, as well as resonant braking. The exact impact is strongly linked to whether the field is predominantly resonant or non-resonant, whether the magnetic field topology is stochastic or contains islands and at the plasma edge whether a region of open field lines is created inside the traditional 2D separatrix location.

In general any weakening of the $E \times B$ shear at the plasma
edge will result in an increase in the power threshold for the
L-H transition [769]. Both DIII-D and AUG have shown a
flattening of the edge $E \times B$ rotation and even a reversal at
high enough magnetic field strength [761, 766]. In AUG, the
reversal of the $E \times B$ shear was linked to the increased power
threshold, while on DIII-D an increase in power threshold was
already observed before the $E \times B$ shear changes signs. Non-
resonant RMPs do not have the same effect on the $E \times B$ velocity
[761] and also do not observe a strong increase in power
threshold. Moreover, the $E \times B$ flows are based on impurity
measurements and particle balance analysis indicated that the
changes in main ion flows could be more pronounced [766].
Finally, to make sure that the changes in $E \times B$ flow are not a
toroidally localized effect as a result of the 3D corrugation of
the separatrix, AUG was able to show that the changes were
consistent at multiple toroidal locations with respect to the
applied RMP spectrum.

6.4.1.3. *Plasma edge profiles.* Aside from altering the $E \times B$ edge flow, 3D fields above a threshold can alter plasma profiles, such as the electron density and temperature as well as the ion temperature through changes in neoclassical and turbulent transport as discussed in sections 6.2 and 6.3. In this section, we focus on the changes observed in L-mode plasmas just ahead of the transition and less on what transport channel caused the effects.
The power needed to transition into H-mode is directly linked to the electron density and above a minimum density, the power threshold increases linearly with an increase in electron density. In general, the application of magnetic perturbations above a minimum threshold reduces the electron density gradient at the plasma edge on AUG [766], DIII-D [761, 770], MAST [762, 763]. Moreover, an increase in electron heat transport is observed as more power is needed to reach similar temperature gradients at the plasma edge as without the application of 3D magnetic fields on DIII-D [770] and AUG [766]. Both of these lead to a flattening of the electron pressure gradient, which is one component along with the toroidal and poloidal rotation in determining the $E \times B$ shear flow. To complicate the physics of the L-H transition further, the profiles are not toroidally symmetric due to the 3D corrugation of the separatrix [761, 766, 770]. DIII-D did not observe much change in the ion temperature or the impurity density [770], while earlier observations on MAST indicated a strong change in the impurity density [763]. The impact of impurities is still an open topic of research with respect to its impact on the L-H transition even without 3D fields. As 3D fields alter impurity transport, see section 6.2.3, and thus the $Z_{eff}$ at the plasma edge it could also affect the LH transition.

### 6.4.2. Turbulence characteristics.
The interplay between turbulence, main plasma flows and turbulent plasma flows, such as ZFs and GAMs are critical for the L-H transition dynamics. Two sets of experiments on DIII-D showed that the RMP induced changes in turbulence before the L-H transition depend on whether the RMPs were resonant or non-resonant [761, 770]. Both experiments observed that the application of RMPs result in a strong increase in density perturbations and a shift to lower frequencies, whereas the NRMPs slightly damp the density fluctuation measurements as measured by the BES at multiple radial locations at the plasma edge. These observations were found to be independent of the 3D corrugation of the separatrix. The changes were not consistent with the changes in turbulence drive as calculated with a quasilinear gyrokinetic code, thus indirectly pointing to a 3D effect on turbulence characteristics. Aside from changes in electro-static perturbations, an increase in magnetic fluctuations is observed on DIII-D with the application of both resonant and NRMPs [761].

#### 6.4.2.1. Direct impact of RMPs on turbulent flows.
As indicated earlier, a crucial aspect for the formation of the transport barrier during the L- to H-mode transition is the sheared flow at the plasma edge, which helps sustain the turbulence decorrelation. Aside from the main plasma flows at the plasma edge, long wavelength $n, m = 0$ zero or low frequency radially localized modes such as ZFs and GAMs have been shown to impact the L-H transition.

As these flows have typically $n, m = 0$ mode numbers, any breaking of axisymmetry of the magnetic field will have an impact on them. The impact of 3D fields without directly connecting to the impact on the L-H transition can be found in

section 6.3. Theoretically, it has been shown that the application of RMPs damps the ZF, which increases the power threshold for the transition [771]. DIII-D experiments indicate that while there is an increase in density fluctuations at very low frequencies both with resonant as well as NRMPs, the poloidal velocity fluctuations are strongly damped at the plasma edge. Deeper inside the plasma, the RMPs actually result in the observation of a GAM [770]. Similarly, DIII-D showed that energy transfer rate between the turbulence and the turbulent flows was reduced by a factor 10, disrupting these flows from contributing to aiding the L-H transition.

#### 6.4.3. Stellarators.
The observation of the improved-confinement regimes and the occurrence of ELMs in helical devices confirms the generic character of L-H-type transitions in toroidal confinement. The first observation of an L-H-type transition in a 3D device was reported in 1991 at W7-AS [772]. Further results followed by at CHS [773], Heliotron-J [774], LHD [775] and TJ-II [776].

While the improvement of the plasma confinement in the case of 3D devices is not as pronounced as in tokamaks with poloidal divertors, it occurs in a somewhat similar fashion. Indeed, density fluctuations can be associated with specific changes in radial electric field shear and perpendicular velocity at the plasma boundary, and a formation of a pressure pedestal (however, not as pronounced as in the case of tokamaks), which leads to an observable increase in plasma density and plasma energy content. The width of the pedestal (typically a few cms) seems to be independent of the machine size, which suggests that the barrier formation is defined by processes from the plasma boundary independent of the size of the core plasma [777]. The typical MHD activity associated with the L-H transition, manifesting as ELMs, has also been observed in W7-AS [778], LHD [779] and also TJ-II [780]. The details of the ELM regimes in 3D devices, repetition frequency and associated particle and power loss to the SOL and ultimately to the divertor targets, depend sensitively on the configurations.

Obviously, if the transition to a high confinement regime is due to turbulence suppression, the plasma would ideally transition to the lowest possible level of transport, which in general is the neoclassic one (surely for tokamaks and optimized stellarators). Therefore, the most important difference between tokamaks and stellarator comes from the presence of a 3D magnetic equilibrium, which often impacts the transition to an H-mode due to an ambipolar radial electric field given by the neoclassical fluxes. This is due to the fact that neoclassical transport to large degree depends on the magnetic equilibrium in 3D configurations [603], see section 6.2.

#### 6.4.3.1. Magnetic configuration.
Regardless of the level of non-axisymmetry of a device, the importance of the details of the magnetic equilibrium on TC has become more and more evident. These aspects are referred to as magnetic configuration effects. Broadly speaking, they are all the factors that determine the plasma shape, for the presence of one or several null-points, plasma triangularity, symmetries, $q$-profiles, and
<figure><img src="image_30.png" /><figcaption>**Figure 27.** Changes in (a) the radial electric field $E_r$ at TJ-II during L-H transition and (b) accompanying modification of the density fluctuations. Reproduced from [782]. © IOP Publishing Ltd All rights reserved.</figcaption></figure>

mirror ratios, for instance. 3D devices, of course, have com- pelled us to understand magnetic configurations effects more urgently than tokamaks, showing inevitably a strong impact of the magnetic configuration on the L-H transition. This is espe- cially apparent in devices with low magnetic shear. At W7-AS, TJ-II (low shear devices), and Heliotron-J, an H-mode-type confinement regime only occurs when the ranges of $\iota$ for the configurations are near values that are slightly less than those of the major natural resonances of the machines [781].

6.4.3.2. *Poloidal rotation and shear at the edge.* Another quantity that shows magnetic configuration dependence dur- ing the L-H transition is the shear of the negative radial electric field, which can increase in this phase. The shear of the neg- ative radial electric field increases at the L-H transition by an amount that depends on the magnetic configuration and heat- ing power. At TJ-II, fluctuation measurements show a reduc- tion in the turbulence level that is most pronounced at the pos- ition of maximum $E_r$ shear [782]. However, as opposed to the tokamak case, the poloidal rotation is non-negligible in the radial force balance equation. High temporal and spatial resolution measurements of the radial electric field and dens- ity fluctuations at TJ-II indicate that turbulence reduction pre- cedes the increase in the mean shear flow, but is simultaneous with the increase in the low frequency oscillating shear flow. These observations are consistent with L-H transition mod- els, developed in a tokamak context, predicting plasma bifurc- ations triggered by ZFs [783].

Similarly as in W7-AS [778], in TJ-II the classical H-mode occurs at high densities with NBI heating and changes in plasma radial electric field, plasma rotation and density fluc- tuations. An example is presented in figure 27. During an L-H transition at LHD in co- and counter-NBI heated discharges $E_r$ is reduced by a factor of 2, which leads to significant increase in negative $E_r$-shear (figure 27(a)) and the perpen- dicular rotation velocity. The fluctuation level (figure 27(b)) shows a factor of five reduction at the location of the max- imum velocity shear.

6.5. *Plasma edge and MHD effects on transport*

As indicated earlier, 3D fields can alter both neoclassical as well as turbulent transport. At the plasma edge these changes in transport are intermixed with atomic physics through colli- sions with non-ionized or partially ionized main ions or impur- ities. Moreover, as RMPs alter the edge stability and are used to mitigate and suppress ELMs, this will impact the average particle and heat flux integrated over multiple ELM cycles. Finally, as the pedestal is characterized by MHD limits, MHD modeling has been employed to investigate the changes in con- finement to go beyond neoclassical and gyro-kinetic driven turbulent transport.

The application of magnetic perturbations leading to ELM suppression or mitigation is very often associated with a decrease in the pedestal plasma parameters—in particular plasma density (the so called ‘pump-out’), see figure 28. This was observed at DIII-D [784–786], later also confirmed at MAST [787], ASDEX Upgrade [227, 787], EAST [788] and KSTAR [789]. This increased transport, linked to both neo- classical effects for ions (section 6.2.3) and turbulence for electrons (section 6.3), seems to play a critical role in the reduction of the pressure gradients in the pedestal region and thus leading to ELM suppression, as discussed in the pedestal chapter of this Nuclear Fusion Special Issue [21].

XGC, 5D kinetic modeling of RMPs was able to repro- duce some of the observations for the pedestal [790]. The simulations evolve with time and are limited to less than 1 ms. As a result of this short time, the simulation starts from experimental profiles, but does not include actual heat and momentum sources and includes a numerical method to radiate the heat in the SOL to keep the separatrix electron temperature constant, as observed in experiments. Modeling shows that the 3D fields break ambipolar neoclassical trans- port, which mostly affects the ions and that for the electrons the changes in magnetic topology are the dominant component. In a follow-up study that included electrostatic turbulence, it was observed that while the density fluctuations increased, there was a decrease in electron temperature fluctuations [791]. The simulations are able to reproduce a density pump-out of much
<figure><img src="image_31.png" /><figcaption>**Figure 28.** Pedestal changes during application of RMPs. Reproduced from [784]. © 2015 IAEA, Vienna. All rights reserved.</figcaption></figure>

higher magnitude than observed experimentally, without an increase in heat flux. Extrapolation from the simulation results, which capture 0.5 ms of the plasma evolution, result in an estimate that the pedestal density can be reduced by 50 % in less than 100 ms.

Recent reduced modeling to study the impact of non-ambipolar neoclassical transport breaking [792, 793], showed the roles this played in determining the density pump-out and the changes in $E \times B$ [794]. The model captures the density pump-out as a result of the thin stochastic edge (from $\Psi_N \sim 0.98-1$) at the plasma edge, where radial currents can arise to compensate electron currents present within stochastic magnetic field regions [542], but as it does not include turbulence it cannot capture the changes in electron heat transport closer to the pedestal top. These results confirmed prior work comparing the impact of RMPs on the toroidal rotation and radial electric field and comparing them to non-ambipolar theoretical predictions [573, 681].

**6.5.1. Fueling through the pedestal/influence of RMPS.** The application of RMPs results in an increase in transport and a decrease in the electron density in tokamaks. This additional outflow of particles and the creation of a tangle structure interacting with the wall can alter the neutral density and thus fueling of the pedestal. To understand where the particles in this density pump-out wind up, a careful particle balance is

performed. Unterberg et al [795] showed with a particle balance analysis that depending on the shape of the plasma in the divertor region, the particles can end up in the cryo-pump or in the wall and SOL and that the pumping efficiency can change [796]. While in low triangularity H-mode discharges in DIII-D the pumping decreases, in the ISS H-mode discharges in DIII-D, pumping increases and remains elevated throughout the RMP application. The changes in pumping were attributed to the distance of the traditional strike-point to the pump plenum entrance and the complex 3D geometry introduced by strike-point striations. Additional measurements show that at the midplane $D_\alpha$ increases by a factor of 2 when compared to the RMP-off phase of the experiment, which can affect the neutral fueling rate of the core plasma [797]. All these changes in particle balance suggest a change in plasma interaction with the walls. Additional modeling using SOLPS showed that while changes in recycling and pumping can lead to a strong reduction in pedestal density, a change in transport was necessary to explain the changes in the pedestal density gradient [798].

**6.5.2. Changes in the pedestal due to islands.** As discussed in earlier sections, RMPs can lead to enhanced turbulent and neoclassical transport reducing pedestal gradients both when ELMs are suppressed or only mitigated. To explain the difference between conditions in which ELMs
are suppressed versus mitigated, based on the strong dependence on the toroidal rotation as well as the location of the resonant surfaces and the measured plasma response as discussed in section 6.2.2, observations indicate the formation of islands prevents the expansion of the pedestal to an unstable width thus suppressing ELMs [661, 799, 800]. The formation of magnetic islands in the pedestal foot can cause further density pump-out [786]. The induced changes to magnetic topology then impact turbulent transport as discussed in sections 6.3.3.2, 6.3.3 and 6.3.3.1. Nonlinear MHD modeling of KSTAR plasmas with RMPs, where the neoclassical transport code PENTRC has been coupled to the nonlinear 3D MHD code JOREK showed that a combination of kink-peeling, tearing response, and NTV (see section 6.2) can explain the pedestal degradation observed in experiments [801].

Experiments at DIII-D with low pedestal collisionalities ($\nu^* < 0.1$) and ITER-similar plasma shapes show clear variation of electron temperature and electron density at the plasma edge [784]. The coherent nature of the oscillations in this region is suggestive of a kink-like helical deformation of the plasma flux surfaces by the RMP. As described in section 6.2.2, the magnitude of the response is limited to certain $q_{95}$ resonant windows due to the density-dependence of the field penetration threshold. Those windows overlap with the ELM suppression $q_{95}$ windows, but especially for the pedestal density the window is wider than for the ELM suppression. In regions with high resistivity and and/or low electron fluid velocity, resistive MHD codes allow for penetration of the RMPs to the rational surfaces and thus formation of magnetic islands. EPED modeling [799] suggests that a magnetic island or stochastic layer caused by the RMPs may enhance transport near the top of the pedestal, which results in broader pedestal width. This model is consistent with the resonant nature of the changes in the pedestal densities and temperatures. Higher amplitude of RMPs were found at DIII-D [786] to cause the pedestal collapse in low confinement H-mode plasmas. It occurred when the applied magnetic perturbations exceeded the penetration threshold in the steep pressure gradient region of the ETB. Nonlinear MHD indicated that penetration in the steep gradient region of the pedestal can occur at a comparable RMP amplitude required for ELM suppression in model ITER equilibria.

Prior nonlinear MHD modeling showed an increase in radial convective $E \times B$ plasma transport to occur in the presence of the RMPs using nonlinear MHD modeling [802, 803]. Nonlinear MHD modeling [804] shows that the reduction in pedestal density observed in RMP experiments is consistent with the presence of $E \times B$ convection cells crossing the separatrix. Such a process can operate when the resonant field amplitudes are reduced by plasma rotation, although the amplitude of $E \times B$ flows is reduced with the increasing plasma rotation velocity at the separatrix. Above some critical rotation, the enhanced density transport is completely suppressed, even though the $E \times B$ flow is only partially suppressed. This could mean that the rotation not only leads to screening of the penetration of the external fields, but also plays a secondary role in the reduction of the transport.

### 6.5.3. Change of transport due to mitigated/suppressed ELMs.
Application of RMPs leads very often to scenarios in which ELMs become more frequent and smaller in size. One of the assumptions on such scenarios was that the enhanced particle transport due to much higher frequency of the mitigated ELMs leads to the density drop at the pedestal [805] and would be detrimental for ITER operating at such scenarios.

In fact, as shown in figure 29, the changes in the density on AUG during experiments with $n = 1$ magnetic perturbation cannot be explained by more frequent mitigated ELMs. Absolute particle losses for natural and mitigated ELMs are identical (figure 29(a)) and depend only on the ELM frequency. At ASDEX Upgrade the particle loss rate ($\Delta n_e \cdot f_{ELM}$) is constant and does not depend on plasma density (figure 29(b)). Measured particle losses suggest that the higher the frequency of ELMs the smaller the particle losses per ELM are. In fact particle losses per second remain more or less constant independent of mitigated ELMs frequency. The application of the RMPs produces an increase in $f_{ELM}$ and corresponding decrease in $\Delta n_{ELM}$ consistent with $f_{ELM} \cdot \Delta n_{ELM} = \text{const}$.

The experiments with plasmas in connected double null or lower single null plasma shapes at MAST showed very similar results [574]. The frequency of the mitigated ELMs in these experiments was strongly related to the amplitude of the maximal resonant component of the applied field. For both shapes, the application of the RMPs lead to an increase in ELM frequency (up to ca. 600 Hz) and the corresponding decrease in $\Delta W_{ELM}$ was consistent with $f_{ELM} \cdot \Delta W_{ELM} = 500$ kW. The particle losses due to mitigated ELMs were smaller than for the natural (non-mitigated) ELMs. The relative loss of pedestal density due to mitigated ELMs ($\Delta n_e/n_{ped} \approx 0.02$) was two times smaller than for the natural ELMs ($\Delta n_e/n_{ped} \approx 0.04$). Similar effects of reduction of the ELM induced particle and energy losses due to mitigation of ELMs with $n = 1$ RMPs at LHD have been also reported [807]. Time resolved electron density profiles at the pedestal show much smaller modulation due to mitigated than due to non-mitigated ELMs. Based on these findings one may conclude that any changes in the pedestal height or width come from transport mechanisms not related to particle transport due to ELM losses.

## 6.6. Summary

Over the last two decades, enormous progress has been made in understanding how the 3D fields affect transport in magnetic fusion plasmas. Particularly, for the case of tokamaks with external magnetic perturbations, where a number of new experiments (e.g. EAST, KSTAR) and improvements to existing devices (e.g. DIII-D, ASDEX upgrade or MAST) allowed for new physics discoveries. The physical picture of 3D effects on transport has evolved from an over-simplistic approach of a stochastic boundary, where electron heat and particle diffusivity are enhanced by additional radial terms [582], to a rather complex interplay between the penetration of the external fields, governed by the plasma parameters, and their effects on both neoclassical and turbulent transport. Experimental results showed that e.g. RMPs cause NTV torque damping in the
<figure><img src="image_32.png" /><figcaption>**Figure 29.** ASDEX Upgrade: (a) Particle losses per ELM plotted vs ELM frequency for a case without magnetic perturbation (black), a resonant *n* = 1 magnetic perturbation (blue) and non-resonant *n* = 1 magnetic perturbation. (b) Particle losses per second due to mitigated and non-mitigated ELMs as a function of line averaged density. Reproduced with permission from [806].</figcaption></figure>

plasma core or change the transport regime from ITG to TEM at the plasma boundary.

The effects of 3D fields require a synergistic approach, as e.g. changes in the neoclassical NTV torque due to external fields can lead to changes in the radial electric field and thus directly affect turbulent transport vortices via the $E \times B$ shear. On the other hand, turbulent fluxes are also involved in the neoclassical transport mechanism, e.g. turbulence-induced poloidal asymmetries can affect the neoclassical impurity transport. The plasma boundary is also significantly affected by the external perturbations, with the experimental and theoretical evidence for a heterogeneous structure of the magnetic field lines intersecting the plasma-facing components. This complex interplay between different physical mechanisms is still not fully understood, but our understanding of how plasma transport is affected by the 3D fields has improved enormously thanks to the efforts of many fusion scientist.

## Acknowledgments

This work was conducted under the auspices of the ITPA Topical Group on Transport and Confinement. The Authors would like to thank all of the colleagues who contributed to improving our understanding of 3D transport over the years. In addition, the authors are very grateful for all of the contributions to this work. We acknowledge the valuable contributions of the late Todd Evans, who lead for many years joint activity on 3D transport within ITPA TC.

# 7. Confinement mode transitions

J.W. Hughes, L. Schmitz, E.R. Solano and E. Wolfrum

## 7.1. Introduction

The phenomenon of ETB formation has been found to dramatically improve the energy confinement in tokamaks and to increase their performance above that of simple low-confinement modes, or L-modes. The most well known and

characterized transition is that from L-mode to H-mode, or the L-H transition, discovered decades ago on ASDEX, and which has become foundational to multiple high energy gain burning plasma concepts. Consequences to confinement from H-mode formation are discussed in section 8. The ETB in H-mode results in a pedestal forming in both edge temperature and density, such that both the core thermal content and particle inventory are superior to that in L-mode. Another confinement regime, the I-mode, was identified subsequently and exhibits primarily a thermal ETB with particle and impurity confinement that is more like that in L-mode than that in H-mode. The physics determining the transport and stability within the pedestal region in H-mode and I-mode, along with impacts on pedestal profile structure, are discussed extensively in a companion article [21].

The ITPA has created a number of activities over the years intended to improve the predictive understanding of access to confinement regimes, with a particular emphasis on issues related to ITER operation. These have included analysis of multi-device databases for power thresholds, as well as joint experiments among tokamak facilities. This section will address general progress in the understanding of physics governing these transitions, with a focus on work that was organized through the ITPA Topical Groups in TC and Pedestal and Edge Physics (PEP), as well as a former Group named Confinement Database and Modeling (CDM). Improved scalings for predicting threshold power for H-mode $P_{th}$ have been a top priority (see section 7.2), naturally, due to questions of power requirements for ITER. Next the ITPA moved toward refining these predictions to account for several distinguishing features of the ITER device, which are shared with a number of other tokamak burning plasma concepts:

* the need to minimize $P_{th}$ through optimization of the density at the L-H transition

* operation with main ion isotope mixes, especially DT

* dominant electron heating

* small values of externally applied torque

* predominantly metal plasma facing components PFCs
* compatibility with elevated edge radiation, $Z_{eff}$ from intrinsic or seeded impurities

* a specific magnetic equilibrium and divertor configuration, not universally present on existing devices

* specifications around intrinsic and applied 3D magnetic fields

* the need to achieve high performance without large ELMs

* the desire to avoid uncontrolled back-transitions to L-mode.

Impacts of these features on L-H $P_{th}$ will be discussed at the global level in section 7.3. Access to the I-mode regime, which has application to the final two bullets, is taken up in section 7.4. Finally, section 7.5 will document a number of investigations into local edge operating space separating confinement regimes and potential edge threshold conditions for transitions. Findings related to local dynamics in turbulence and flows are discussed in section 7.6, and finally efforts to model and simulate L-H transitions are shown in section 7.7. Throughout section 7 discussion will focus on confinement transitions on tokamaks. Section 6 contains additional discussion of L-H transition phenomenology on stellarators, in addition to an extensive discussion of the impact of magnetic perturbations on the L-H transition in tokamaks.

## 7.2. Global H-mode power threshold scaling

Projecting access to H-mode on future tokamaks has historically been attempted using scalings developed from empirical databases. Typically the L-H transition in existing devices is characterized by a power threshold requirement $P_{th}$, which is treated as a dependent variable in a regression analysis. At the time of the Update to the IPB (UIPB), several established power laws for $P_{th}$ had been developed, and ITER auxiliary power requirements were largely set by these [3].

Scalings for L-H power threshold in common use in 2007 were largely supplanted following publication of the present ITPA scaling, expressed in equation (2) of [1] and given here without uncertainty estimates:

$$P_{th} = 0.0488 n_{e20}^{0.717} B_{T}^{0.803} S^{0.941} \quad (20)$$

where $P_{th}$ is in units of MW, $n_{e20}$ is the line averaged electron density in units of $10^{20} m^{-3}$, toroidal magnetic field $B_T$ is in units of T, and S is the plasma surface area in m$^2$. The scaling was determined through regression on 1024 individual $P_{th}$ measurements from Alcator C-Mod, ASDEX Upgrade, DIII-D, JET, JFT-2M, and JT-60U. These data represented a subset of the larger ITPA threshold database, determined in order to select discharges with characteristics most applicable to ITER operation. Notably, the included discharges are all single null with $B \times \nabla B$ directed toward the active X-point. The data are also chosen to have density values which are greater than the empirically determined density that minimizes $P_{th}$, or $n_{th,min}$, which is specific to each device. Plasmas with radiated power fractions above 50% are excluded as well. These criteria are among the constraints on the data which eliminate abnormally high threshold values. As is customary the $P_{th}$ in [1] is given by the loss power $P_L$, which is determined by subtracting from

<figure><img src="image_33.png" /><figcaption>**Figure 30.** Empirical loss power at the L-H transition in various devices versus the threshold power given by equation (20). Data from spherical tokamaks (MAST, NSTX) and helical devices (e.g. CHS) were not used to generate the scaling. Reproduced from [1]. © IOP Publishing Ltd All rights reserved.</figcaption></figure>

the total heating power the time variation of stored energy and any known fast ion losses. Core radiated power losses were not subtracted.

Figure 30 shows the level of agreement between the power law in equation (20) and the empirical data from the conventional aspect ratio devices used to derive it. The data organize well over a significant range of device size, field and density, albeit with significant scatter. While this scaling appears to require modification in a number of experimental circumstances, as described in section 7.3, the expression is routinely used, with caveats, to project H-mode access in burning plasma devices including ITER [1, 808], DEMO [809] and SPARC [810]. Spherical tokamaks (MAST, NSTX) and a helical device (CHS), none of which were incorporated in the scaling derivation, are shown to depart significantly from the scaling predictions. In addition to CHS, several inherently 3D toroidal devices have achieved H-mode ETBs, demonstrating similar phenomenology to tokamaks [775, 783], however, their power thresholds are not considered here.

## 7.3. Common influences on H-mode threshold not captured by scalings

As noted above, multi-machine scalings do not capture dependencies on a number of features seen to affect power thresholds on individual devices, and which have potential importance for burning plasma devices. Several of these features are considered here.

### 7.3.1. The density dependence of H-mode power threshold.
As noted above, the ITPA database studied in [1] demonstrated a minimum in $P_{th}$ as a function of density on several devices. The $n_{th,min}$, was estimated on each machine in the ITPA database, and only L-H data having $n > n_{th,min}$ were used to generate the scaling in equation (20). Inspection of $n_{th,min}$ data in [1] suggested this critical density may have a positive dependence on $B_T$ and a negative dependence on machine size R. Subsequently, detailed examination of the
density dependence on multiple machines showed consistent minima in $P_{th}$ and the prominence of a 'low-density branch', in which the threshold power decreases with increasing density [760, 811–813]. These results confirmed this leading order departure from the ITPA scaling. Specifically, C-Mod [812] and JET operated with its ITER-like wall [813] individually identified a positive, nearly linear, scaling of $n_{th,min}$ with $B_T$. A tendency of $n_{th,min}$ to increase with plasma current $I_p$ has been observed on JET-ILW [814] and AUG [815].

Much of the leading order behavior of $n_{th,min}$ can be captured by a model expression developed by Ryter et al [815] (and later generalized by Bilato et al [816]), inspired by evidence on AUG that a critical surface-integrated edge ion heat flux $Q_{i,edge}$, rather than a total power, must cross the separatrix in order to facilitate the L-H transition. Based on the assumption that sufficient electron-ion coupling is needed to achieve adequate power in the ion channel, an expression is derived from a simultaneous solution of $P_{th}$ in equation (20), the L-mode energy confinement time $\tau_{E,L}$ [817], and an empirically determined critical value of $\tau_{E,L}/\tau_{ei}$, where $\tau_{ei}$ is the volume-averaged electron-ion energy exchange time. The resulting scaling,

$$n_{th,min} \approx 0.7 I_p^{0.34} B_T^{0.62} a^{-0.95} (R/a)^{(0.4)} \quad (21)$$

was found to agree relatively well with both AUG data and those from other devices reported in the literature [1, 812, 813]. Figure 31 illustrates the level of agreement with existing data and some indication of its potential predictive ability, in this case for ITER. Equation (21) has also been used to project the optimum density for accessing L-H transitions on SPARC [810].

Other explanations have been proposed for the transition between low and high density branches. For example, numerical modeling provides indications that a critical ion pressure gradient at the plasma edge is needed to gain H-mode access, and that this is related to generation of ZFs and mean flow shear. Collisionality-dependent damping of ZFs may play a role in determining the shape of the $P_{th}$ versus $n_{th,min}$ curve. Power threshold modeling which accounts for these processes finds that the calculated $P_{th}$ attains a minimum at low density and maximum ion heating, but increases at low density and increased electron heating [818] (see discussion in section 7.7). Other modeling work has identified changes in dominant turbulent modes in the edge in a collisionality scan, with a transition from ITG and TEMs toward resistive ballooning modes (RBMs) as density passes from the low to high density branch [819]. In the simulation this results in a changing critical edge temperature for turbulence stabilization and a local transition threshold. Yet another model, partially tested on JET, conjectures that L-H transitions occur when perpendicular transport time is comparable to the parallel Alfvénic time, and the transition between branches of $P_{th}$ is explained by asserting the scrape-off layer transitions between sheath-limited and conduction-limited [820]. Models such as these offer tests that could be performed on multiple devices with the aim of deriving additional physics-based predictions of $n_{th,min}$.

<figure><img src="image_34.png" /><figcaption>Figure 31. Empirical density at minimum $P_{th}$ (here denoted $n_{e,min}$ but called $n_{th,min}$ in our text) plotted versus the scaling expression in equation (21). Reproduced courtesy of IAEA. Figure from [815]. © 2014 EURATOM.</figcaption></figure>

### 7.3.2. Modification for main ion species.
The sensitivity of the power threshold to main ion species is an important consideration for burning plasma devices. Most experience determining $P_{th}$ today is based on DD plasmas, and multi-machine scalings all have this constraint built in (e.g. [1]). Naturally, burning plasma demonstration devices and pilot plants will instead target nearly balanced mixtures of D and T in order to optimize fusion gain. In addition, new burning plasma facilities can require initial operation in H or He plasmas, in order to gain experience without generating excessive levels of neutron fluence. This has been a particularly prominent feature of ITER, which prompted an ITPA Joint Experiment, TC-4, as well as experiments on individual facilities, to explore the main ion species effects on several tokamaks [808]. Examples of $P_{th}$ observed in density scans with varied main ion species are shown in figure 32 for AUG and JET operating with metallic walls. Capturing the trends with density is found to be necessary due to the potential variability of $n_{th,min}$ with the choice of main ion.

First we consider the hydrogenic isotope scaling of $P_{th}$. When applying $P_{th}$ scalings to DT plasma scenarios that employ significant amounts of tritium, it is customary to apply a $1/A_{eff}$ isotope scaling, where $A_{eff}$ is the average atomic mass of the hydrogenic species. This implies a reduction in $P_{th}$ of $2.0/2.5 = 0.8$ when projecting DD $P_{th}$ results onto a 50:50 DT scenario. This approach has been in common use dating back to 1990's threshold experiments in H, D, T and DT plasmas on carbon-wall JET [823], and has found support through additional isotope studies on a variety of devices comparing H and D plasmas [765, 811, 824–827].

Ongoing comparisons on DIII-D, exploiting a carbon wall, continue to show substantially larger $P_{th}$ in H plasmas than in D plasmas [760, 824, 825]. There are, however, indications that this gap in $P_{th}$ closes at higher density values [827]. This density-dependence variation complicates the goal of scaling
<figure><img src="image_35.png" /><figcaption>**Figure 32.** AUG (top): power threshold versus density, normalized by a $B_T^{0.8}$ dependence, for D, H and He with the W wall. $P_{scal}$ is taken from equation (20). Reproduced courtesy of IAEA. Figure from [765]. Copyright (2013) IAEA. JET: loss power at the L-H transition as a function of line density (lower horizontal axis) and Greenwald fraction (upper horizontal axis) for H, D and He plasmas at 1.8 T (middle) and D, T and D-T plasmas at 3 T (bottom) with horizontal target strike points. Colored vertical lines mark $n_{th,min}$ for each species. Dashed blue curves represent the ITPA threshold scaling equation (20), and the solid blue curve (TC26) is the JET D threshold scaling for this configuration. Middle: green triangle symbols correspond to pure RF L-H transitions; ones with a black boundary have added NBI blips. Reproduced from [821]. © EURATOM 2021. CC BY 4.0. Bottom: all are NBI heated plasmas; DT points are gold stars, D blue squares, T magenta diamonds. Next to each DT datapoint is the tritium concentration before the transition. Reproduced from [822]. © EURATOM 2023. CC BY 4.0.</figcaption></figure>

$P_{th}$ with isotope mass, and is a widespread observation, as will be seen below.

A significant increase in $P_{th}$ when going from D to H plasmas was confirmed on AUG with a tungsten wall, as illustrated in figure 32, top. AUG studies indicate that higher edge heat transport in hydrogen plasmas results in the higher threshold

power [828–831], and that this is also reflected in a higher edge ion heat flux observed at the L-H transition in H plasmas, when compared with D plasmas [828, 831, 832]. $P_{th}$ data from He plasmas are shown in the figure as well, and will be discussed later.

JET has made comparisons of $P_{th}$ in H and D plasmas operated with W/Be PFCs [821, 833]. Figure 32, middle, shows trends for JET-ILW $P_{th}$ in a specific configuration (horizontal strike point target, 1.8 T, 1.7 MA) comparing H, D and He plasmas [821]. A strong dependency of $n_{th,min}$ on plasma species is seen. For densities below $n_{th,min}(H)$, $P_{th}(H)$ is significantly higher than $P_{th}(D)$, and $P_{th}(He)$ is much larger than $P_{th}(D)$ and $P_{th}(H)$ at medium densities (see the more detailed discussion on helium below). The difference in $P_{th}$ between H and D plasmas in JET-ILW is much larger for NBI-heated plasmas, reaching up to a factor of 4. Investigations are ongoing to determine whether this is due to an effect of NBI torque on $P_{th}$ (as observed on DIII-D [825]), or to the partition between ion and electron energy fluxes at the L-H transition (as observed for example on AUG [834]), or to radiation from Cu influx in NBI-heated plasmas.

Recent JET-ILW studies of $P_{th}$ during the Tritium and DT campaign [822] illustrate the isotope dependencies of $P_{th}$ in a higher field configuration (horizontal strike point target, 3.0 T, 2.5 MA), for which it is relatively easy to document the change in $n_{th,min}$ and $P_{th,min}$ with isotope. Shown in figure 32, bottom, is $P_{th}$ as a function of density for D, DT and T plasmas. The value of $n_{th,min}$ is lowest for T, followed by DT then D plasmas. A strong increase of $P_{aux,min}$ with $A_{eff}$ is observed, suggesting it may be easier to enter H-mode in T-rich plasmas in future DT fusion devices and experiments. In the high density branch thresholds increase monotonically with decreasing isotope mass, with $P_{th}(T) < P_{th}(DT) < P_{th}(D)$, yet a quantitative study of this dependency based on the new data has not been undertaken yet.

Altogether the accumulated results from various machines reveal no simple dependence on isotope mass, in scans of hydrogenic composition. The $1/A_{eff}$ scaling for $P_{th}$ should be considered a rule of thumb with significant uncertainty, with actual scalings departing quantitatively on each machine. TCV for example reports a H:D $P_{th}$ ratio of 7:4 rather than 2:1 [826], and AUG has shown similar departures [828, 831]. Further, the density dependencies noted above complicate the ability to scale $P_{th}$ with $A_{eff}$, as does the variability with divertor geometry and input torque (discussed more below).

Initial mixed isotope campaigns on JET began to deliver data on $P_{th}$ in tritium-containing plasmas [836–838]. In those experiments Ohmic T H-modes at low density agreed reasonably well with a $P_{th} \sim 1/A_{eff}$ scaling, but ICRH-heated plasmas at higher density did not. This was, in part, explained by a hardware problem in the RF system [837]. New measurements of $P_{th}$ in plasmas of pure and mixed species, taken with good RF conditions, are reported in [838]. A comparison of various isotope mixtures in this study, including DT and HT, with pure isotope plasmas revealed that $P_{th}$ can vary up to 65% for the same effective isotope mass $A_{eff}$ at otherwise constant
parameters [838]. It is found that different L-mode transport levels are present with different isotope compositions having the same $A_{\text{eff}}$ (e.g. DD and HT), necessitating different power levels to reach the critical edge temperatures as needed for H-mode access.

Other isotope dependence studies have been performed in mixed hydrogen-deuterium plasmas. Transitions in 50:50 H-D mixtures in JET-ILW resulted in $P_{\text{th}}$ and $n_{\text{th,min}}$ values intermediate between the pure hydrogen and deuterium cases [833]. It was further concluded that, at least in the high density branch, $P_{\text{th}}$ variation in HD mixtures was non-linear, with significant threshold variation at less than 20% or more than 80% H concentration, with little variation in between [839]. A different variability was observed in AUG HD mixtures: $P_{\text{th}}$ and the edge ion heat flux are constant at the level of pure deuterium for $n_{\text{H}}/(n_{\text{H}} + n_{\text{D}})$ between 0 and 0.5 and monotonically increase to the level of pure hydrogen for $n_{\text{H}}/(n_{\text{H}} + n_{\text{D}})$ between 0.6 and 1 [831].

Lastly we consider helium plasmas, which are potential targets for early operation of burning plasma devices, as they may be useful in creating operational experience prior to introducing high neutron fluxes. Introducing He into tokamaks has been seen to influence $P_{\text{th}}$ in a variety of ways. Often it is seen that He plasmas have a significantly larger $P_{\text{th}}$ than in D, and prior to about 2008, indications from experiments were that $P_{\text{th}}$ in He plasmas was simply about 40% higher than that in D [811, 840]. Subsequently, as part of the ITPA TC-4 Joint Experiment, the ratio of the thresholds $P_{\text{th,He}}/P_{\text{th,D}}$ was found to range between 1.0 and 1.8 on C-Mod, DIII-D, JET, MAST and NSTX, and was shown to depend on the choice of density on a number of these devices [808]. Similar findings were later reported on EAST [841]. Notably, the discrepancy between $P_{\text{th}}$ in He and D is often seen to decrease at higher density values [808, 841, 842]. An illustration of this can be seen in figure 32, middle. On JET, for densities below the $n_{\text{th,min}}$ for He, $P_{\text{th}}$ is significantly higher for He than for D, but at higher density there is little difference [821]. Although less pronounced, a similar effect was observed in DIII-D (see [760], figure 1). The observation that in JET-ILW $n_{\text{th,min}}(\text{He})$ is larger than $n_{\text{th,min}}(\text{H})$ results in similar minimum $P_{\text{loss}}$ for both species (partly due to enhanced W sputtering and higher radiation in He). Surprisingly, although AUG found $P_{\text{th}}$ to be higher in He than in D with its carbon wall, following the transition to tungsten PFCs, the He power threshold was similar to the one in deuterium plasmas, even exhibiting the same non-monotonic density dependence [765, 811]. This is shown in figure 32, top.

Given the higher $P_{\text{th}}$ commonly associated with H plasmas, and the potential desire to start up next step devices in hydrogen, experimental attempts have been made to lower the threshold with He doping. The results range from significant to negligible. In JET-ILW, helium-4 fueling into H plasmas was performed, resulting in a ~25% reduction of the threshold with up to about 10% He concentration [839]. DIII-D shows a similar level of reduction in $P_{\text{th}}$ but requiring larger concentrations of He (~30%) [843]. On AUG $P_{\text{th}}$ stays constant on the level of pure hydrogen in hydrogen plasmas with helium doping up to $c_{\text{He}} = 20\%$ [834, 844].

<figure><img src="image_36.png" /><figcaption>**Figure 33.** $P_{\text{th}}$ as a function of injected torque on DIII-D for various target densities and heating methods for hydrogen, deuterium and helium. The threshold power in this case is determined by subtracting the core radiated power from the loss power. L-mode densities are indicated in parentheses. Open symbols denote discharges that failed to transition to H-mode at the applied power. Reproduced courtesy of IAEA. Figure from [825]. Copyright (2010) IAEA.</figcaption></figure>

### 7.3.3. Influence of torque and wall material.
When considering extrapolation of $P_{\text{th}}$ scalings to burning plasma devices, it is important to consider additional qualities which distinguish these future machines and which are not universally represented on present devices. In particular, data used to generate scalings such as equation (20) originate from multiple devices, many of which have features not expected for ITER, DEMO, SPARC, and fusion pilot plants. Examples include (1) significant torque driven by NBI and (2) plasma facing components made predominantly of carbon.

In order to determine the influence of externally applied NBI torque on $P_{\text{th}}$, the balance of co-current and counter-current beams was adjusted on power threshold experiments on DIII-D [825, 845]. Examples of $P_{\text{th}}$ response to input torque are illustrated in figure 33 for several different main ion species plasmas. H-mode access was facilitated in all cases studied (favorable vs. unfavorable $B \times \nabla B$ direction, different main ion species) by having externally applied torque be more counter-current. At zero input torque, power threshold is modest and the L-H transition may be facilitated by an increased poloidal flow shear. These results suggest $P_{\text{th}}$ should not be worse in a low-torque burning plasma device.

Presently used scalings for $P_{\text{th}}$ such as equation (20) were developed using data from mostly carbon wall machines, creating questions about their applicability to metal wall machines. Subsequently, two devices reported changes in $P_{\text{th}}$ following changes from carbon PFCs to metal PFCs. AUG showed an approximately 25% reduction in $P_{\text{th}}$ following changeover from carbon to tungsten [765]. This reduction was seen for both D and H independently, while the reduction for He discharges was much more significant. The AUG data in figure 32 were collected with W PFCs, and for D
<figure><img src="image_37.png" /><figcaption>**Figure 34.** Threshold power in Alcator C-Mod as a function of line density for strike point on the vertical plate divertor (blue) and with strike point on the horizontal target in the slot (red). Reproduced courtesy of IAEA. Figure from [846]. Copyright (2013) IAEA. Loss power at L-H transition in JET-ILW plasmas with vertical target and corner strike point configurations (red squares) and with horizontal target configurations (black triangles), compared to the ITPA 2008 scaling, equation (20). Underlying gray triangles represent data from the ITPA database used to generate the scaling. Reproduced courtesy of IAEA. Figure from [836]. © EURATOM 2022.</figcaption></figure>

plasmas with $n > n_{th,min}$ the $P_{th}$ values are notably smaller than the ITPA scaling would predict. JET also compared its H-mode access with its ITER-like wall (W, Be) to prior results with C PFCs [813]. At plasma densities above $n_{th,min}$, $P_{th}$ in D plasmas was reduced by ~30%, and by ~40% when the radiation from the bulk plasma is subtracted, when going from JET-C to JET-ILW. One can see the prevalence of JET-ILW deuterium $P_{th}$ values below the ITPA scaling curve in figure 32. Similarly, $P_{th}$ data from Alcator C-Mod, which operated with molybdenum PFCs for its entire existence, showed optimal $P_{th}$ significantly below the values given by equation (20), as seen in figure 34. Possible explanations for the reduction in $P_{th}$ with C replacement by W are described in section 7.5.

The significant differences between carbon and metal walls with respect to the H-mode power threshold have suggested the formulation of multi-machine scalings that include only devices with low carbon content. An ITPA activity (TC-26) has been initiated to this end, examining data sets from AUG, C-Mod and JET. The complete results of the database analysis are to be published, although the subset of data from JET-ILW has been used to produce an alternate scaling [836]. Beginning from the form of the expression in equation (20), a linear relationship between $P_{th}$ and plasma surface area $S$ was assumed and dependencies on field and density fitted to data for which $n > n_{th,min}$. In aggregate a stronger density dependence is found than in the 2008 ITPA scaling, along with a striking separation in terms of divertor geometry. When JET-ILW is operated with a horizontal divertor target, $P_{th}$ is systematically below the scaling prediction, as for the D points in figure 32. However when the outer strike point is placed on or near the vertical target plate, elevated $P_{th}$ is observed. Direct comparisons of the JET-ILW threshold to the 2008 ITPA scaling database are shown in figure 34. Divertor effects such as this turn out to be not uncommon on tokamaks, as further discussed in section 7.3.4.

### 7.3.4. Divertor configuration and magnetic topology.
As tokamaks move toward demonstrating net fusion power, the emphasis on divertor power handling becomes crucial, and reactor concepts must develop improved divertor designs, sometimes incorporating advanced divertor concepts. In addition future concepts move increasingly farther from single null equilibria and toward double null configurations to facilitate exhaust management. As noted above, the specifics of the divertor geometry, including the placement of strike points and X-points, can have a sizable impact on the power threshold for H-mode on a given device. This was recognized at the time of the Update to the IPB [3], based in part on $P_{th}$ comparisons in JET with and without a divertor septum and with scans of X-point height [847]. There a decrease in $P_{th}$ was observed as the X-point moved closer to the divertor floor, while strike points were on mostly horizontal targets. Similar trends have also been reported on DIII-D with the outer strike point on a horizontal target, although equilibrium shaping was also varied [760]. MAST also performed experiments demonstrating a substantial reduction of $P_{th}$ as the X-point was lowered [848]. Subsequently, L-H transitions on TCV under Ohmic heating showed a clear trend of reduced $P_{th}$ with X-point closer to the vessel floor [849].

While the above experiments indicated that longer outer divertor leg length correlated with higher H-mode threshold power, an inverse trend was reported on Alcator C-Mod [850]. However, the substantial decreases in $P_{th}$ on C-Mod corresponded also to the movement of the outer strike point from a vertical to a horizontal target, and also to a configuration with greater neutral baffling [846] (see figure 34). These results in addition to the JET-ILW results with vertical vs. horizontal targets [836], may indicate a greater importance for divertor
shaping in setting $P_{\text{th}}$. Despite the notable influence of diver-
tor geometry and strike point placement on many devices in
at least one configuration on TCV, little effect was seen on $P_{\text{th}}$
from variation in both the major radius of and the poloidal flux
expansion at the strike point [851].

Systematic studies of the effects of divertor configura-
tion and placement of X-points and strike points have been
attempted under Joint Experiments originating in the ITPA
PEP Topical Group. The first was PEP-28, ‘Physics of H-
mode access with different X-point height’, inspired mainly
by the JET, DIII-D and MAST results obtained prior to 2011.
Increasingly, as findings emerged on these and other devices
suggesting complex dependencies on divertor geometry and
plasma shaping, PEP-28 transitioned to PEP-39, ‘Impact of
the SOL and divertor conditions on the L-H threshold’. Work
continues in this area on multiple devices, with a focus on
single null discharges with the $\nabla B$ drift in the favorable direc-
tion for H-mode access, i.e. with $B \times \nabla B$ directed toward the
active X-point.

Tokamaks commonly observe that power threshold in SN
discharges is significantly elevated when the direction of $B \times
\nabla B$ is reversed to point away from the active X-point. Typical
experience suggests a factor of ~2 increase [852–854]. An
interesting exception was noted in JET, in which $P_{\text{th}}$ for access-
ing H-mode was seen to agree relatively closely across a $B_{\text{T}}$
direction switch [855]. Notably, operating JET in the unfavor-
able direction did, in some cases, increase the power require-
ments to reach an H-mode with an improved density ped-
estal and large ELMs. Some other devices have associated
similar phenomena with I-mode (see section 7.4), a confine-
ment regime that is distinct from L-mode and H-mode, and
which tends to be observed at intermediate power levels when
operating with unfavorably directed $B \times \nabla B$. In scans of the
magnetic balance on various devices, increases in $P_{\text{th}}$ can be
observed in SN plasmas that are near the DN point, and trans-
itioning from the favorable to unfavorable case [852, 856,
857]. However, there is also evidence of improved H-mode
access in nearly exact DN configurations, as first reported on
MAST [858] then in a joint study among MAST, NSTX-U and
AUG [859], and in experiments on EAST [860]. Power reduc-
tions in connected DN configurations on the low aspect ratio
devices were found to be a factor ~2× while on AUG the $P_{\text{th}}$
decreased by only ≈30% in DN.

### 7.3.5. 2D equilibrium considerations.
Details of equilib-
rium shaping are not captured by typical $P_{\text{th}}$ scalings, other
than indirectly through the surface area dependence (see
equation (20)). Direct tests of the influence of plasma trian-
gularity $\delta$ on $P_{\text{th}}$ have been attempted, although design of
such is complicated by the correlation of plasma equilibrium
shape changes with X-point movement and strike point place-
ment within divertors, aspects which as seen above have a
large impact on $P_{\text{th}}$. For example, indications from JET that
H-mode access was facilitated at increased $\delta$ could not read-
ily be decoupled from transitions from vertical to horizontal
divertor targets [861]. Later experiments with the JET-ILW

suggested that, for a given $\delta$ associated with the lower cham-
ber divertor, increasing the upper $\delta$ away from the divertor
actually raised the power flux needed to trigger the L-H trans-
ition [813]. Increasing power requirements with higher trian-
gularity was also shown in a scan of $\delta$ from 0.2 to 0.8 on DIII-
D [853].

Whereas most tokamak experience consists of plasmas
with non-negative values of triangularity, a growing body of
research into negative triangularity (NT) has come about in
recent years [862]. NT plasmas have demonstrated improved
energy confinement over their positive triangularity analogs
without the need to form an H-mode pedestal, as demonstrated
on TCV [863–865] and DIII-D [866, 867]. Indeed the L-H
transition can be suppressed up to much higher levels of loss
power than in positive triangularity [867, 868]. Single null
discharges with negative, but nearly zero, average triangular-
ity on AUG exhibit elevated $P_{\text{th}}$ relative to positive triangu-
larity cases, when $B \times \nabla B$ is in the unfavorable direction, but
show $P_{\text{th}}$ similar to PT cases when the drift direction is favor-
able [869]. The mechanism behind the suppression of H-mode
access for sufficiently negative values of $\delta$ is actively being
investigated. One promising explanation for the suppression is
the tendency of NT plasmas to be limited to low edge gradients
by high-n ballooning instabilities, which prevents the forma-
tion of significant $E \times B$ shear and pedestal turbulence sup-
pression [870]. This model has enabled prediction of H-mode
suppression for a wide range of possible NT reactor paramet-
ers [871]. Additional discussion of NT is found in a companion
article [21].

### 7.3.6. Influence of non-axisymmetric fields.
Departures from axisymmetry exist in real toroidal devices, and these are known to have effects on H-mode access. One case of special interest for tokamaks is the influence of toroidal field ripple. The finite number of toroidal field coils in a tokamak results in a ripple in $B_{\text{T}}$, which is known to influence the degree of ion orbit loss and which could impact flows and radial electric field in the edge of a plasma as it approaches an L-H transition. In order to assess the impact of ripple on H-mode physics, JET utilized a feature of its TF configuration to scan its ripple field from 0.08% to 1.10%, a range intended to span the anticipated ripple field in ITER at its nominal operating field [861]. Although significant changes in toroidal velocity at the edge were observed in the scan, there was no impact on $P_{\text{th}}$. Subsequently, DIII-D performed experiments with a set of in-vessel coils intended to simulate the level of TF ripple commensurate with the presence of ferromagnetic test blanket module inserts on ITER [760]. A local ripple value of 3% was achieved and $P_{\text{th}}$ for H-mode was evaluated for both NBI and ECH heated discharges, at low values of input torque, consistent with ITER and other burning plasma targets. The presence of the applied ripple field had no appreciable effect on access to H-mode.

In addition, a wide variety of 3D magnetic perturba-
tions may be applied in tokamak plasmas with the intent to
affect features such as core rotation, particle inventory and
the prevalence of ELMs in the H-mode pedestal. The influ-
ence of applied 3D perturbations has been studied on several
devices, including DIII-D [760, 761, 770], AUG [764–766],
MAST [762, 763], NSTX [767] and KSTAR [768]. Tokamak
experiments generally observe increases in P<sub>th</sub> with increas-
ing strength of RMPs, with indications that non-resonant
MPs have less impact on power thresholds [760–763, 766].
Experiments on AUG determined that while the increase on
P<sub>th</sub> was maximized for the perturbation alignment planned
for ELM suppression on ITER, the radial field perturbation
level needed to realize this elevated threshold power actu-
ally exceeds the level required for ELM suppression [766].
If the AUG results extrapolate to ITER, access to H-modes
with RMP becomes promising. An overview of the effects of
RMPs is found in section 6, where discussion is included of
the effects of these perturbations on L-mode plasma condi-
tions before the L-H transition, and on how RMPs could alter
the edge turbulence and flow characteristics in the vicinity of
an L-H transition.

L-H transitions are also observed in toroidal devices with
significant intrinsic 3D fields. Details of observations in stel-
larators are discussed in section 6.

7.4. I-mode

When tokamaks operate with the unfavorable ∇B drift, increasing their power requirements for H-mode, the I-mode confinement regime is commonly observed. I-mode is an ELM-suppressed confinement regime with energy confinement significantly above that in L-mode, yet having L-mode-like particle and impurity confinement properties. I-mode phenomena were observed in the 1990s both in Alcator C-Mod [872] and in ASDEX Upgrade [873], but was recognized as a distinct confinement regime only in the following decade [549, 874]. In addition to C-Mod and AUG, I-mode has been reported on DIII-D [875], EAST [876] and HL-2A [877]. KSTAR has also identified a candidate I-mode regime [878]. I-mode access seems to be favored in conditions in which H-mode access is hindered, i.e. when P<sub>th</sub> is elevated.

The L-I transition is typically obtained in single null dis-
charges with B × ∇B directed away from the primary X-
point, the configuration in which H-mode P<sub>th</sub> is elevated, as
explained above. The L-I transition is characterized by an
increase in energy confinement time, the development of a
temperature pedestal, and the absence of a significant density
pedestal or prompt change in particle inventory. The interested
reader may find a significant discussion of the fully developed
I-mode pedestal, in a companion article [21] which discusses
pedestal structure and stability, fluctuation characteristics, and
transport properties in the regime.

Experience with accessing I-mode is considerably less than
that for H-mode, and the development of global I-mode power
thresholds is less advanced. Nonetheless, some trends have
emerged from P<sub>th</sub> studies on single machines. The L-I trans-
ition power generally increases with plasma density [879,
880], although there are indications of a reversal of this trend
at lower density values, reminiscent of the L-H low density

<figure><img src="image_38.png" /><figcaption>**Figure 35.** Normalized power $P_{\text{loss}}/\bar{n}_e S$, vs. $B_T$. First plot shows a multi-machine database from AUG, C-Mod and DIII-D. Solid triangles are L-I transitions, while open circles are I-modes. Constant and linear magnetic field scalings are shown for reference. Reproduced courtesy of IAEA. Figure from [881]. © 2016 EURATOM. Second plot shows I-mode data from EAST, with some example power law curves for $B_T$. All data are with unfavorable (to H-mode) $\nabla B$ drift. Reproduced courtesy of IAEA. Figure from [876]. Copyright (2020) IAEA.</figcaption></figure>

branch [835, 881]. There are some indications that I-mode
access may also be facilitated at decreased plasma current, or
higher edge *q* [874, 879], although this could merely reflect
a correlation with density. At a given density, I-H transitions
do not necessarily occur at larger power levels than the L-I
transition [881]. I-modes have to date been verified on com-
pact and medium sized tokamaks only, and so the machine
size scaling is fairly uncertain. If one assumes a linear scaling
with *S*, similar to most H-mode *P*<sub>th</sub> expressions, and a linear
scaling with line density, the experience on multiple devices
suggests a relatively weak dependence of the I-mode *P*<sub>th</sub> with
*B*<sub>T</sub> [876, 881]. Figure 35 shows normalized loss power for L-I
and I-mode data, taken from AUG, C-Mod, DIII-D and EAST.
Taken together, the results suggest a much weaker *B*<sub>T</sub>-scaling
of the L-I threshold than for the H-mode power threshold
in discharges with *B* × ∇*B* directed away from the X-point.
Subsequent extensions of the C-Mod data set to *B*<sub>T</sub> = 8T rein-
forced this trend [882]. Attempts to estimate a field depend-
ence for I-mode power threshold, i.e. *P*<sub>th,L-I</sub> ∝ *B*<sub>T</sub>^α, have pro-
duced estimates of α well below 0.5 [875, 876, 880, 882],
a much weaker dependence than H-mode access scalings,
e.g. Equations (20), (22). Enhanced access at high field has positive implications for I-mode access in burning plasma regimes.

A small number of I-mode discharges have been obtained with $B \times \nabla B$ directed toward the primary X-point. A specific divertor configuration on C-Mod was discovered that had a slightly elevated H-mode power threshold, and in which an I-mode window was found [874]. Nonetheless the vast majority of I-mode results to date utilize the oppositely directed $\nabla B$ drift. Access to I-mode does not appear to be limited to devices with particular wall materials (e.g. C vs. W) or to a particular form of heating (e.g. NBI vs. ICH vs. ECH vs. lower hybrid). Recently, I-mode has been successfully extended to He plasmas [883, 884] and to H plasmas. The threshold power for I-mode in H plasmas on AUG is higher than for D plasmas, though not necessarily a full factor of 2 [884].

## 7.5. Local (edge) threshold conditions for confinement mode transitions

Following the first experimental indications that H-mode was associated with the sudden formation of an ETB, threshold conditions associated with edge parameters were sought in experimental data. Critical values of edge temperature have been reported, and linked to both the L-H and H-L transitions in a given machine. Scalings of critical edge $T_e$ at the L-H transition have also been reported. The importance of $E \times B$ shear in producing the stabilization of turbulence needed to induce an L-H bifurcation has been emphasized in a number of experiments.

### 7.5.1. Plasma edge operational space and critical parameters.
In addition to conducting database studies of engineering parameters to produce global scalings, it is instructive to examine data sets of local edge parameters. Mapping out the operational space of the edge in a given device, and potentially on combinations of devices, can inform the boundaries between confinement regimes in local terms and facilitate comparison with transition models.

TheUIPB reported on efforts to compare C-Mod and DIII-D edge data to theoretical predictions for critical parameters at the L-H transition, based upon the finite beta stabilization of drift-RBMs [3]. Extended work on C-Mod examined the operational space in terms of ballooning-normalized pressure gradient $\alpha_{MHD}$ and a diamagnetic term $\alpha_{dia}$ and found a separation between L-mode and H-mode points in qualitative agreement with predictions from theory, for both Ohmic [885] and RF-heated plasmas [812]. On AUG a clear separation between L-mode and H-mode can be shown directly in the operational space of separatrix electron temperature and density. In this picture, the dominant turbulence is more drift wave like at low densities and more ballooning like at high densities. The transition into H-mode occurs in both cases if the energy transfer from the turbulence to the mean flow exceeds the turbulence drive [886]. Figure 36 illustrates the different operational limits of the AUG L-mode and H-mode domains

<figure><img src="image_39.png" /><figcaption>**Figure 36.** L-mode and H-mode separatrix operational space for AUG plasmas with $B_t = 2.5$ T and $I_p = 0.83$ MA, in terms of electron density and temperature. Different operation limits/regimes are indicated by different symbols. The red line shows the boundary of the L-mode density limit. The blue line indicates the L-H transition condition based on the transfer of turbulence energy to mean flows exceeding the turbulence drive; the black line indicates the MHD resistive ballooning boundary. Error bars are plotted only for a subset of the data for clarity in the figure. Reproduced from [886]. © EURATOM 2021. CC BY 3.0.</figcaption></figure>

for $B_t = 2.5$ T, $I_p = 0.83$ MA, plotted in terms of the separatrix (LCFS) density and electron temperature. Here, in addition to the MHD resistive ballooning limit and the L-mode density limit, the experimentally observed L-mode and H-mode operational domains are delineated well by the calculated criterion for H-mode access, when the energy transfer from the turbulence spectrum exceeds the turbulence drive.

There are numerous experimental indications that gradients in edge kinetic profiles influence access to H-mode. Several of these in turn are consistent with the hypothesis that a critical $E \times B$ shear is necessary to trigger L-H transitions, as discussed in section 7.5.2. This is due to the leading dependence of radial electric field $E_r$ on the diamagnetic term proportional to the ion pressure gradient. On AUG, at density values high enough to ensure electrons and ions are coupled, the H-L back-transition occurs at the same electron pressure gradients as the L-H transition [887, 888]. The threshold power for the H-L back-transition exhibits the same increase at low densities as the L-H transition, as described in section 7.3.1. At modest to high densities on Alcator C-Mod, a critical edge $T_e$ is also observed at the L-H transition [812, 854].

Observations in JET-ILW indicate that, at least in the high density branch, the favorable isotope effect on $P_{th}$ is a consequence of the favorable isotope effect on transport at the L-mode edge. At a given line density, $n_e$ and $T_e$ profiles at the transition for H, D, H and T mixtures, T, and DT plasmas match, despite very different power threshold values and relation to $n_{th,min}$ in each case [833, 838]. This picture is also supported by experimental work on AUG [831] and theoretical investigations [829, 830].

The impact of PFC material on $P_{th}$, described in section 7.3.3, was examined on AUG through comparison of kinetic profiles. A reduction in $P_{th}$ by about 25% was
observed after the plasma facing components have been changed from carbon to tungsten [765]. A possible explanation for this reduction in power threshold could lie in the changed density profiles, which are steeper in plasmas with W PFCs [889]. The higher particle and energy reflection coefficient of W when compared to C has been shown to steepen the density profile [890]. Another possible contributor to the reduction in $P_{\text{th}}$ when going from C to W PFCs is the reduction in low Z impurities and reduced edge $Z_{\text{eff}}$. Gyrokinetic modeling of pre-L-H conditions on JET-ILW suggests a prevalence of instabilities with growth rates that decrease as $Z_{\text{eff}}$ decreases, which can be linked to experimentally reduced $P_{\text{th}}$ at sufficiently high collisionality [891]. In contrast, at low collisionality (higher edge temperature), where ITG/TEM modes are expected to dominate, recent DIII-D experiments and corresponding gyro-fluid/kinetic simulations indicate an upshift of the ITG critical gradient, and hence reduced instability growth rates and thermal fluxes with increasing $Z_{\text{eff}}$, resulting in reduced $P_{\text{th}}$ [892]. This could be an important consideration for discharges with impurity seeding for dissipative divertor operation, also.

### 7.5.2. Radial electric field and $E \times B$ shear.
The impact of sheared $E \times B$ flow on stabilizing plasma turbulence and improving confinement in magnetized plasmas is widely accepted, and is thought to play a key role in the origin of pedestal formation in tokamaks [893]. Typically, the radial electric field well at the edge of the confined plasma shows a distinct minimum, where the minimum corresponds to the strongest $E \times B$ velocity (see for example [549, 894, 895]). Examples of fine scale variation in $E_r$ have been observed in some circumstances, as will be discussed in section 7.6. More generally, however, smooth profiles are reported, and observed differences in low-power plasmas are often invoked to explain changes in observed $P_{\text{th}}$, for example in the case of a reversal in the $\nabla B$ drift direction [896, 897]. Under the assumption of a critical shear in the poloidal rotation from $E \times B$, it is possible to predict scalings for L-H power threshold that are linear in density and field (see for example [898]). Numerous experiments have sought to determine whether a critical value of sheared $E \times B$ velocity can indeed explain thresholds on a given device.

On AUG a threshold in the $E \times B$ velocity minimum, serving as a proxy for the $E \times B$ velocity shear in the pedestal region, has been found at the L-H transition in favorable drift direction plasmas with a single X-point geometry [832, 899]. Experimental indications are that, for discharges of a sufficient density (i.e. above the low density branch), the plasma transitions into H-mode if this critical velocity is exceeded. Under these conditions the evolution of the $E_r$ near the L-H transition is dominated by the $\nabla p_i/n$ term [889, 897, 899], and emphasis is placed on the accurate identification of the ion pressure gradient in L-mode under various conditions. The $E \times B$ velocity threshold at AUG was found to be highly uniform across a range of toroidal magnetic field strengths, densities and for both D and H isotopes, with $P_{\text{th}}$ varying by a factor

of 3 for this database. Using these findings, a simple derivation of the $P_{\text{th}}$ scaling is proposed giving a physics interpretation of the $B_T$, density, isotope and surface area dependence of $P_{\text{th}}$ [832].

As discussed in section 7.3.1, a linear dependence on density of the ion heat flux at the L-H transition can explain the non-monotonic density dependence of the L-H threshold power [815]. On AUG, results are persistent in H plasmas, helping to explain the isotope effect on L-H power [828]. Based on this key role of the ion heat channel and the L-mode scaling of the energy confinement time, a phenomenological model for the L-H power threshold was developed that explains the dependence of $P_{\text{th}}$ on plasma current $I_p$ in the low density branch on AUG [816]. In the context of this model it is shown that the assumption of a critical velocity shear (shear suppression paradigm) leads to the prediction of the ion heat flux being linearly dependent on density (key role of ion heat flux).

On the other hand, JET-ILW observations contradict the hypothesis of a critical $E \times B$ edge shear flow at the L-H transition across the range of parameters explored (including variations in density, ion mass, heating method and divertor configuration) [895]. No significant change in the edge $E \times B$ shear flow is measured along the power ramp preceding the L-H transition in D plasmas [836, 900]. In D plasmas, but not in He plasmas, there are indications that the inner shear layer, farther from the separatrix, may play a role in setting the threshold, rather than the outer shear layer [895]. The measured $E_r$ profile in the SOL shows only a modest dependence on divertor configurations and cannot explain the different power thresholds with horizontal and vertical targets described in section 7.3.4 (although a significant impact on $E_r$ inside the confined region was seen) [900]. These experimental results contrast with earlier findings from modeling in which lower $P_{\text{th}}$ is correlated with $E_r$ spikes in the near SOL [901].

### 7.5.3. Can transition thresholds be explained by a critical edge ion heat flux?
The aforementioned work on the critical ion heat flux for L-H transitions arose from studies of the so-called 'low density branch' (section 7.3.1), in which electrons and ions are not well coupled by collisions and in which the heat transport of electrons and ions can be separately determined. For example, this is done in AUG experiments by solely heating the electrons via electron wave heating (ECRH). It could be shown that the ion heat flux at the plasma edge exhibits a linear dependence on the line averaged density, whereas the electron heat flux does not [765, 815]. Examples for two values of plasma currents are shown in figure 37. The linear dependence of the ion heat flux at the L-H transition with density can explain the non-monotonic density dependence of the L-H threshold power: the L-H transition mechanism possibly requires a critical edge ion heat flux and due to insufficient coupling of electrons and ions at low density, a much higher total input power is needed to reach the required edge ion heat flux. With pure ECRH heating the threshold power
<figure><img src="image_40.png" /><figcaption>**Figure 37.** Comparisons of threshold power calculated using loss power with edge ion heat flow $Q_{i,edge}$. First plot shows data at the L-H transition with $P_{L-H}$ corrected for the $B_T^{0.8}$ dependence. Reproduced courtesy of IAEA. Figure from [815]. © 2014 EURATOM. Second plot shows data from L-I transitions. Reproduced courtesy of IAEA. Figure from [835]. © 2017 EURATOM.</figcaption></figure>

is also higher at low densities than with additional NBI heating, confirming the role of the ion heat flux across the plasma edge [815].

Extended tests of the critical heat flux hypothesis were made by examining Alcator C-Mod discharges heated with ion cyclotron power (ICRH), and calculating the contributions of power flow into the ion and electron channels [902]. Results were found to be consistent with a linear scaling of $Q_{i,L-H}$ with density. A fit to a combined data set of AUG and C-Mod data resulted in a power law expression for the critical edge ion heat flux at the L-H transition

$$q_{i,L-H} = 0.0021 \times \bar{n}_e^{1.07 \pm 0.09} B_T^{0.76 \pm 0.20} \quad (22)$$

where $q_{i,L-H}$ is in units of MW m⁻², $\bar{n}_e$ is the line averaged electron density in units of 10¹⁹ m⁻³, and $B_T$ has units of T. Multiplying both sides of this equation by surface area S yields a power law expression for the surface-integrated ion heat flux $Q_i$ with dependencies not unlike ‘nBS’ scalings for $P_{th}$ such as equation (20). Figure 38 illustrates the quality of the fit for AUG ECH-heated plasmas and C-Mod ICRH-heated plasmas. Some NBI-heated plasmas on AUG are found to have

<figure><img src="image_41.png" /><figcaption>**Figure 38.** Comparison of the observed ion heat flux $q_i$ at the L-H transition to a scaling ($q_i^{\text{fit}}$) derived using ECH-heated AUG data (black diamonds) at 2.3–2.5 T and C-Mod data (maroon circles) at 4.0–7.8 T. NBI-heated plasmas from AUG and JET are on average underpredicted by the scaling. Reproduced from [903]. © IOP Publishing Ltd All rights reserved.</figcaption></figure>

higher $q_{i,L-H}$ than predicted by equation (22). This is also, on average, the case for NBI-heated deuterium discharges on JET [903], also shown in figure 38. The role of torque, discussed in section 7.3.3, provides a possible explanation for this discrepancy. A role for toroidal velocity in determining $Q_{i,L-H}$ was identified by Ryter et al [815]. When significant co-current NBI torque is applied, a reduction in the magnitude of $E_r$ naturally occurs from the increased $v \times B$ in the radial force balance. This can result in a higher ion heat flow being needed to reach a sufficient $\nabla p_i / n$ and critical velocity shear, enabling the L-H transition. However, as noted in section 7.5.2, JET evidence points against a critical velocity shear value defining access to the L-H transition.

Analysis of I-mode threshold data on AUG also agreed with the hypothesis of a critical $Q_i(n)$ [835], as figure 37 illustrates. In this analysis, the $Q_i$ is strictly linear with line density with no offset, while an offset density is needed to fit the I-mode $P_{th}$ trend [880]. Independently performed gyrofluid simulations targeted at AUG I-mode edge conditions additionally suggest requirements on local edge conditions, namely low collisionality near the separatrix and an ion-to-electron temperature ratio well above unity [904].

## 7.6. Dynamics near the L-H transition

As seen above, the community has advanced its characterization of global threshold conditions for transitioning between L-mode and higher confinement regimes, and has also identified a number of potential critical requirements on edge parameters. For a more complete understanding of the physics origin of these transitions, it is instructive to study the dynamic interactions between edge flows and turbulence, leading up to transitions. At the time of the UIPB [3], it had become recognized that mean flows could be accompanied by ZFs that arise from intrinsic turbulence and Reynolds stress. Also, the damping of turbulence can reduce the ZF drive. The energy transfer from turbulence to ZFs and back is then an important
consideration in the overall picture of turbulence suppression by sheared flows [769, 905–921]. Oscillatory behavior in turbulence and flows is a fundamental feature of predator-prey models that take these processes into account, and oscillations have indeed been observed between turbulence amplitude and flow amplitude in tokamak experiments [905, 908, 922–925]. The generation of GAM or low frequency ZFs is found to be significant in many instances. This section will provide a summary of recent advances in understanding these dynamics, and potentially the triggers for the transition.

**76.1. Oscillatory behavior near L-H transitions.** On AUG limit cycle oscillations (LCO, also called ‘I-phase’ (distinct from ‘I-mode’ discussed in section 7.4) are observed when close to the L-H power threshold. At very low density, Doppler reflectometry data are interpreted in a way that GAM shear flow can be dominant in the first phase of the L-H transition giving rise to a turbulence-flow interaction, which generates the LCOs [922]. LCOs were also observed in DIII-D, associated with low frequency (~2 kHz) poloidal flow oscillations consistent with the measured Reynolds stress drive, before edge pressure gradient oscillations become dominant leading up to the transition from LCOs to H-mode [905, 913]. By way of example, figure 39 shows contour plots of the normalized density fluctuation level $\bar{n}/n$ and the $E \times B$ velocity, as measured by a multi-channel Doppler Backscattering system in DIII-D, in comparison to the $D_\alpha$ recycling light evolution [769]. The data demonstrate clearly the periodic formation of the well structure in the $E \times B$ velocity during LCO, starting at the LCFS, and the concomitant periodic reduction in normalized edge turbulence level $\bar{n}/n$, as well as the radial inward expansion of the edge flow layer. At the transition from L-mode to LCOs, a 90° phase shift between $E \times B$ velocity and density fluctuation level has been observed in several devices, including TJ-II [783] and DIII-D [905]. At later times during LCO, and in a wider density range, plasma flow and pressure gradient are in phase, indicating that the diamagnetic component of the $E \times B$ flow dominates under those conditions. Near the LCO-H-mode transition, LCOs show a behavior more similar to type-III ELMs [926, 927], with magnetic precursors on Mirnov signals sometimes present.

In contrast to the DIII-D measurements, the turbulence-driven flow contribution is measured to be small in AUG [928] and in JFT-2M [917, 918], demonstrating the dominant neoclassical contribution to the background flow in the vicinity of the L-H transition. Another interpretation of the LCOs in I-phase stems from measurements of up-down asymmetric parallel current fluctuations occurring during the I-phase in ASDEX Upgrade. They are interpreted as a response to strongly ballooned plasma transport coupled with the Stringer spin-up mechanism [929].

DBS measurements on JET have revealed radial structure in $E_r$ profiles, within the well, in L-mode near the high density H-mode transition. The structures were shown to be quite stationary and were found consistent with ZFs. A conjecture is that stationary ZFs play a stronger role in mediating the

L-H transition on JET than oscillatory GAMs or LCO mechanisms [930]. In most JET L-H transitions achieved with slow power ramps, a low frequency $n=0$, $m=1$, up-down, magnetic oscillation is observed in the H-phase, named the M-mode at JET [931]. A multi-diagnostic comparison of JET and AUG I-phase shows that the M-mode and the I-phase are the same [932] While M-mode frequency has been shown on JET to scale with the poloidal Alfvén velocity, I-phase frequency scaling is more complex [926] and no universally accepted scaling has been established.

The interaction of edge turbulence with edge flows is also significant for H–L back-transitions [906, 909]. Substantial power hysteresis has been observed in DIII-D between the L–H transition threshold power (separatrix loss power) and the required ‘holding’ power below which an H-L back-transition is initiated. These observations contrast recent results in AUG, where a significant L-H/H-L power hysteresis was only obtained if the density difference between the pre-L-H transition L-mode plasma and the pre-H-L transition H-mode plasma was taken into account [897]. In regimes where LCOs are present, the H-L back-transition is typically characterized by transients resembling type I or type III ELMs, although these transients can be triggered at edge pressures well below peeling-ballooning stability limits. Following a significant reduction of the pedestal pressure gradient, LCOs are often observed, with the cycle frequency increasing before edge turbulence levels rise and an LCO-L-mode transition occurs. The time evolution observed in the L-LCO-H-mode progression is therefore reversed in the H-LCO-L back-transition sequence,

<figure><img src="image_42.png" /><figcaption>**Figure 39.** Example Limit Cycle Oscillations (LCO) from DIII-D: (a) $D_\alpha$ recycling emission from the divertor; (b) radial variation of density fluctuation level as measured by Doppler Backscattering (DBS); and (c) radial variation of $E \times B$ velocity measured via DBS. Reproduced from [769]. © 2017 IAEA, Vienna. CC BY 3.0.</figcaption></figure>
<figure><img src="image_43.png" /><figcaption>**Figure 40.** Profile in normalized poloidal flux $\psi$ of (a) Reynolds stress gradient and (b) poloidal velocity shearing rate, in the vicinity of an L-H transition on DIII-D. Black is at a time about 2 ms before the transition, red is at the time just before the transition and blue is at the time after the transition. Reprinted figure with permission from [910], Copyright (2014) by the American Physical Society.</figcaption></figure>

and a reciprocal evolution of the LCO frequency is observed during the back-transition [926].

### 7.6.2. Energy transfer between turbulence and flows.
Direct measurements of the Reynolds stress gradient across the L-H transition and the energy transfer from edge turbulence into directional poloidal flows have been carried out for example in DIII-D [910], and C-Mod [914]. On DIII-D in edge plasmas nearing the L-H transition, rapid (<1 ms) increases are measured in turbulence amplitude, Reynolds stress gradient and poloidal flow. This burst is seen to immediately precede the transition, and his highly localized to the region 1–2 cm inside the plasma separatrix. Figure 40 shows examples of this profile evolution. The connection of this to the L-H transition is further reinforced by the infered non-linear energy transfer from the turbulence into ZFs. Prior to the L-H transition, the normalized energy transfer rate was shown to exceed both the effective turbulence growth rate $\gamma_{eff}$ and the turbulence decorrelation rate just inside the LCFS. At smaller minor radii, the energy transfer rate remained below $\gamma_{eff}$, indicating that the L-H transition is triggered near the LCFS in this case [910]. Measurements on C-Mod yielded similar findings, with the ratio of Reynolds work (which generates low-frequency flow) to the turbulence drive exceeding unity shortly before an L-H transition, and a subsequent drop in the turbulence kinetic energy and an increase in poloidal flow, leading finally to a rapid buildup of an edge pedestal [914].

On the HL-2A tokamak, Langmuir probe measurements were used to document the production of sheared $E \times B$ flow by turbulence, revealing a stronger shear layer and turbulence stress as heating power approached threshold values [923]. The dominant transfer in this case was to a GAM oscillation, which eventually peaked and was replaced by ZFs. A similar finding was obtained on C-Mod in scans of power in I-mode, approaching an I-H transition [919]. In the C-Mod case, competition between GAMs and ZF was not found to directly impact the L-H transition, and a critical rate of ZF generation was deemed necessary in both ion $B \times \nabla B$ drift directions.

Altogether these measurements indicate that turbulence-flow interactions are instrumental for the initial trigger dynamics of the L-H transition as well as LCO precursor states. A

more detailed overview of recent experimental and modeling progress on turbulence-flow interactions is given in [769].

### 7.6.3. Macroscopic transients.
If the edge pressure gradient is transiently increased by a heat pulse from the core plasma, or if the edge particle source is manipulated, L-H transitions can also be triggered without self-organizing turbulence-flow interactions. L-H transitions triggered by sawteeth or core MHD activity resulting in an outward propagating heat pulse (e.g. [933]) are examples for this behavior. Small pellet injection in DIII-D [934] or supersonic molecular beam injection (SMBI) in KSTAR [906] have demonstrated that L-H transitions can be initiated via steepening of the edge density gradient alone (due to the associated localized increase of the edge ionization source). The pressure-gradient-driven $E \times B$ shear flow due to increased diamagnetic flow is sufficient in these cases to induce and sustain the H-mode transport barrier. The L-H power threshold was reduced by more than 25% via deuterium pellet injection in DIII-D [934]. SMBI has also been used successfully in KSTAR, although repeated SMBI injections were required to maintain the H mode state [906].

### 7.7. Heuristic, fluid-based, and gyrokinetic modeling of the L-H transition

Heuristic reduced models aimed at describing the interaction dynamics of turbulence and flows across the L-H transition date back to the early 1990s and 2000s [935–937]. These heuristic models have evolved from 0D Lotka–Volterra predator–prey models to 1D simulations including radial particle and heat radial transport and the radial ion force balance. An intermediate oscillatory limit-cycle state that precedes the final transition to sustained turbulence suppression in H-mode was first described in [937]. This model includes ZF shear as well as the effect of diamagnetic shear resulting from pressure profile evolution, and describes predator-prey behavior that precedes the H-mode transition locked in by increasing diamagnetic shear. More advanced 1D models have been recently developed, including equations describing neoclassical and anomalous radial heat and particle transport. Realistic radial particle and power deposition profiles have been prescribed in this context, and the evolution of the poloidal ion (mean) flow (subject to neoclassical ion flow damping) has been included [818, 906, 915, 916, 938, 939].

The model detailed in [906] is focused on turbulence driven by the ITG modes, and includes a simple critical-gradient ITG model. TEM modes destabilized by electron temperature and density gradients are also heuristically accounted for. Including electron and ion transport separately has been necessary to model the non-monotonic behavior of the L-H power threshold with density, as previously discussed in section 7.3.1. Earlier AUG results suggested that the increase of the L-H threshold power at low density is caused by increased electron thermal loss [815, 828, 899] (see figure 37). These AUG experiments employed selective electron heating, which allowed decoupling of electron and ion thermal transport at low density. The AUG results show clearly that the
<figure><img src="image_44.png" /><figcaption>**Figure 41.** Plot of L-H transition power threshold $P_{thr}$ from modeling, versus the ion heating fraction $H_{mix}$, and plasma density $n$. The selection of $n$ and $H_{mix}$ in the sample was chosen based on the experimental trend of dominant electron heating at lower densities and dominant ion heating at higher densities. A subset of modeling results for $H_{mix} = 1$ (pure ion heating) is also included. Reprinted from [818], with the permission of AIP Publishing.</figcaption></figure>

L-H power threshold increased with electron power deposition and electron temperature at the separatrix. However, the ion diamagnetic L-mode flow and flow shear (determined by the local ion temperature together with the ion pressure gradient and mean shear flow) decreased with decreasing density in L-mode before the L-H transition. These findings demonstrate that the increased L-H power threshold at low density is a consequence of additional electron thermal loss. The 1D modeling results shown in figure 41 were accordingly obtained by varying both the L-mode density and the mix of electron and ion heating power [818]. The calculated threshold power displays a minimum at low density and dominant ion heating but increases at low density and increased electron heating power. The calculated power threshold therefore shows trends in agreement with experimental results, and the modeling results confirm the observed power threshold increase at low density in electron heat-dominated plasmas [815, 816].

Reduced fluid modeling of the L-H transition has primarily investigated ideal or resistive interchange turbulence [938, 940], or resistive ballooning turbulence [941–944] in L-mode plasma conditions leading up to the L-H transition. Regarding RBMs, the vorticity equation and the continuity and ion momentum equations have typically been solved, including parallel resistivity. The Braginskii equations (or sets of reduced equations based on the Braginskii equations) have also been used to simulate L-mode-LCO and L-H transitions. The dominant L-mode turbulence modes at the plasma edge are expected to depend strongly on collisionality. RBMs are expected to dominate at high collisionality in the Pfirsch-Schlüter regime, and TEM/ITG turbulence may be dominant at low collisionality, where trapped particle effects are expected to contribute significantly to the turbulence drive. Fluid models, most appropriate at high collisionality (Pfirsch-Schlüter regime), assume $k_{\perp} \rho_s \ll 1$ for the dominant modes. Earlier work, using 3D simulations in toroidal flux-tube geometry and the complete set of Braginskii equations, has already shown a transition driven by turbulence-generated transient poloidal flow, followed by an increase of the ion pressure

gradient, reflected in the formation of a pronounced electric-field well from the diamagnetic contribution to the $E \times B$ velocity, which maintains turbulence suppression via the associated sheared flow [941].

In an example of a flux-driven 3D simulation, a plasma edge barrier was obtained in which the $E \times B$ flow is driven by the radial gradient in neoclassical friction [942, 944]. Neoclassical (collisional) friction was mostly a driver for the shear flow before formation of the ETB formation, whereas Reynolds stress was observed to intermittently act as both a source and a sink. After the L-H transition the Reynolds stress was mainly an energy sink (transferring flow energy back to the turbulence), but neoclassical friction remained a source of $E \times B$ shear flow. Intermittently, LCOs were also present, characterized by transients in which the $E \times B$ flow collapses and fluctuation levels increase.

Electrostatic resistive ballooning turbulence has been modeled by a reduced fluid equation set (vorticity and pressure advection equations) [943] using the BOUT++ code as a framework. A temporal sequence of the L-H transition was observed. This simulation is appropriate for the collisional regime, where the normalized ion collisionality reaches values of order 10–100 near the LCFS. A transient, periodic LCO phase was observed before the L-H transition, where the pressure gradient increased transiently after the $E \times B$ velocity during the LCO-H-mode transition. The turbulence-flow energy-transfer coefficient increase beyond unity preceded the increase of the ion pressure gradient across the transition. The conventional turbulence quench criterion, $\omega_{E \times B} > \gamma_L$, where $\gamma_L$ is the dominant linear growth rate, is however only fulfilled after positive feedback due to the increasing pressure gradient further increases the $E \times B$ shearing rate. The dependence of the ion pressure gradient on heating power confirms this transition as a first-order phase transition: the pressure gradient increases more rapidly with increasing heating power beyond a critical threshold.

While still very challenging, initial encouraging simulation results from first principles Gyrokinetic modeling of the L-H transition have also been obtained [920], indicating a potential synergy of turbulent and neoclassical processes in the transition trigger phase. In this work, a global full-f code (XGC) has been used to evolve the edge plasma turbulence and the electric field leading up to the L-H transition, starting from a C-Mod L-mode plasma equilibrium. After an initial phase of GAM oscillations associated with oscillating Reynolds stress drive, X-point orbit loss is found to produce positive $E \times B$ shear that, synergistically with ZF shear, suppresses edge turbulence.

## 7.8. Outlook and open questions

Since the UIPB, the characterization of access to enhanced confinement regimes has improved considerably on existing devices, allowing us to refine our projections toward burning plasmas, particularly those to be developed using the tokamak. The strength of the multi-machine $P_{th}$ approach, embodied by the work described in section 7.2, allows us to target to first order the power requirements of devices such as ITER and SPARC. Meanwhile, deviations from the ITPA scaling
of equation (20) have been systematically addressed through extensive experimentation, allowing the community to better prepare to access H-mode on future devices.

The density dependence (section 7.3.1) of $P_{\text{th}}$ will have a substantial impact on the realization of H-mode in burning plasma devices. Because of unavoidable limitations on the amount of available auxiliary power, it is desirable to optimize L-H transitions so that they minimize external power. Therefore, determining with greater certainty the L-mode density values near the minimum of $P_{\text{th}}$, and the correct value of $P_{\text{th}}$, become of crucial importance to successful operation. As seen above, the density behavior of $P_{\text{th}}$ is sometimes seen to link closely to the isotopic mix (section 7.3.2), though not in a way that has been reproduced widely. This creates an important need for future experimentation and modeling.

As seen above, increases in $P_{\text{th}}$ at low density are often attributed to a shift in the balance of heating, or in the balance of edge power flow, from the ions to the electrons. This may present a challenge for some future fusion devices. For example, ITER will be electron-heat dominated, especially in its early phases of operation, and could be affected significantly by the low-density branch. The L-H power threshold is not easily predicted in plasmas where the electron loss power (across the LCFS) exceeds the ion loss power. Current DIII-D experimental results in low collisionality, ITER-similar-shape (ISS) plasmas indicate that the electron and ion loss powers are similar [843], and that equation (20) and similar multi-machine scalings do not perform well in this regime. In addition, ZF break-up in the collisionless regime could alter or prevent the L-H transition trigger sequence and prevent access at collisionality relevant to these kinds of high power density plasmas.

Extrinsic impurity seeding is expected to be a prevalent feature of burning plasma devices, in order to manage the interaction of high heat flux plasmas with divertors, both conventional and advanced. Impurity seeding could have a significant impact on threshold power, both through changes in power balance near the plasma edge and through increases in edge collisionality via higher $Z_{\text{eff}}$. A study on the power threshold with varying levels of N seeding during the LH transition did not change the power threshold in ASDEX Upgrade with a W wall [765]. However, these impacts have not been widely studied, and merit significant further investigation.

Another substantial influence on transition thresholds is the combination of magnetic configuration, divertor configuration and equilibrium parameters of a particular tokamak plasma (sections 7.3.4 and 7.3.5). This area presents a potentially significant gap in predictive capability toward future devices. This will be especially true as devices implement advanced divertors and/or true double null configurations for improved power handling. NT concepts are included in consideration here, and are gaining interest in the community. As seen above, a variety of divertor and shaping effects on L-H $P_{\text{th}}$ have been reported on devices, with no clear unifying model to explain the results. Additional experimental work is desirable in varied configurations, including analysis of the local edge conditions, both in low power L-mode and at threshold conditions.

An opportunity exists here as well to examine the dynamics of turbulence and flow properties for comparison.

In recent years a greatly improved understanding of the effect of 3D fields on $P_{\text{th}}$ has been obtained (section 7.3.6). One of the highest impact areas is the application of RMPs for purposes of ELM mitigation. In the event that a naturally ELM-suppressed regime cannot be obtained, RMPs are a candidate means of preventing large ELMs, and indeed are a crucial component of ITER operation. Other forms of ELM control, such as pellet pacing and vertical kicks, have been examined less thoroughly in the context of $P_{\text{th}}$.

To the above gaps it should be added that burning plasmas will extend tokamak parameters to extremes not explored on present devices. Most notably they will operate with low normalized ion gyro-radius $\rho^*$, which is predicted to lead to reduced edge $E \times B$ shear. This can be seen if one considers that the L-mode 'seed' $E \times B$ flow can be approximated by the pressure-gradient-driven ion diamagnetic flow $v_{E \times B} = -(1/neB)\nabla p = (\rho/a)c_s(a/\lambda_p) = \rho^*c_s(a/\lambda_p)$ where B is the total magnetic field, a is the minor plasma radius, $c_s$ is the ion sound speed, and $\nabla p$ is the radial ion pressure gradient. While the $\rho^*$ dependence of the $E \times B$ flow and flow shear has been predicted by gyrokinetic simulations [945, 946], no experiments have so far conclusively confirmed the theoretically expected $\rho^*$ dependence.

By comparison to L-H thresholds, The L-I threshold is much less well characterized, having been studied in earnest for a much shorter time, and on fewer devices. A scaling of the L-I $P_{\text{th}}$, analogous to the ITPA scaling for L-H, is a highly desirable goal, and likely within reach given the number of devices now having reported I-mode access. However it will be important to go beyond simplified power law scalings for the L-I threshold, because I-mode thresholds exhibit very similar impacts as L-H $P_{\text{th}}$: elevation at low density, inverse dependence on isotope mass, and possibly divertor effects to name a few. A second consideration, which will be critical if I-mode is to be used as a safely ELM-suppressed scenario on fusion devices, is the projection of the I-H threshold.

Insofar as high fusion gain will depend on sustained high confinement operation, back-transitions to L-mode are generally to be avoided in burning plasmas. Other motivations for avoiding uncontrolled H-L transitions include the desire to maintain stable equilibria and to avoid excessive demands on equilibrium shaping coils. While proximity to H-L back-transition is often expressed as a ratio of discharge power to a $P_{\text{th}}$ scaling, this is not quantified well in most experiments. Indeed a significant hysteresis has been observed in a number of devices, with $P_{\text{th,H-L}} < P_{\text{th,L-H}}$ for example.

The keys to a power requirement projection with high confidence may lie in local edge parameters at threshold (7.5), and/or in the dynamics of turbulence and flows leading to the transition trigger (7.6). As seen above, experiment and modeling have progressed in connecting these with global parameters, for example in the way $P_{\text{th}}$ can be related through comprehensive transport analysis to the edge ion heat flow, which in turn may be correlated with the diamagnetic contribution to the $E \times B$ shearing rate. Significant progress has
been made addressing microscopic turbulence–flow interac-
tions experimentally and via heuristic modeling of the trans-
ition dynamics, including ZFs and mean flows. However, a
physics-based model of the L-H power threshold scaling is
ultimately desired, with the capability to extrapolate power
requirements for H-mode access to future burning plasma
experiments. It is therefore crucial to link microscopic trans-
ition physics to the power-threshold scaling in such a way that
the prominent parameter dependencies of the power threshold
are correctly reflected.

Over the last decade, simulations and modeling of the L–H transition have been more widely attempted, including the use of heuristic models as well as more complete fluid models. Results based on global 2D/3D electromagnetic fluid codes have also been reported in [947, 948]. For certain aspects of transition physics, reduced models are suitable and very valuable for simple parameter scans. First principles fluid simulations will however ultimately be required, based on the complete set of Braginskii equations, to account for the flow and turbulence dynamics parallel to the magnetic field in the scrape-off layer as well as the outer core plasma, in realistic toroidal geometry. Implementing the correct boundary conditions for the treatment of the transition between the closed and open field-line region is a major challenge, as well as the computationally efficient inclusion of sheath boundary conditions in the SOL. Several fluid and gyrokinetic simulation results indicate an increased poloidal flow concomitantly with an increasing ion pressure gradient across at the transition [916, 939]. Other simulations do not show a clear trigger signature prior to the increase of the pressure gradient and diamagnetic flow [938]. Simulations more closely matched to experiments for a wider range in plasma density and collisionality range may further illuminate and connect these observed variations in trigger physics and their parameter dependence.

Global gyrokinetic simulations of the L–H transition still face significant obstacles. These simulations have to accommodate large parameter variations across the simulation domain, and also require interfacing domains with different parallel boundary conditions in the outer core and SOL regions. The density, electron and ion temperature, and ion/electron collisionality can vary strongly across the plasma boundary, in particular once pedestal formation starts. Radial temperature and density profile scale lengths in the pedestal can be very short, on the order of several ion gyro-radii. Gyrokinetic modeling of the L-mode edge plasma just inside the LCFS has become feasible and already provides significant insight into L-mode turbulence characteristics and transport fluxes in low-density and high density plasmas with dominant electron or ion heating, and their isotopic dependence, with encouraging results.

Acknowledgments

The authors wish to express their gratitude to G. Birkenmeier,
M. Cavedon, E. Delabie, J. Hillesheim, C. Maggi, Y. Martin,
U. Plank, F. Ryter, C. Silva, P. Vincenzi and M. Willensdorfer
for their input and feedback on the material contained in

this section. Authors would like to warmly thank all of
the colleagues who have contributed to the Transport and
Confinement and the Pedestal ITPA activities on L-H trans-
itions, for the many interesting and valuable presentations and
discussions during the ITPA meetings, and for the very fruitful
collaborations that the ITPA has fostered.

8. Global confinement

G. Verdoolaege, S. M. Kaye, C. Angioni and M. Maslov

8.1. Introduction

Energy confinement studies based on empirical scaling expres-
sions form a useful complement to predictions of turbulent
transport from first principles. Obtained through regression
analysis on multi-machine data sets, they offer a conveni-
ent instrument for extrapolating plasma performance to new
machines, such as ITER [3, 274] or DEMO [949]. In gen-
eral, scaling laws are based on very few model assumptions,
often employing a log-linear model for the functional form of
the regression model, with normally distributed measurement
errors. As such, global confinement scaling can be considered
a primarily data-driven activity for quantifying the depend-
ence of confinement on machine parameters. In addition, scal-
ing laws provide a reference for assessing the performance of
experiments in terms of global confinement and are used as
boundary conditions for modelers.

In the present overview of progress in the area of global
confinement scaling, we concentrate on the energy confine-
ment scaling in tokamak H-mode plasmas. Usually, these
scalings are derived from the Global H-Mode Confinement
Database, maintained by the ITPA since 2001. Specifically, we
concentrate on the most recent iteration, DB5.2.3, which con-
tains data from 19 devices of different sizes and shapes, includ-
ing ASDEX Upgrade with the full tungsten wall and JET with
the ITER-like wall. Recently, a subset of this version of the
database has been used to derive the ITPA20 scalings [950].
In this section, the database and the latest multi-machine scal-
ings are briefly discussed and compared with earlier data and
results. In particular, we compare with the IPB98(y,2) scal-
ing and with the corresponding database version DB2.8, since
IPB98(y,2) is currently still the de facto standard for global
H-mode energy confinement scaling [274]. Additional details
regarding the data and its analysis can be found in [950].

8.2. *The global H-mode confinement database*

The ITPA global H-Mode Confinement Database is the result of a joint effort between teams from the various tokamaks that have contributed data since the inception of the database [951]. It contains carefully curated data corresponding to global plasma quantities measured during brief time slices in H-mode operation, usually with ELMs of various types, although some non-stationary ELM-free plasmas were also included. The selection criteria for the overall database and for the ‘standard’ subsets used for fitting scaling laws are summarized in [950] and described in further detail in [951,
952]. The latest version of the database (DB5.2.3) is available online, along with a brief description of its variables, as well as selection criteria for various subsets of the data [953]. For later reference we also mention version 2.8, on which IPB98(y, 2) is based [274], version 3, documented in [952] and the significantly expanded version 4 [954, 955]. A further subset of the standard set has at times been considered for analysis. Referred to in [950] as the 'ITER-like subset', it contains data from devices with certain ITER-like characteristics in terms of size and/or shape, $q_{95}$ and $Z_{eff}$ (ASDEX Upgrade, Alcator C-Mod, COMPASS-D, DIII-D, JET, JFT-2M, JT-60U and PBX-M).

In moving from DB4 to DB5, special care was taken to add reactor-relevant data in previously sparsely populated areas of the operational space. In addition, 866 time slices were added from JET with the fully metallic ITER-like wall (JET-ILW) during the stationary phase of H-mode discharges [221, 956] and 825 H-mode slices from ASDEX Upgrade (AUG) with the full tungsten wall (AUG-W) [223]. In the analysis of DB5 described in [950], the data from JET-ILW and AUG-W, together with the entries from Alcator C-Mod (molybdenum), form the fully metallic ('High-Z') subset of the data.

Of the more than 200 database variables (including many selection and indicator variables), the most relevant, for the majority of the activities on global confinement scaling, are the thermal energy confinement time $\tau_{E,th}$ (s), the plasma current $I_p$ (MA), the on-axis vacuum toroidal magnetic field $B_t$ (T), the central line-averaged electron density $\bar{n}_e$ (10$^{19}$ m$^{-3}$), the thermal power lost due to transport through the LCFS $P_{l,th}$ (MW), the major radius $R_{geo}$ (m), the elongation of the LCFS, defined as $\kappa_a = V/(2\pi R_{geo}\pi a^2)$ (with $V$ (m$^3$) the plasma volume inside the LCFS), the inverse aspect ratio $\epsilon = a/R_{geo}$ and the effective atomic mass $M_{eff}$ of the plasma. In addition, the global confinement scaling can be expressed in dimensionless variables [957]. This includes the normalized ion gyroradius $\rho_*$, normalized plasma pressure $\beta_t$, normalized collisionality $\nu_*$ and cylindrical safety factor $q_{cyl}$, defined as in [274] and [950].

In comparison with DB2.8, the standard set of DB5.4.3 contains many more data points (7537 vs. 1310), going to higher power, current and density. In addition, DB5 contains data from the low-aspect ratio machines MAST and NSTX, which can be useful for determining the $\epsilon$ scaling [958]. The analysis also pointed out that $\rho_*$ in the new database reaches a lower median, compared to DB2.8, partly owing to the addition of reactor-relevant data.

As touched upon below, a well-known issue in many scaling efforts with data from fusion devices relates to dependence between predictor variables, which to some extent can be described by pairwise correlations on a logarithmic scale. This issue persists in DB5, although it has been slightly mitigated with respect to DB2.8. Still, significant correlation remains between the plasma current and power loss, current (or power) and major radius, and also between the shape parameters $\kappa_a$, $\epsilon$ and $\delta$.

### 8.3. Global H-mode confinement scaling

As is the case for most scaling laws for various quantities in fusion plasma physics, the IPB98(y, 2) scaling relation has the form of a power law [274]:

$$ \tau_{E,th} = \alpha_0 I_p^{\alpha_I} B_t^{\alpha_B} \bar{n}_e^{\alpha_n} P_{l,th}^{\alpha_P} R_{geo}^{\alpha_R} \kappa_a^{\alpha_\kappa} \epsilon^{\alpha_\epsilon} M_{eff}^{\alpha_M}. \quad (23) $$

In [950], the average triangularity $\delta$ of the LCFS was added as a predictor variable, using a factor $(1+\delta)^{\alpha_1+\delta}$, which allows for plasmas with zero triangularity.

Invoking scale-invariance of the equations describing the thermal energy transport, the scaling (23) can be cast in dimensionless form:

$$ \Omega_i \tau_{E,th} = \alpha_{0,D} \rho_*^{\alpha_p} \beta_t^{\alpha_\beta} \nu_*^{\alpha_\nu} q_{cyl}^{\alpha_q} \kappa_a^{\alpha_{\kappa,D}} \epsilon^{\alpha_{\epsilon,D}} M_{eff}^{\alpha_{M,D}}, \quad (24) $$

where $\Omega_i = qB_t/M_{eff}$ is the effective ion cyclotron frequency, with charge $q=e$ for hydrogenic species. If triangularity is included, a factor $(1+\delta)^{\alpha_1+\delta}$ has to be added on the RHS of (24). The requirement of scale-invariance imposes a constraint on the exponents in the engineering scaling (23). Usually, the high-$\beta$ Kadomtsev constraint is investigated [950]:

$$ \alpha_K \equiv 4\alpha_R - 8\alpha_n - \alpha_I - 3\alpha_P - 5\alpha_B - 5 = 0. \quad (25) $$

The following equivalent form of the dimensionless scaling is also common in the literature:

$$ \tau_{E,th} \propto \tau_B \rho_*^{2+\alpha_p} \beta_t^{\alpha_\beta} \nu_*^{\alpha_\nu} q_{cyl}^{\alpha_q} \kappa_a^{\alpha_{\kappa,D}} \epsilon^{\alpha_{\epsilon,D}} M_{eff}^{\alpha_{M,D}}. $$

Here, $\tau_B$ is the Bohm time, given by $\tau_B = a^2/\chi_B$, with $\chi_B \propto T/B_t$ the Bohm diffusivity and $T$ (eV) the volume-averaged temperature, assuming equal electron and ion temperature. Theory suggests gyroBohm transport ($\alpha_p = -3$), as opposed to the less favorable Bohm scaling ($\alpha_p = -2$), although global confinement scalings derived by regression analysis do not always show a clear preference.

### 8.4. Methodological issues

It has long been recognized that regression analysis based on multi-machine data poses a multitude of challenges [274, 950, 952, 959]. Obtaining parameter estimates, predictions and error bars with a least squares technique is straightforward enough, but easily leads to misleading conclusions, either because assumptions are not fulfilled or due to a variety of uncertainties that may not have been properly taken into account. Multiple efforts have been directed at improving the reliability of the regression estimates, with some of the most recent activities described in [950]. There, apart from the implementation based on the commonly used weighted least squares (WLS) method (motivated by imbalance of the amount of data contributed by the various devices), a robust Bayesian technique was applied, as well as a new regression method rooted in information geometry [960].
One important cause for uncertainty, recognized since long, relates to the functional form used to model the trends of the confinement time. While a power law like (23) certainly has the advantage of simplicity and robustness, more flexible models have been proposed and implemented. For instance, there have been strong indications that log-nonlinear models, with interaction terms on a logarithmic scale, are better suited at fitting the trends of the confinement time reflected by the data (see references in [274]). Models with two terms have also been considered, both with log-linear ('offset-linear') terms (e.g. [961]) and log-nonlinear ('offset-nonlinear') terms (e.g. [962]). In H-mode, one of the terms can be interpreted as modeling pedestal confinement, influenced by conduction and ELM stability, while the other describes the core transport. Techniques learning the functional form from the data have been explored too [963].

Physical mechanisms that may influence confinement but that are not well represented by any (combination) of the predictor variables contribute to the residual uncertainty of the scaling. Plasma rotation and electron-to-ion heating ratio are known to affect the confinement, but insufficient data about these effects is available in the database. Plasma shaping also has an influence, e.g. through the plasma triangularity, particularly at high Greenwald fraction $f_{GW} = \bar{n}_e/n_{GW}$, with $n_{GW}$ the Greenwald density.

A particularly thorny issue, which often arises when fitting fusion scaling laws in general, relates to the uncertainty on parameter estimates and predictions induced by dependencies between predictor variables ('multicollinearity' between the log-transformed variables). Part of the analysis of DB5 was aimed at quantifying the effect of these dependencies on the uncertainty of parameter estimates in the confinement scaling, even though predictions based on new data (e.g. for ITER) should be relatively impervious to this issue. The problem has often been investigated by means of principal component analysis (PCA) [951], which allows characterizing the shape of the data cloud in logarithmic space. Techniques like principal component regression or partial least squares offer a systematic approach to regularization of the regression problem in the face of multicollinearity. An alternative approach was proposed in [950], attempting at quantifying the impact of predictor dependencies on uncertainties of parameter estimates and predictions. By varying the estimated coefficients according to their joint posterior distribution in a Bayesian context, hence obeying the correlation structure, tolerances could be set on the coefficients corresponding to an acceptable goodness-of-fit. These tolerances were interpreted as uncertainties in the ITPA20 scaling laws (see below).

Finally, it is common practice to first estimate the scaling in engineering form and then convert via a set of algebraic relations to the dimensionless scaling. However, as shown in [950], this can lead to nonsensical results due to error accumulation. The alternative is to fit the scaling (24) in dimensionless space, by means of appropriate regression techniques.

## 8.5. The ITPA20 confinement scalings

### 8.5.1. Engineering scalings.
This section focuses on the multi-machine confinement scaling expressed in engineering variables. In [950], single-device scalings were explored as well, leading to the observation that the full-metal devices AUG-W and JET-ILW may exhibit somewhat weaker power degradation compared to the situation with carbon-based plasma-facing components. The multi-machine scalings reported in [950] were obtained from the standard set STD5 in DB5.2.3, consisting of measurements in ELMy H-mode plasmas. Here, we concentrate on the corresponding parameter estimates and predictions $\hat{r}_{E,th,ITER}$ for the thermal confinement time in ITER. The predictions, to be interpreted, as usual, with due caution, are based on the following values in the baseline inductive $Q = 10$ scenario in ITER: $I_p = 15$ MA, $B_t = 5.3$ T, $\bar{n}_e = 10.3 \times 10^{19} \text{ m}^{-3}$, $P_{1,th} = 87$ MW (based on 40 MW auxiliary heating power and 400 MW fusion power), $R_{geo} = 6.2$ m, $\delta = 0.48$, $\kappa_a = 1.7$, $\epsilon = 0.32$ and $M_{eff} = 2.5$ [964]. Using the definitions of the dimensionless variables given in [950], this corresponds to $\rho_* = 0.0020$, $\beta_t = 2.24$, $\nu_* = 0.014$ and $q_{cyl} = 1.94$. More recent studies relying on 50 MW auxiliary power and 500 MW fusion power suggest a conservative lower limit on $P_{1,th}$ of about 100 MW [75, 965]. In order to facilitate comparison with the predictions obtained from IPB98(y, 2), here most predictions are for the former estimate of the power loss, $P_{1,th} = 87$ MW (e.g. in tables 1 and 2). In addition, we provide some predictions for the more recent estimate of $P_{1,th} = 100$ MW.

The parameter estimates for the scaling of the global energy confinement in ELMy H-mode plasmas in DB5.2.3-STD5, obtained with WLS, are given in table 1 (the error bars are discussed below). The average triangularity is included as a predictor variable in the new scalings. The quality of all regression fits is good, as gauged by the root-mean-square error (RMSE), the coefficient of determination $R^2$ and the median absolute percentage error (MdAPE). Estimates for the ITER-like subset are also given, and compared with those given by the IPB98(y, 2) scaling (which did not include $\delta$) [274].

The most notable differences of the results on the full data set DB5.2.3-STD5 in ELMy H-mode, in comparison with IPB98(y, 2), are the slightly weaker dependence on density (as observed in individual machine scans) and machine size. There are also some differences in terms of the shape parameters, but these turn out to be less significant in light of the uncertainties, as explained below. The triangularity appears to have some influence on confinement. In addition, at $3.07 \pm 0.46$ s, the prediction for ITER is at least 17% lower than that obtained with the IPB98(y, 2) scaling on DB2.8 (3.62 s). With the updated power loss estimate, $P_{1,th} = 100$ MW, the prediction is $2.79 \pm 0.42$ s.

Focusing on the ITER-like subset, one notices the stronger scaling with current, as opposed to almost no dependence on magnetic field and an even weaker density dependence. However, the most striking difference with the scaling on
the full STD5 set in ELMy H-mode and, a fortiori, with IPB98(y, 2), is the weaker dependence on major radius. This is compensated to some extent by the stronger scaling with current and reduced power degradation, yielding a confinement prediction for ITER of 2.90 ± 0.46 s, or 2.65 ± 0.42 s for $P_{l,th}$ = 100 MW.

The robust Bayesian regression method applied in [950] yielded somewhat different estimates, with a slightly lower ITER prediction of about 2.6 s (25% lower than the prediction from IPB98(y, 2)). Using the procedure outlined below, a 95% credible interval with equal tail probabilities of [1.86 s, 3.63 s] was obtained for the prediction, or [-28%, +41%]. These intervals should be regarded as a guideline, rather than a strictly quantitative uncertainty measure.

The reason for the reduced dependence on machine size is currently being investigated [966]. So far this has resulted in the following observations. First, starting from DB2.8, various additions to the database have contributed to decreasing the size dependence, as observed also in DB4. Hence, whereas the addition of data from AUG-W and JET-ILW in the latest version of the database, DB5.2.3, may have resulted in a decreased size dependence, it is certainly not the only reason. Second, a subset of ELMy H-mode DB5.2.3-STD5 could be identified, about 17%, that contributes strongly to lowering $\alpha_R$, whereas the other points have little influence. In dimensionless space, the ITER operational point lies relatively far from that subset, which is characterized by either higher $\nu_*$ or higher $\rho_*$.

In other words, the ITER operational point appears to lie in a region characterized by stronger size scaling, reminiscent of that in IPB98(y, 2). Third, correlations between (logarithmic) predictor variables are of similar strength in the low-$\alpha_R$ subset, versus the subset formed by the other points in DB5.2.3-STD5. Therefore, multicollinearity cannot be solely responsible for lowering $\alpha_R$. Nevertheless, due to the multicollinearity, the confinement predictions obtained by regression in the total DB5.2.3-STD5 data set are similar to those resulting from a scaling fitted to a subset with the low-$\alpha_R$ points removed. This confirms the observations above that the lower size scaling is compensated by other dependencies, resulting in only slightly reduced predictions for ITER.

A confinement enhancement factor $H_{20} \equiv \tau_{E,th}/\hat{\tau}_{E,th}$ was defined based on the ITPA20-IL scaling in table 1 [950]. Figure 42 shows plots of (a) $H_{98}(y, 2)$ and (b) $H_{20}$ against Greenwald fraction $f_{GW}$, distinguishing the points per device. As is well known, and noted e.g. in [3], there is a decreasing trend of the residuals $H_{98}(y, 2)$ with increasing Greenwald fraction. The plot indicates that this is particularly the case for the purely metallic devices [967, 968]. The decreasing trend can also be seen in the plot of $H_{20}$, but it is much weaker. Hence, based on the latest version of the database, the ITPA20 scalings are recommended. Nevertheless, based on data from AUG and JET, a clear decrease remains of $H_{20}$ with increasing normalized separatrix density $n_{e,sep}/n_{GW}$ [950]. Still, regressions based on a reduced data set, omitting the points at high separatrix density, did not affect the scalings substantially.

An important aspect of global confinement scaling concerns estimates, that are of practical use, of the degree of uncertainty on the parameter estimates and predictions. For

instance, one could aim at uncertainty estimates that convey a realistic idea of the variability to be expected in trends of the confinement time across existing and future experiments. As outlined in [274] and described in more detail in [969], the conventional uncertainty estimates, routinely computed by statistiks software and merely resulting from the scatter of the data around a given functional model, are not very useful, because not all assumptions that they rely on are fulfilled. In particular, uncertainty regarding the functional form of the regression model and dependencies between the predictor variables are known to allow variations of confinement trends and predictions that are much larger than suggested by the conventional error bars. In [969], various approaches were considered for quantifying this additional uncertainty, e.g. by measuring the effect of removing individual devices from the data set (jack-knife resampling) or by gauging the spread of predictions due to various log-nonlinear models. This led to confidence intervals that are several times larger than the conventional ones.

In [950], the focus lay on the uncertainty due to predictor multicollinearity. The uncertainty estimates were based on a multivariate normal approximation to the joint posterior distribution of the scaling parameters. It was observed that samples from a much broader normal distribution with the same correlations, as depicted in figure 43, did not appreciably deteriorate the goodness-of-fit. For instance, increasing $\alpha_I$ while decreasing $\alpha_B$ with respect to the ITPA20(-IL) scaling can still yield an acceptable fit to the data, due to their negative correlation. The error bars (one standard deviation) obtained with this analysis, shown in table 1, are about 10 times wider than the conventional ones. Moreover, the relative error on the individual exponents is smallest for $P_{l,th}$ (9%), $I_p$ (11%) and $R_{geo}$ (27%), whereas the other exponents have a considerably larger relative uncertainty. In fact, given these error bars, several of the predictor variables ($B_t$, $\bar{n}_e$ and the shape parameters) do not have a clearly discernible influence on the confinement times in the database considered.

### 8.5.2. Dimensionless scaling.
Within the uncertainties obtained by the Bayesian analysis, both the ITPA20 and ITPA20-IL scalings obey the high-$\beta$ Kadomtsev constraint quite well. However, algebraic transformation between the engineering and dimensionless form of the scaling—the conventional practice followed also for IPB98(y, 2)—did not yield acceptable results due to error accumulation. Instead, the Bayesian regression technique proposed in [950], which allows errors in all variables, was used to estimate the scaling directly in dimensionless space.

The results of this analysis are shown in table 2. Compared to IPB98(y, 2), where $\alpha_\rho = -2.7$, i.e. close to gyroBohm scaling, the $\rho_*$ scaling is less favorable in the revised scalings. On the other hand, the negligible dependence on $\beta$ and weak inverse scaling with $\nu_*$ are more favorable for extrapolation to ITER (see also [970, 971]). This explains why, all-in-all, the prediction for ITER (2.53 s) is very similar to that obtained from the engineering scaling using the same Bayesian regression technique. However, the 95%
<table><caption>**Table 1.** Parameter estimates and ITER predictions for the IPB98(y, 2) scaling, as well as the scalings obtained from the DB5.2.3-STD5 data set for ELMy H-mode plasmas and its ITER-like subset, including triangularity. The error bars correspond to a single standard deviation.</caption><thead><tr><th></th><th>α<sub>0</sub></th><th>α<sub>I</sub></th><th>α<sub>B</sub></th><th>α<sub>n</sub></th><th>α<sub>P</sub></th><th>α<sub>R</sub></th><th>α<sub>1+δ</sub></th><th>α<sub>κ</sub></th><th>α<sub>ε</sub></th><th>α<sub>M</sub> (s)</th><th>τ̂<sub>E,th,ITER</sub> (%)</th><th>MdAPE</th><th>RMSE</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>IPB98(y, 2)</td><td>0.0562</td><td>0.93</td><td>0.15</td><td>0.41</td><td>-0.69</td><td>1.97</td><td>—</td><td>0.78</td><td>0.58</td><td>0.19</td><td>3.62</td><td>3.7</td><td>0.15</td><td>0.98</td></tr><tr><td>ITPA20</td><td>0.053<br>+0.030<br>-0.018</td><td>0.98<br>±0.19</td><td>0.22<br>±0.18</td><td>0.24<br>±0.11</td><td>-0.669<br>±0.059</td><td>1.71<br>±0.32</td><td>0.36<br>±0.39</td><td>0.80<br>±0.38</td><td>0.35<br>±0.66</td><td>0.20<br>±0.17</td><td>3.07<br>±0.46</td><td>6.1</td><td>0.18</td><td>0.96</td></tr><tr><td>ITPA20-IL</td><td>0.067<br>+0.059<br>-0.032</td><td>1.29<br>±0.16</td><td>-0.13<br>±0.17</td><td>0.147<br>±0.097</td><td>-0.644<br>±0.061</td><td>1.19<br>±0.27</td><td>0.56<br>±0.36</td><td>0.67<br>±0.63</td><td>-0.000<br>±0.66</td><td>0.30<br>±0.16</td><td>2.90<br>±0.46</td><td>6.0</td><td>0.17</td><td>0.95</td></tr></tbody></table>

<table><caption>**Table 2.** Similar to table 1, but expressed in dimensionless variables.</caption><thead><tr><th></th><th>α<sub>0</sub></th><th>α<sub>ρ</sub></th><th>α<sub>β<sub>i</sub></sub></th><th>α<sub>ν</sub></th><th>α<sub>q</sub></th><th>α<sub>1+δ</sub></th><th>α<sub>κ,D</sub></th><th>α<sub>ε,D</sub></th><th>α<sub>M,D</sub> (s)</th><th>τ̂<sub>E,th,ITER</sub> (%)</th><th>MdAPE</th><th>RMSE</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>IPB98(y, 2)</td><td>4.24 × 10<sup>-7</sup></td><td>-2.69</td><td>-0.90</td><td>-0.0081</td><td>-2.99</td><td>—</td><td>3.29</td><td>0.71</td><td>0.96</td><td></td><td></td><td></td><td></td></tr><tr><td>ITPA20-dim</td><td>0.087 × 10<sup>-6</sup><br>+0.51 × 10<sup>-6</sup><br>-0.075 × 10<sup>-6</sup></td><td>-2.24<br>±0.34</td><td>0.20<br>±0.21</td><td>-0.469<br>±0.061</td><td>-0.70<br>±0.37</td><td>0.36<br>±0.64</td><td>1.24<br>±0.47</td><td>-1.70<br>±0.51</td><td>0.53<br>±0.30</td><td>3.07<br>±0.82</td><td>16</td><td>0.32</td><td>0.91</td></tr><tr><td>ITPA20-IL-dim</td><td>2.1 × 10<sup>-6</sup><br>+19 × 10<sup>-6</sup><br>-1.9 × 10<sup>-6</sup></td><td>-1.97<br>±0.33</td><td>0.12<br>±0.22</td><td>-0.426<br>±0.063</td><td>-1.03<br>±0.43</td><td>0.14<br>±0.64</td><td>1.89<br>±0.85</td><td>-0.26<br>±0.64</td><td>0.62<br>±0.37</td><td>2.53<br>±0.88</td><td>16</td><td>0.31</td><td>0.85</td></tr></tbody></table>

credible interval is [-47%, +88%]—too large for confident extrapolation.

## 8.6. Application of the confinement scalings

We end the section with a number of recommendations and remarks about application of global confinement scalings, some of which are relevant to scaling laws in general. The method owes part of its popularity to its conceptual simplicity. Experiments can be benchmarked and new predictions can be made very easily by substitution in a very simple scaling formula<fnref n="22" />. Moreover, even though the method hinges on various assumptions and choices in the analysis stage, very few physical model assumptions are made. Therefore, despite important recent advances in the understanding of energy transport in fusion plasmas and integrated modeling activities, global confinement scalings still offer a useful quick and easy rule-of-thumb for comparison and prediction across devices.

When applying the confinement scaling expressions, it is essential to keep in mind a number of important limitations to avoid misunderstandings.

* Power-law scalings offer a very simple approximation to confinement trends. It is known that these global confinement scalings do not include various physical effects, either because the corresponding variables are not known or, quite

possibly, not available in the database, or for only a few devices. Likewise, as the scalings are based on global data, it is clear that certain profile effects cannot be captured.

* The parametric dependencies of the confinement suggested by scaling laws are averages across a multi-machine database and a broad variety of plasma conditions. One should therefore not expect to see a complete correspondence with specific single-device scans in all circumstances. This effect, referred to in statistics as 'Simpson's paradox' [972], is at least partly responsible for differences in dependencies between the multi-machine scalings and those observed in individual tokamaks. This has been experienced for instance during the preparations for the DTE2 experiments in JET, as reported in [973].

* It does not make sense to vary one of the exponents in the scaling expression, even within its error bar, while keeping the others fixed. The exponents are strongly correlated, as reflected by the joint posterior distribution: when one exponent changes, the others will change too. For the same reason, one can generally not ignore dependencies expressed by exponents that are negligible within their error bar.

Several among the above limitations are not due to a deficiency of the idea of scaling laws per se, but are related to the data quality or to assumptions, some of which could relatively easily be relaxed or altered.

Future activities in this area could benefit greatly from additional multi-machine data with open-access availability,

<footnote n="22">²² In a Bayesian perspective, predictions require marginalization of the parameters, but in the present case substitution yields a good first-order estimate.</footnote>
<figure><img src="image_45.png" /><figcaption>**Figure 42.** Confinement enhancement factors (a) $H_{98}(y, 2)$ and (b) $H_{20}$ versus Greenwald fraction $f_{GW}$ for the DB5.2.3-STD5 ELMy H-mode data. Reproduced from [950]. © EURATOM 2021. All rights reserved.</figcaption></figure>

e.g. regarding possible ‘hidden variables’ affecting the confinement (separatrix density, plasma rotation, etc), including profile effects. A distinction between ‘standard’ and ‘hybrid’ plasmas could also be made [974], either with two separate power laws or with a single, more flexible model.

Finally, it is hoped that some of the statistical techniques and practices, developed and applied over many years in the context of global confinement scaling, will find their use in uncertainty quantification (UQ) and model comparison involving other models of fusion plasma behavior, either semi-empirical (e.g. machine learning models) or theory-based.

## 9. Reduced core transport modeling

J. Citrin, Y. Camenen, G. Dif-Pradalier, R. McDermott, J. Rice and G. Staebler

This section describes advances in core turbulent transport models used within integrated tokamak modeling for physics interpretation of present-day experiments, scenario optimization, and extrapolation to future devices. The focus is on the models that apply the quasilinear approximation and have saturation rules calibrated by and compared with nonlinear gyrokinetic simulations. We review the latest updates
<figure><img src="image_46.png" /><figcaption>**Figure 43.** Normal approximation to the joint posterior distribution for the exponents in the engineering scaling in the DB5.2.3-STD5 ELMY H-mode ITER-like data set, including triangularity. Reproduced from [950]. © EURATOM 2021. All rights reserved.</figcaption></figure>

on the models and their underlying theory, provide validation examples in various regimes, review extrapolations to ITER and other devices, as well as present-day ‘Predict First’ integrated modeling. New frontiers are summarized, including advances in NN surrogate models, and challenges to quasilinear transport modeling based on insights from higher-fidelity simulation.

## 9.1. Introduction

One of the key roles of the ITPA TC Topical Group (T&C TG) is prediction of core confinement and detailed plasma core kinetic profile information for ITER scenarios, and applying theory-based predictions to support the ITER research plan. The L-mode, H-mode and hybrid H-mode regimes are the focus since these are of primary interest to ITER operations. Alpha particle transport and stability is the subject of the chapter on energetic particles and is not addressed here. The neoclassical transport of thermal species is less important in ITER because of the low collisionality and higher magnetic field compared to present day experiments. As a consequence, gyrokinetic turbulence plays an even larger role in the ion thermal transport of the hybrid regime of ITER. While nonlinear gyrokinetics provides an accurate framework to model tokamak turbulence, its computational burden precludes routine and extensive application for tokamak scenario studies. Such modeling demands time-dependent evaluation of particle, heat, and momentum (rotation) equations as well as current diffusion, together with all sources and sinks, collisional and turbulent transport, over multiple confinement times. Reduced-order physics models are thus necessary to

reduce the computational burden, while still remaining theory based. Even quasi-linear models of the turbulent fluxes are highly non-linear functions of the plasma profiles. In this chapter the focus is on equilibrium plasmas that are not highly self-organized, but self-organized states like internal transport barriers can be simulated with quasi-linear transport models [975, 976]. Even cold pulse inversion experiments, that were long thought to be beyond local transport models, have been successfully simulated with a stiff quasi-linear transport model [117, 977, 978].

The prevailing method for developing theory-based reduced-order tokamak turbulence models invokes the ‘quasi-linear’ approximation. The computational bottleneck is then the solution of the linear instability spectrum. Nonlinear physics is captured by analytical saturation rules, a crucial part of the transport model, aided by calibrations to databases of nonlinear gyrokinetic simulations. Transport fluxes from quasilinear models are up to six orders of magnitude faster to compute than local $\delta f$ nonlinear gyrokinetics, and enable more routine applications for interpreting transport phenomena in present-day experiments, extrapolating to future scenarios or devices, and machine design.

Significant efforts have taken place within the ITPA T&C TG in developing, validating, and applying quasilinear turbulence models. This subsection summarizes highlights of advances in the development and application of quasilinear turbulent transport models over the past ~15 years, with an emphasis on the models that rely on validation against local $\delta f$ nonlinear gyrokinetics. These concepts are further elucidated in section 9.2, which provides an overview of the advances in high fidelity core turbulent transport modeling, which anchor the reduced models. Section 9.3 summarizes the progress in
reduced turbulent model development, including a justification of the quasilinear approximation, updates on specific reduced models, and on validation and applications within integrated tokamak modeling. A discussion on open questions and new frontiers, including challenges to the underlying assumptions of local quasilinear models, based on higher-fidelity simulation, is provided in section 9.4. For additional recent reviews, see [273, 979].

## 9.2. Advances in core turbulent transport theory and simulation

This section briefly summarizes advances made in nonlinear tokamak turbulence simulations and validation thereof. We focus primarily on the gyrokinetic framework. However, we note that important advances have also been made in the gyrofluid nonlinear framework [980, 981], hybrid gyrofluid-gyrokinetic schemes [982], and projecting velocity space onto a Laguerre-Hermite basis, which with sufficient moments can interpolate between gyrofluid and gyrokinetic formulations [983].

Immense understanding has been achieved on the basic paradigms and mechanisms of nonlinear gyrokinetic turbulence. This has been achieved through the proliferation of verified numerical schemes and simulation frameworks for solving the gyrokinetic equations, numerical diagnostics for interpretation, the explosion of High Performance Computing (HPC) capacity to run the codes, and dedicated experimental validation against turbulence diagnostics and power balance inference. Justified by successful validation against experiments, nonlinear gyrokinetic simulations then serve as the ground truth against which the reduced models are calibrated and verified. The reduced models are an approximation to the gyrokinetic theory and can be used to validate the theory with experimental data. Discrepancies of the reduced model with the data can then be checked with nonlinear gyrokinetic simulations to determine if there is missing physics in the theory or just inaccuracy of the reduced model. This is a very efficient validation workflow that has built confidence in the reduced model predictions for a range of tokamak discharge conditions and has identified plasma conditions where the reduced models need improvement. Section 9.2.1 reviews the advances in the simulation frameworks, focusing on the *δf* local (flux-tube) approximation; see section 9.4 for an overview on the relaxation of this approximation. Section 9.2.2 reviews the validation of nonlinear simulations against turbulence measurements.

### 9.2.1. Advances in nonlinear GK simulation and theory.
The gyrokinetic theory is now on a sound physical and mathematical basis [163]. The development of the gyrokinetic field theory has in particular proven instrumental to establish the energy and momentum conservation properties of the system of equations to all orders in $\rho_*$, see e.g. [401] and references therein. We focus here on the *δf* local limit (flux-tube) of the gyrokinetic equations, which is the reference for the construction of reduced transport models used in tokamak simulation

frameworks. In this limit the computational expense, while sizeable, makes it feasible to generate nonlinear simulation databases with physics such as kinetic electrons, finite-$\beta$, and multiple ions species, for reduced transport model validation. This limit only retains terms up to first order in the small parameter $\rho_* = \rho_i/L_p$ where $\rho_i$ is the ion Larmor radius and $L_p$ is the equilibrium pressure gradient scale length. It assumes that the perturbed distribution function, $\delta f$, is small compared to the background distribution function $F$: $\delta f/F \sim \rho_* \ll 1$. In the simulations, the perturbed distribution function is evolved while the background is kept fixed (gradient-driven approach). The scale-length of the fluctuations in the direction perpendicular to the magnetic field is further assumed to be small compared to that of the background profiles and magnetic equilibrium. The local treatment typically converges to the global one, for which profile variations are retained, when $\rho_i/\Delta < 1/300$ with $\Delta$ the width of the strong gradient region expressed in Larmor radius units [984]. For smaller devices, e.g. with TCV parameters ($B=1.4$ T and $R=0.9$ m), the applicability of the local assumption becomes subtle and can be regime dependent, see e.g. [985]. A local treatment is appropriate for the majority of phenomena in core plasmas with ASDEX Upgrade $\rho_*$ or smaller ($B \sim 2.5$ T and $R=1.65$ m) but may be marginal in the pedestal region.

Due to the gradient-driven approach, the $\delta f$ local limit is challenged in conditions where meso-scale phenomena such as avalanches or staircases dominate the dynamics. In practice, the $\delta f$ local approach reaches its applicability limits for edge/pedestal turbulence and close to the turbulence threshold. These limitations are further discussed in section 9.4.

In the last 20 years, there have been many advances in numerical schemes and available computational power to solve the $\delta f$ local gyrokinetic system of equations with reduced assumptions and higher physics fidelity. Modern $\delta f$ flux-tube codes like CGYRO [309], GENE [310], GKV [986], GKW [30], GYRO [142] or GS2 [302] can now routinely run with an arbitrary number of kinetic species (including electrons), multi-species collisions, electromagnetic perturbations ($B_\|$ and $B_\perp$), sheared flows, toroidal rotation effects (Coriolis and centrifugal effects) and shaped magnetic equilibrium. A number of refinements have been introduced to improve the physics description in these codes: advanced collision operators [311, 987-989], poloidal variations of the background distribution function due to centrifugal effects [990], neoclassical correction to the background distribution function [468, 469], arbitrary background distribution functions for fast particles [284], profile shearing effects [434], and higher order parallel derivatives [991]. The last two effects in this list are formally next order in $\rho_*$ and an attempt to capture specific global effects relevant for momentum transport in a flux-tube treatment.

The extension of gyrokinetic code capabilities together with the large increase in HPC resources has driven large progress in the understanding of turbulent transport in tokamaks. Increasingly challenging regimes have been addressed. The interplay between ion Larmor radius (ITG/TEM) and electron Larmor radius scale (ETG) turbulence has been investigated in multi-scale simulations [364, 371, 375, 377, 992] confirming
that large scale eddies at ion-scale impact electron-scale turbulence and showing that, conversely, electron-scale turbulence can also damp ion-scale ZFs with a direct impact on the saturation level, as well as perturb trajectories of resonant trapped electrons, reducing the TEM turbulence level. These simulations, which are typically 100 times more expensive than ion-scale-only simulations, laid the basis for reduced models capturing multi-scale interactions [381]. nonlinear simulations of high performance plasmas, characterized by a high plasma beta and a large fast ion pressure, revealed the role of fast-ion-enhanced electromagnetic (EM) stabilization of ITG turbulence, related to an increase in relative ZF excitation [285, 316, 337, 344]. Progress in the understanding of micro-tearing turbulence, prone to occur at high beta and in spherical tokamaks, was achieved thanks to high radial resolution simulations, required to resolve the resistive layer at the resonant surface [993, 994]. Traditionally, $\delta f$ local flux-tube simulations focused on the so-called gradient region, typically $0.3 < r/a < 0.8$. Recently, there has been a growing effort to extend this domain towards the L-mode edge [830], pedestal [995–997] and the inner core [998].

### 9.2.2. Validation of nonlinear GK against turbulence measurements.
Experimental validation is a key justification for claiming that nonlinear gyrokinetic simulation is sufficient ground truth for guiding the development of reduced order models. In this regard, significant progress has been made, especially for drift-wave turbulence, in direct comparison of turbulence measurements against nonlinear gyrokinetic simulation synthetic diagnostics.

A multi-level validation hierarchy for model validation has been defined, depending on the degree of diagnostic processing required [999–1001]. The first level, meaning the most direct measurements, are fluctuation levels of electron density and temperature, and their power spectrum. The second level are cross-phases between fluctuating quantities (such as $\delta T_e - \delta n_e$) and turbulence correlation lengths, both requiring the processing of either multiple diagnostics or multiple measurement locations. The third level are the turbulent particle, momentum, and energy fluxes. These latter quantities, while the most indirect, are the most frequently compared since they do not depend on dedicated turbulence diagnostics. They require kinetic profile measurements, magnetic equilibrium reconstruction, and source calculations using dedicated models. These same measurements and calculations (including energetic particle calculations which can be source model output) are also used to determine the set of inputs for the gyrokinetic simulation. Measurement and model uncertainties propagate both into the gyrokinetic code output uncertainty, as well as the power balance uncertainty. See [317, 1002] as examples. Ideally, validation should be constrained by attempting simultaneous matching of multiple variables in the hierarchy. As an example, see [1003] for power balance, correlation length,

$\delta T_e/T_e$, and the $\delta n_e - \delta T_e$ cross-phase in AUG, where multi-variable matching was achieved apart from a systematic discrepancy in $\delta T_e/T_e$.

Limited examples of validation studies for various variables across the hierarchy are provided below, all for low-k (ion-scale) turbulence. For significantly more comprehensive reviews, see [1004, 1005] and references therein. Note that many of these examples, while cited to illustrate validation of specific diagnostic measurements, do incorporate multivariate validation.

BES can provide localized 2D electron density fluctuations, allowing comparison of absolute values of $\delta n_e/n_e$, as well as its spectral shape in both the $k_\theta$ and $k_r$ dimensions. Radial and poloidal correlation lengths can be constructed via processing of spatially separated pairs of signals. These quantities have all been compared successfully against GYRO simulations at DIII-D [1006]. These same quantities can also be obtained with Doppler and fast-sweeping reflectometers, leading to agreement between GYRO and Tore-Supra fluctuation measurements, across multiple radial positions [1007]. Frequency spectrum reflectometry was also compared to GENE simulations with a synthetic reflectometry diagnostic for Tore-Supra discharges across the LOC-SOC transition, leading to the agreement in the appearance of distinct markers in the reflectometry frequency spectra, identified as TEMs [1008, 1009].

The correlation electron cyclotron emission (CECE) diagnostic measures $\delta T_e$ fluctuations. Examples of comparison to gyrokinetic simulations include: at DIII-D [1010, 1011], showing good agreement particularly at mid-radius; at C-Mod, where measured relative changes in $\delta T_e$ and $\delta n_e$ across a LOC-SOC scan agree with simulation predictions of a turbulence regime shift, although for many cases the absolute level of $\delta T_e/T_e$ does not agree [991, 1012]; and at TCV reproducing the trends observed for positive and NT discharges [1013]. Radially resolved CECE allows for $\delta T_e/T_e$ correlation length measurements, as successfully carried out and compared with GENE simulations at AUG [1003]. See figure 44.

Spatial alignment and simultaneous measurement with reflectometer and ECE diagnostics can be used for electron density and temperature fluctuation cross-phase measurements. This is a direct experimental test of the validity of the quasilinear approximation for the cross-phase. This technique was developed on DIII-D [1010]. The cross-phase measurement on DIII-D observed trends consistent with crossing a predicted TEM critical threshold [1014]. The same technique applied to AUG L-mode low density electron heated cases found remarkable agreement between simulation and measurements [1003].

In DIII-D, GYRO runs with input parameters at mid-radius, within experimental uncertainty of the nominal parameters, led to simulations with transport fluxes consistent with power balance, and density fluctuations from a synthetic diagnostic in agreement with the measured electron density level [1015].