2.1. Radial coordinates

The $\texttt {raw}$ profiles and other calculated plasma profiles are cast as functions of the radial coordinate: either $R$ (the radial distance from the axis of JET-ILW) or $\psi _N$ (the poloidal-magnetic-flux label). The normalisation convention of $\psi _N$ and the radial shift applied to each pulse will be described below.

Due to uncertainties in the exact underlying equilibrium and in the location of the separatrix, the HRTS profiles are shifted radially to pin them at a specific prescribed value of the electron temperature at the separatrix,

(2.1) \begin{equation} T_e({R^{\mathrm{Sep}}})\equiv {T_{e}^{\mathrm{Sep}}}, \end{equation}

a relationship used to determine the location of the separatrix $R^{\mathrm{Sep}}$ . The same separatrix location applies to both the electron-temperature and density profiles. The value of $T_{e}^{\mathrm{Sep}}$ is estimated to be $100\;\mathrm{eV}$ in JET-ILW using the two-point model for heat flux, as described in Chapter 5.2 of Stangeby (Reference Stangeby2000). The two-point model is a simplified model that describes the transport of heat along the scrape-off layer (SOL) between a point on the divertor and a point in the outboard plasma midplane. In this context, the model treats the SOL as being in a dominantly conductive (as opposed to convective) regime, which is consistent with the usage of stationary inter-ELM plasma profiles. This model yields a scaling of the SOL heat flux $Q_e \propto T_e^{7/2}({R^{\mathrm{Sep}}})$ , which implies that $T_{e}^{\mathrm{Sep}}$ is very insensitive to changes in $Q_e$ . The EDGE2D simulations consistently show that $T_{e}^{\mathrm{Sep}}$ varies in the range $80{-}110\;\mathrm{eV}$ (Simpson et al. Reference Simpson, Moulton, Giroud, Groth and Corrigan2019). As shown by Frassinetti et al. (Reference Frassinetti2020), this uncertainty of the value of $T_{e}^{\mathrm{Sep}}$ has a negligible effect on the radial location of the separatrix as a result of the steepness of the profiles around $R = {R^{\mathrm{Sep}}}$ . This can also be generally inferred from the shapes of the electron-temperature profiles shown in figure 1(a).

The poloidal-magnetic-flux label $\psi _N$ is a normalised poloidal-magnetic-flux coordinate $\psi (R)$ . It is defined to be the poloidal-magnetic flux subtended between the magnetic axis and the radial position $R$ . Consistent with this definition, on the magnetic axis, $\psi _{\mathrm{axis}}=0$ ; and $ \psi (R)$ is obtained by reconstructing the equilibrium magnetic field using the MHD code HELENA (Huysmans et al. Reference Huysmans1999). For this, HELENA is provided with the plasma profiles and the shape of the plasma boundary reconstructed using EFIT (Lao et al. Reference Lao, John, Stambaugh, Kellman and Pfeiffer1985). Then $\psi _N$ is obtained by normalising $\psi (R)$ so that $\psi _N = 1.0$ at the radial location of the separatrix, viz.,

(2.2) \begin{equation} \psi _N = \frac {\psi (R)}{\psi ({R^{\mathrm{Sep}}})} . \end{equation}

2.2. mtanh fits

The pedestal profiles that are the subject of our analysis are functions fit to the measured raw profiles. The general form of the fits is

(2.3) \begin{equation} \mathtt {mtanh}(x,{h_{\mathrm{Ped}}},{s_{\mathrm{Core}}}) = \frac {{h_{\mathrm{Ped}}}}{2}\left [\frac {(1+x \; {s_{\mathrm{Core}}} )e^x - e^{-x}}{e^x+e^{-x}}+1 \right ], \end{equation}

where $x = ({\psi _{\mathrm{Pos}}}- \psi _N)/{w_{\mathrm{Ped}}}$ and all the free parameters will be defined below. Such mtanh fits are commonly used in many studies of H-mode physics and, thus, they offer the advantage of a generalised convention for pedestal profiles, while being differentiable fits with relatively few free parameters. These free parameters also represent formal definitions of several important properties of the pedestal. Namely, these are: the profile value at the top of the pedestal $h_{\mathrm{Ped}}$ ( $= {T_{e}^{\mathrm{Ped}}}$ and $={n_{e}^{\mathrm{Ped}}}$ for the temperature and density, respectively), the position of the middle of the pedestal $\psi _{\mathrm{Pos}}$ , the width of the pedestal $w_{\mathrm{Ped}}$ ( $={w_{{T_e,}\mathrm{Ped}}}$ and $={w_{{n_e,}\mathrm{Ped}}}$ ), and the position of the pedestal top ${\psi _{\mathrm{Top}}} = {\psi _{\mathrm{Pos}}} - {w_{\mathrm{Ped}}}/2$ ( $= {\psi _{{T_e,}\mathrm{Top}}}$ and $= {\psi _{{n_e,}\mathrm{Top}}}$ ). The core slope to which the profile tends at $\psi _N \ll {\psi _{\mathrm{Top}}}$ is determined by ${h_{\mathrm{Ped}}} {s_{\mathrm{Core}}} / 2 {w_{\mathrm{Ped}}}$ , parametrised by $s_{\mathrm{Core}}$ . The profiles derived using an mtanh fit, as per (2.3), will be referred to as $\texttt {fit}$ profiles, and they represent the nominal plasma profiles used for the EUROfusion database (Frassinetti et al. Reference Frassinetti2020).

Importantly, the fits are performed taking into account the finite radial resolution of the HRTS diagnostic, as described by Frassinetti et al. (Reference Frassinetti, Beurskens, Scannell, Osborne, Flanagan, Kempenaars, Maslov, Pasqualotto and Walsh2012). Therefore, each fit profile is computed from the raw profile using deconvolved fits. For this, the fit electron-density profile is obtained by minimising the residual between the raw $n_e$ values and the fit $n_e$ profile, convolved with the spatial instrument function of the HRTS diagnostic. Then, the fit electron-temperature profile is obtained by minimising the residual between the raw $T_e$ values and the fit $T_e$ profile weighed by the fit $n_e$ profile and convolved with the instrument function of the HRTS diagnostic. This is because the density profiles affect the inference of the temperature from the HRTS data. The instrument function of HRTS represents a ‘scattering kernel’ with a full-width-half-maximum of approximately $11\;\mathrm{mm}$ in the pedestal region. The scatter is caused by the HRTS diagnostic imaging a finite volume of the plasma around each measurement location, so the variations of the profile across this volume must be taken into account.

2.3. Gradients

Fitting the $\texttt {raw}$ profiles to a differentiable function allows for gradients to be meaningfully calculated. The gradients are an essential ingredient for studying the local quasi-equilibrium that the plasma settles to during the inter-ELM period. They must be related to the heat flux crossing the pedestal because they provide the drive for the turbulence that controls transport across the pedestal. It is important to acknowledge that the choice of an mtanh fit is a source of systematic bias in estimating the gradients of the profiles, however, this fit has been used to characterise the properties of pedestal shapes across devices for more than 20 years, which helps make our study easily generalisable to other databases.

The gradients of the $T_e$ and $n_e$ profiles are calculated with respect to the radial coordinate $R$ . For any quantity $A(R)$ , we define

(2.4) \begin{equation} L_A = A(R) \left ({\frac {\mathrm{d}A}{\mathrm{d}R}}\right )^{-1} \end{equation}

as the corresponding gradient scale length.

Figure 1 is an overview of pulse $\#90339$ : the plasma profiles (panels aandb), the value of the gyro-Bohm heat flux (shown in panel c and defined below), the profile gradients (panels d and e) and the gradient-length-scale ratio $\eta _e = {L_{n_e}} / {L_{T_e}}$ (panel f). The radial gradients $\mathrm{d}T_e/\mathrm{d}R$ and $\mathrm{d}n_e/\mathrm{d}R$ in figure 1(d) are normalised to the pedestal-top values $T_{e}^{\mathrm{Ped}}$ and $n_{e}^{\mathrm{Ped}}$ , respectively. This normalisation is done in order to show the qualitative behaviour of the raw gradients. For our analysis throughout this study, we will focus on the gradient length scales $L_{T_e}$ and $L_{n_e}$ , defined in (2.4). The most important trends manifest in figure 1 are the increase by approximately two orders of magnitude of the normalised gradients $R/{L_{T_e}}$ and $R/{L_{n_e}}$ towards the separatrix, and the typical values of $\eta _e = \mathcal{O}(1) - \mathcal{O}(10)$ . The finite values of $R/{L_{T_e}}$ and $R/{L_{n_e}}$ outside the separatrix region is a result of the $\texttt {mtanh}$ convention that all $\texttt {fit}$ profiles tend to zero in the SOL. These tendencies of the gradients of pulse $\#90339$ are qualitatively representative of the behaviour of most pulses across the database.

The gyro-Bohm heat flux is defined as

(2.5) \begin{equation} {Q_{e,\mathrm{gB}}} = n_e T_e {v_{\mathrm{th},e}} \frac {\rho _e^2 }{R^2}, \end{equation}

where $ {v_{\mathrm{th},e}} = \sqrt {2 T_e / m_e}$ is the electron thermal velocity, $\rho _e = m_e {v_{\mathrm{th},e}}/eB$ is the electron gyroradius, $m_e$ is the electron mass, $e$ is the electron charge, and $B$ is the magnitude of the magnetic field. The gyro-Bohm heat flux (2.5) maintains values of $\mathcal{O}(\mathrm{MW}\,\mathrm{m}^{-2})$ across the profiles inside of the half-way point between the density-pedestal top and and separatrix (defined as the `steep-gradient point' in § 2.4); $ {Q_{e,\mathrm{gB}}}$ decreases rapidly towards the separatrix, approaching $0$ together with the $T_e$ and $n_e$ profiles.

2.4. Pedestal regions and relevant locations

It is useful to define formally some important radial locations and regions for our profiles. Using the parameters of the $\texttt {mtanh}$ fits in $\psi _N$ coordinates, the pedestal-top positions of $T_e$ and $n_e$ are $\psi _{{T_e,}\mathrm{Top}}$ and $\psi _{{n_e,}\mathrm{Top}}$ , respectively. These pedestal-top positions are defined as

(2.6) \begin{equation} {\psi _{\mathrm{Top}}} = {\psi _{\mathrm{Pos}}} - \frac {{w_{\mathrm{Ped}}}}{2}. \end{equation}

We refer to the region $\psi _N \lt {\psi _{{T_e,}\mathrm{Top}}}$ as the core of the plasma. This definition with respect to $\psi _{{T_e,}\mathrm{Top}}$ holds across the database as a result of the relative shift between the tops of the $n_e$ and $T_e$ pedestals (Frassinetti et al. Reference Frassinetti2021): the top of the density pedestal is radially further out than the top of the temperature pedestal. Hence, the low temperature gradients in the regions within $\psi _{{T_e,}\mathrm{Top}}$ guarantee low electron-density gradients. We will also use this pedestal-shift property in § 2.5: the separatrix position will be conventionally placed at $\psi _N={\psi _{\mathrm{Sep}}}=1.0$ such that $T_e = 100 \; \mathrm{eV}$ as per the normalisation described in § 2.1, and, consequently, the region of $\psi _N\gt 1.0$ is referred to as the SOL.

The region between the separatrix and $\psi _{{T_e,}\mathrm{Top}}$ is the pedestal. Because most of the pulses obey ${\psi _{{T_e,}\mathrm{Top}}}\lt {\psi _{{n_e,}\mathrm{Top}}}\lt 1.0$ , the interval between $\psi _{{n_e,}\mathrm{Top}}$ and $1.0$ will be referred to as the steep-gradient region, since both the electron-temperature and density gradients are very large there. In the interval between $\psi _{{T_e,}\mathrm{Top}}$ and $\psi _{{n_e,}\mathrm{Top}}$ , the electron-density gradient is much smaller, which justifies making a qualitative distinction between these two neighbouring regions of the pedestal. We will identify this distinction again in § 5, and we will discuss the physical difference between the two regions in the sections thereafter.

The profile values at the top of the $T_e$ pedestal are $T_{e}^{\mathrm{Ped}}$ and $n_{e}({\psi _{{T_e,}\mathrm{Top}}})$ ; at the top of the $n_e$ pedestal they are $T_{e}({\psi _{{n_e,}\mathrm{Top}}})$ and $n_{e}^{\mathrm{Ped}}$ . These pedestal-top points (large dots in figure 1 a,b) are the locations where the gradient scale lengths change from the high-gradient regime of the pedestal to the low-gradient regime characteristic of the core. As a representative location in the steep-gradient region, we define the ‘steep-gradient point’ location as

(2.7) \begin{equation} {\psi _{\mathrm{Steep}}} = \frac {{\psi _{{n_e,}\mathrm{Top}}}+1}{2} , \end{equation}

i.e., half-way between the electron-density top and the separatrix. This location will be considered representative of the steepest region of the pedestal, as it will be far enough away from $\psi _{{n_e,}\mathrm{Top}}$ and $\psi _{{T_e,}\mathrm{Top}}$ to not be influenced by core turbulence. Further out from the steep-gradient point, towards (or beyond) the separatrix, the uncertainties of $R/{L_{n_e}}$ and $R/{L_{T_e}}$ become comparable to the values of the gradients themselves, as obtained by uncorrelated variation of the $\texttt {mtanh}$ parameters within their confidence intervals (Field et al. Reference Field, Chapman-Oplopoiou, Connor, Frassinetti, Hatch, Roach and Saarelma2023). This is because the normalisation of the gradients against the asymptotically vanishing values of $n_e$ and $T_e$ in the SOL results in diverging errors. We shall, therefore, use the steep-gradient point as indicative of the pedestal’s steep-gradient region.

2.5. Rescaled radial coordinate

In order to group together qualitatively similar regimes of the pedestal, we use the landmarks that we defined above, i.e., the separatrix position $\psi _{\mathrm{Sep}}$ , the density-pedestal-top position $\psi _{{n_e,}\mathrm{Top}}$ and the temperature-pedestal-top position $\psi _{{T_e,}\mathrm{Top}}$ , to define a new ‘renormalised’ radial coordinate $X_{\mathrm{Ren}}$ . This will map each point in the density-top and steep-gradient regions to a value $X_{\mathrm{Ren}}$ that depends on its position relative to these landmarks.

Namely, a point whose flux label is $\psi _N \in [\min ({\psi _{{T_e,}\mathrm{Top}}},{\psi _{{n_e,}\mathrm{Top}}}), 1 ]$ will have the radial coordinate ${X_{\mathrm{Ren}}} \in [ 0, 2 ]$ defined by

(2.8) \begin{equation} {X_{\mathrm{Ren}}} \left (\psi _N\right ) = \begin{cases} \dfrac {\psi _N - 1}{{\psi _{{n_e,}\mathrm{Top}}} - 1} & \text{if } 1 \geqslant \psi _N \geqslant {\psi _{{n_e,}\mathrm{Top}}}, \\[15pt] 1 + \dfrac {\psi _N - {\psi _{{n_e,}\mathrm{Top}}}}{{\psi _{{T_e,}\mathrm{Top}}} - {\psi _{{n_e,}\mathrm{Top}}}} & \text{if } {\psi _{{T_e,}\mathrm{Top}}} \geqslant \psi _N \gt {\psi _{{n_e,}\mathrm{Top}}}. \end{cases} \end{equation}

This is a piecewise linear transformation that maps $\psi _{\mathrm{Sep}}$ to $0$ , $\psi _{{n_e,}\mathrm{Top}}$ to $1$ , $\psi _{{T_e,}\mathrm{Top}}$ to $2$ , and $\psi _{\mathrm{Steep}}$ to $0.5$ . Because of the relative shift between the pedestal-top locations in the database (Frassinetti et al. Reference Frassinetti2020), all but $22$ pedestals obey the pedestal-shift condition $({\psi _{{n_e,}\mathrm{Top}}} \gt {\psi _{{T_e,}\mathrm{Top}}})$ in the subset relevant to this study, as per § 3.1. For the pedestals that do not obey this condition, the map (2.8) will place the entire pedestal profile in the $[ 0, 1 ]$ interval. The map (2.8) will be used in §§ 6 and 7 to uncover the overall structure of pedestal gradients.

More complex functional forms of $X_{\mathrm{Ren}}$ that maintain the separation of $\psi _{\mathrm{Sep}}$ , $\psi _{{n_e,}\mathrm{Top}}$ , and $\psi _{{T_e,}\mathrm{Top}}$ , but are also, e.g., differentiable at ${X_{\mathrm{Ren}}} = 1$ , offer little advantage while being more cumbersome.

3.1. Data selection and ELMs

In order to contextualise the data representing our H-mode pulses, it is useful to understand how the pedestals in the database are measured and how they relate to the stability of ELMs.

The pedestal profiles are constrained by the peeling–ballooning MHD stability limit, which depends on the pressure gradient and the current density (Wilson et al. Reference Wilson, Connor, Field, Fielding, Miller, Lao, Ferron and Turnbull1999). Edge-localised modes are thought to be triggered when a plasma profile strays past this stability limit, and they result in a sudden reduction of the energy density on spatial scales larger than the size of the pedestal (Snyder et al. Reference Snyder, Wilson, Ferron, Lao, Leonard, Mossessian, Murakami, Osborne, Turnbull and Xu2004). An ‘ELM cycle’ contains the ELM itself and the recovery period of the plasma profile until the occurrence of the next ELM. The classification of ELMs depends on their frequency, associated energy loss and on whether the pedestal is peeling-unstable or ballooning-unstable (Connor, Kirk & Wilson Reference Connor, Kirk and Wilson2008; Murari et al. Reference Murari, Pisano, Vega, Cannas, Fanni, Gonzalez, Gelfusa and Grosso2014). ‘Giant’ type-I ELMs are those occurring throughout the pulses in our database.

The EUROfusion pedestal database contains data from type-I ELMy H-mode pulses in JET, and the pedestal profiles are taken from measurements from the last $20\,\%$ of each ELM cycle, as described in § 2. This choice ensures that the pedestal profiles are in near-steady state over the course of the measurement, and that they represent optimal operation of the plasma, just below the ELM stability threshold.

While the EUROfusion pedestal database contains a wide range of pulses, we will narrow the scope in order to limit the variation of the underlying MHD instability of the plasma. We will focus on deuterium-only plasmas (so the effective ion mass will be $M_{\mathrm{eff}}=2$ atomic units). Furthermore, methods of ELM control or mitigation affect the equilibrium of the plasma, meaning that plasma pressure is reduced to a subcritical, ELM-stable regime. Therefore, we chose pulses that had no methods of ELM control or mitigation applied, such as impurity seeding (added impurities producing a regime with high-frequency, low-amplitude ELMs), pellet injection (a method of plasma fuelling inside the ETB, triggering ELMs) or vertical kicks (vertical displacements of the plasma MHD equilibrium, also triggering ELMs).

This means that, for the current analysis, we will work with a subset of the database containing $1251$ pulses.

3.2. Parameter values

The database also contains a wide range of experimental parameters, so we limit ourselves to a physically relevant subset of engineering parameters and magnetic-equilibrium-reconstruction parameters. The list of these parameters is provided in Appendix A.1 and an overview of their distribution is given in figure 2.

Figure 2.

Histograms of the distributions of the engineering and equilibrium parameters described in Appendix A.1. The number of pulses with each strike-point configuration is also shown. The strike points are formatted as ‘inner/outer’, as explained in Appendix A.1.

Figure 3.

Histograms of the distributions of relevant $T_e$ and $n_e$ profile values. Their definitions are given in § 2.2.

Because of the wide scatter in the values of these parameters, we anticipate the necessity for data augmentation (Appendix B.2) for effective training of $T_e$ -predicting neural networks in § 4. We note that the poloidal beta, $\beta _{\theta }$ , and the electron-electron collisionality, $\nu ^*$ , are physically important in determining the nature of local turbulence in the pedestal; however, the calculation of these parameters directly references $T_{e}^{\mathrm{Ped}}$ , as per Frassinetti et al. (Reference Frassinetti2020). This becomes an issue as the neural networks of § 4 manage to trivially invert the calculation of these parameters. This results in excellent $T_e$ predictions with a priori knowledge of the $T_e$ pedestal. The inclusion of such information into our neural networks defeats the point of our study, although they are worth mentioning amongst the parameters that are usually considered important for characterising the physical environment of a pedestal.

Some of these parameters will also be used in § 8 and Appendix D for predictions using heat-flux scalings and for the calculation of $ {Q_{e,\mathrm{gB}}}$ .

For our analysis of the plasma profiles themselves, we will make use of the $\texttt {raw}$ and $\texttt {fit}$ profiles described in § 2. These profiles are not normally a part of the EUROfusion database, so we sourced them directly from processed pulse files (PPFs) for each pulse. We have access to the electron-density and temperature profiles and the corresponding $R$ and $\psi _N$ for each pulse, as described in § 2.

The distribution of $\texttt {mtanh}$ parameters (defined in § 2.2) and other relevant values of the $T_e$ and $n_e$ profiles are shown in figure 3. The value of the temperature at the density-pedestal top, $T_e({\psi _{{n_e,}\mathrm{Top}}})$ , will be an important measure of the accuracy of our temperature-pedestal reconstruction over the steep-gradient region. The density at the separatrix, ${n_{e}^{\mathrm{Sep}}} = n_e(\psi _N = 1.0)$ , and the ratio ${n_{e}^{\mathrm{Sep}}} / {n_{e}^{\mathrm{Ped}}}$ between the separatrix density and the pedestal-top density are strongly correlated with temperature-pedestal parameters, as was anticipated in Field et al. (Reference Field, Chapman-Oplopoiou, Connor, Frassinetti, Hatch, Roach and Saarelma2023) and will be confirmed in figure 22.

4.1. Full-profile electron-temperature predictions

We first make predictions of the full temperature-pedestal profile using the information from the full density-pedestal profile and various database parameters. Randomly located radial cropping is used for the $n_e$ profile during augmentation to reduce bias, ensuring that networks remain independent of specific radial data points while preserving overall profile features. The random crop keeps $80\,\%$ of the pedestal information, i.e., $L$ is close to the full-pedestal extent, as detailed in figure 23 of Appendix B.2.

Four groups of parameters are defined for this comparison (found under the heading ‘Full-pedestal groups’ in table 1). Figure 4 illustrates the prediction results across these parameter groups, showing the mtanh parameters that describe the reconstructed $T_e$ pedestals for the $251$ pulses in the test set. The results in figure 4 correspond to the groups of parameters listed in table 1 as follows:

1. (a-d) ‘None’ – only the radial coordinate and $n_e$ profile, with no additional pulse parameters;

2. (e-h) ‘No-EFIT’ – $R$ and $n_e$ profiles and engineering parameters only, excluding magnetic-equilibrium-reconstruction parameters;

3. (i-l) ‘EFIT’ – $R$ and $n_e$ profiles and magnetic-equilibrium-reconstruction parameters, excluding engineering parameters;

4. (m-p) ‘All Parameters’ – all available engineering and EFIT parameters alongside $R$ and $n_e$ profiles.

Figure 4 and similar diagnostic plots in what follows display the coefficient of determination

(4.1) \begin{equation} r^2 = 1- \frac {S_{\mathrm{res}}}{S_{\mathrm{tot}}}, \end{equation}

where, for a diagnostic $y \in \left \{ T_e({\psi _{{n_e,}\mathrm{Top}}}), {T_{e}^{\mathrm{Ped}}},{w_{{T_e,}\mathrm{Ped}}},{\psi _{{T_e,}\mathrm{Top}}}\right \}$ , $S_{\mathrm{res}} = \sum (y_{\mathrm{meas}}-y_{\mathrm{pred}} )^2$ is the sum of the prediction residual, $ S_{\mathrm{tot}} = \sum (y_{\mathrm{meas}}-\langle y_{\mathrm{meas}} \rangle )^2$ is the total variance of the measured data, and $\langle y_{\mathrm{meas}} \rangle =(1/N)\sum y_{\mathrm{meas}}$ is the average experimental value. Here, $y_{\mathrm{meas}}$ is the experimentally measured value of a diagnostic $y$ and $y_{\mathrm{pred}}$ is its predicted value. A value of $r^2$ close to $1$ suggests an excellent reconstruction, low values suggest a poor agreement. These values serve as our measure of agreement between the experimental and predicted $T_e$ profiles, and, in particular, they represent a direct measure of profile goodness-of-fit for $T_e({\psi _{{n_e,}\mathrm{Top}}})$ because $\psi _{{n_e,}\mathrm{Top}}$ is common for both profiles.

The ‘None’ group yielded largely random predictions of the $T_e$ -pedestal’s width $w_{{T_e,}\mathrm{Ped}}$ and top $T_{e}^{\mathrm{Ped}}$ , demonstrating insufficient predictive power without parameter input.

The ‘No-EFIT’ group performed significantly better, with marked improvements in the predictions of $T_e({\psi _{{n_e,}\mathrm{Top}}})$ , $T_{e}^{\mathrm{Ped}}$ , and $\psi _{{T_e,}\mathrm{Top}}$ , suggesting that engineering parameters are highly relevant to the $T_e$ profile.

The ‘EFIT’ group showed enhanced accuracy of the prediction of the pedestal width $w_{{T_e,}\mathrm{Ped}}$ , and comparable performance in predicting $T_e({\psi _{{n_e,}\mathrm{Top}}})$ and $T_{e}^{\mathrm{Ped}}$ as the ‘No-EFIT’ group. However, this group predicts the pedestal-top location $\psi _{{T_e,}\mathrm{Top}}$ more poorly compared with the ‘No-EFIT’ group.

The ‘All Parameters’ group provided the most accurate predictions, achieving close alignment with experimental profiles across all our metrics for pedestal amplitudes and position. This is confirmed by the high $r^2$ values for the $T_e({\psi _{{n_e,}\mathrm{Top}}})$ , $T_{e}^{\mathrm{Ped}}$ , and $\psi _{{T_e,}\mathrm{Top}}$ reconstructions. The prediction of $w_{{T_e,}\mathrm{Ped}}$ also improved, compared with the ‘None’ and ‘No-EFIT’ groups, underscoring the complementary roles of EFIT and engineering parameters in predicting $T_e$ profiles.

Examples of these predictions are shown in figure 5 for pulse $\#90339$ . The outputs of the models using the ‘All Parameters’, ‘No-EFIT’, and ‘EFIT’ groups closely follow the experimental $T_e$ and $R/{L_{T_e}}$ profiles.

4.2. Local electron-temperature predictions

We further explore the $T_e$ predictions based on local values of the density $n_e$ and normalised density gradient $R/{L_{n_e}}$ within a narrow radial window. This approach emulates predictions using physical models that use local parameters within the pedestal. Here, neural networks are trained to predict $T_e$ values over narrow radial intervals, using the density information at these positions. The radial extent used at each location is $L = 2.5\,\%$ of the sampled radial interval (Appendix B.2). By choosing many randomly positioned samples, the full $T_e$ profile is then assembled from local predictions.

Four parameter groups are defined under the heading ‘Local-value groups’ in table 1. The resulting mtanh parameters of these reconstructions are given in figure 6, arranged as follows:

1. (a-d) ‘None’ – only the radial coordinate $R$ , $n_e$ , and $R/{L_{n_e}}$ profiles, with no additional pulse parameters;

2. (e-h) ‘No-EFIT’ – engineering parameters and $R$ , $n_e$ , and $R/{L_{n_e}}$ profiles, excluding magnetic-equilibrium-reconstruction parameters;

3. (i-l) ‘No- $R$ ’ – all parameters, $n_e$ , and $R/{L_{n_e}}$ , but excluding the radial coordinate $R$ ;

4. (m-p) ‘All Parameters’ – all available engineering and EFIT parameters, and profiles including the radial coordinate $R$ .

Figure 6.

Database predictions using neural networks informed of the local values of $n_e$ and $R/{L_{n_e}}$ . These correspond to each of the four groups of parameters in table 1: (a--d) no database parameters used as inputs, with the exception of $R$ and $n_e$ ; (e--h) only engineering parameters alongside $R$ and $n_e$ ; (i--l) all available parameters and the $n_e$ values, omitting the radial location $R$ ; (m--p) all available parameters used, alongside $R$ and $n_e$ . The colour bar represents the mean-square error between the predicted $T_e$ and the experimental fit profile. The plots follow the same layout as in figure 4.

Figure 7.

Pulse $\#90339$ $\texttt {fit}$ profiles, compared with neural-network local predictions of the $T_e$ values. The layout is the same as in figures 1 and 5. The $\texttt {fit}$ profiles are shown as solid lines, the predictions with different parameter groups listed in table 1 are shown as dash-dotted (‘None’), dashed (‘All Parameters’), and dotted (‘No- $R$ ’) lines. The vertical lines and dots represent the landmarks of the $\texttt {fit}$ profiles (see § 2.4 and figure 1), the crosses represent the pedestal top of the $T_e$ predictions. The full predicted $T_e$ profiles are a result of assembling many local predictions of $T_e$ values at different locations.

The ‘None’ group yields near-constant pedestals, reflecting the failure of local $n_e$ values alone in capturing pedestal variation over our database. This is a case where insufficient information is provided, resulting in the network yielding an output with little variation. The parameters for the $\texttt {mtanh}$ fits have large negative values, symptomatic of this strategy’s inability to reconstruct the pedestal shape using local predictions with no added parameters.

The ‘No-EFIT’ group enhances predictions of $T_e({\psi _{{n_e,}\mathrm{Top}}})$ , $T_{e}^{\mathrm{Ped}}$ , and $\psi _{{T_e,}\mathrm{Top}}$ with a strong correlation with the experimental values, although the predictions of the pedestal width remain inaccurate.

The ‘No- $R$ ’ group demonstrates the critical role of the radial position $R$ for precise pedestal predictions; without $R$ , $T_{e}^{\mathrm{Ped}}$ is over-predicted by a factor of about $1.5$ , and the pedestal shapes are predicted poorly, indicating that $R$ could act as a proxy for larger-scale spatial effects. This means that the local shape of the density profile alone is not sufficient to predict how tall the $T_e$ profile must be in that location.

The ‘All Parameters’ group achieves high accuracy across all metrics, comparable with the results using the full-pedestal profile in § 4.1. The pedestal width, however, is reproduced with a large inaccuracy, suggesting that $w_{{T_e,}\mathrm{Ped}}$ is linked to global properties of the $n_e$ profile. This link was also noticed by Saarelma et al. (Reference Saarelma, Connor, Bilkova, Bohm, Field, Frassinetti, Fridstrom and Kirk2023, Reference Saarelma2024), who found $w_{{n_e,}\mathrm{Ped}}$ and $n_{e}^{\mathrm{Ped}}$ to be related to the $T_e$ profile based on the penetration of neutrals and their ionisation through the pedestal region.

Examples of these predictions for $\#90339$ are presented in figure 7, showing the effectiveness of local predictions with full parameter inclusion, and a high scatter in the temperature gradient of the reassembled profiles. Overall, the local profile predictions provide slightly poorer pedestal reconstructions than the global ones of § 4.1, and these results are generally far more easily degraded by omitting information.

4.3. Assessment of most important database parameters

Figure 8.

Normalised mean-square error of neural-network predictions of (a) full $T_e$ profiles and (b) local $T_e$ profiles, averaged over combinations of $N$ parameters that include each given parameter (horizontal axis). The normalisation is with respect to the mean-square error of the all-parameter networks, shown as the black dashed line at the value of 1 (the all-parameter networks are shown in figure 4 m–p for full-profile predictions and in figure 6 m–p for local-value predictions). The bright bars indicate the test-set errors, while the pale bars represent the training-set errors. The red bars show the error when only the single parameter marked on the horizontal axis is used. The pink bars display the average error when the parameter is used in two-parameter combinations, with further bars showing averages across combinations of more parameters. The parameters are arranged left to right in ascending order of the single-parameter test error, demonstrating the relative importance of each parameter for accurate full-profile predictions.

For a quantitative assessment of parameter importance, we measure and compare the prediction accuracy of $T_e$ profiles using various parameter combinations as inputs. We scan sequentially through combinations of $N$ parameters from table 1 and identify those that correlate most with the success of the $T_e$ -pedestal reconstruction.

Because of the difference in the sensitivity to parameter inclusion and in the performance of full-profile and local-value predictions, we scan parameters for each case. The results of these parameter scans are presented in figure 8, where we rank the performance of networks that use $N$ parameters as inputs, with $N$ indicated by different colours, as shown in the legends. The diagrams present the average performance for each parameter on the horizontal axis, when $N-1$ other parameters are included. Our metric for the performance of each network is the total mean-square error across all pulses, normalised to the error obtained from the respective ‘All Parameter’ groups from §§ 4.1 and 4.2. The error for any given network is defined as

(4.2) \begin{align} \mathcal{E} = \sum _{\mathrm{Pulses}} \sum _{i}\left [T_{\texttt {fit} }(R_i) - T_{\mathrm{Pred}}(R_i)\right ]^2, \end{align}

where $\{R_i\}$ denote the evenly spaced radial locations where the profiles are sampled numerically, $T_{\texttt {fit} }(R_i)$ is the experimental fit profile, and $T_{\mathrm{Pred}}(R_i)$ is the predicted value at each location. Bright colours represent the sum over the test pulses, pale colours represent the training pulses.

For full-profile predictions (figure 8 a), for $N \geqslant 2$ , the deuterium gas fuelling rate $\varGamma _{D}$ becomes one of the highest-performance parameters in spite of its low importance in the $N=1$ group. Overall, the inclusion of $\varGamma _{D}$ , the separatrix loss power $ {P_{\mathrm{Sep}}}$ , the plasma current $ {I_{\mathrm{Plasma}}}$ , and the strike-point configuration has proved essential for maximising the accuracy of neural networks with $N\geqslant 4$ . Furthermore, the magnetic-equilibrium parameters have less impact than the engineering parameters, suggesting that there is redundancy in the system: the $n_e$ profile contains sufficient information to infer some features of the MHD equilibrium.

The importance of the strike-point configuration and $\varGamma _{D}$ in the full-profile models suggests that changing the magnitude and/or location of the neutral ionisation source (and also the efficiency of the pumping) and, hence, the particle flux through the pedestal, might affect the nature of the turbulence, and, consequently, the steady-state $T_e$ profile that is consistent with this changed transport. One known pathway through which the information about $T_e$ can be inferred lies in the ratio ${n_{e}^{\mathrm{Sep}}}/{n_{e}^{\mathrm{Ped}}}$ , which is not contained in the $n_e$ profiles: profiles with a higher ${n_{e}^{\mathrm{Sep}}}/{n_{e}^{\mathrm{Ped}}}$ have a higher mean $\eta _e$ (figure 13) and are, therefore, more turbulent (§ 8), causing lower $T_{e}^{\mathrm{Ped}}$ values; in turn, ${n_{e}^{\mathrm{Sep}}}/{n_{e}^{\mathrm{Ped}}}$ is strongly related to $\varGamma _{D}$ (Frassinetti et al. Reference Frassinetti2021) and to the strike-point configuration.

For local predictions (figure 8 b), the radial coordinate $R$ has proved to be the most important parameter across all metrics used, consistent with the findings of § 4.2. The inclusion of $R$ and the magnitude of the magnetic field $B$ on the axis has turned out to be the most effective two-parameter choice, however, for $N\geqslant 3$ , $ {P_{\mathrm{Sep}}}$ , $\varGamma _{D}$ , $ {I_{\mathrm{Plasma}}}$ , and the strike-point configuration have proved to be the best-performing set. The importance of the strike-point configuration and $\varGamma _{D}$ in this case is unsurprising, as they are essential for the formation of the $n_e$ pedestal (Frassinetti et al. Reference Frassinetti2020; Saarelma et al. Reference Saarelma, Connor, Bilkova, Bohm, Field, Frassinetti, Fridstrom and Kirk2023, Reference Saarelma2024), supplying information that would be contained in the full shape of the $n_e$ .

Overall, these scans of parameter combinations suggest that the $ {P_{\mathrm{Sep}}}$ , $ {I_{\mathrm{Plasma}}}$ , $\varGamma _{D}$ , and the strike points are the parameters strongest correlated to the correct prediction of the $T_e$ -pedestal shapes. The difference between the ‘All Parameter’ groups in the full-profile and local predictions suggests that successful predictions rely on both global and local plasma characteristics. These global characteristics are contained in the full geometry of the $n_e$ profile, and omitting this information degrades the quality of the predictions.

To conclude, the ML approach has proved quite successful in reconstructing the temperature pedestal given the density profile alongside other available parameters. Each of the eight neural networks presented in §§ 4.1 and 4.2 correspond to entries 39–46 in table 3, where their performances are compared with the other results in this study, stressing the success of such methods. Besides the practical importance of this success for modelling endeavours aiming to help design and operate future devices, the success of our local methods of pedestal reconstruction in § 4.2 also suggests that the search for a physics-based model of local transport is not a hopeless enterprise.

5.1. Instabilities and ETG turbulence

Figure 9.

Loci of the data from all pulses in terms of $({R/{L_{n_e}}}, {R/{L_{T_e}}})$ coordinates: panels (a,c) contain the subsets of each profile spanning the steep-gradient region; panels (b,d) contain profiles between the $T_e$ -pedestal top and the $n_e$ -pedestal top; panels (a,b) contain all pulses considered by us, with their position in $\psi _N$ used to colour code the points; the colour in panels (c,d) shows the number of pulses where profiles cross each bin in the $( {R/{L_{n_e}}}, {R/{L_{T_e}}})$ space. All data points lie above the slab-ETG stability threshold of ${R/{L_{T_e}}} = 0.8 {R/{L_{n_e}}}$ (Jenko, Dorland & Hammett Reference Jenko, Dorland and Hammett2001), depicted as the lower blue dash-dotted line in (a,b). The upper blue line represents ${R/{L_{T_e}}} = 1.2 {R/{L_{n_e}}}$ , as explained in § 5.1.

Besides the density and temperature gradients, which can trigger instabilities, the pedestal region features a large $\boldsymbol{E} \times \boldsymbol{B}$ flow shear. This has a suppressive effect on microinstabilities (Zhang & Mahajan Reference Zhang and Mahajan1992; Hahm & Burrell Reference Hahm and Burrell1995; Terry Reference Terry2000), quenching those of them that have growth rates smaller than the shear rate. Amongst the instabilities that are fast enough to overcome this suppression, the most relevant ones are modes with electron-scale radial extent, which cause heat transport in the electron channel (Told et al. Reference Told, Jenko, Xanthopoulos, Horton and Wolfrum2008; Jenko et al. Reference Jenko, Told, Xanthopoulos, Merz and Horton2009; Hatch et al. Reference Hatch, Told, Jenko, Doerk, Dunne, Wolfrum, Viezzer and Pueschel2015, Reference Hatch, Kotschenreuther, Mahajan, Valanju, Jenko, Told, Görler and Saarelma2016, Reference Hatch, Kotschenreuther, Mahajan, Valanju and Liu2017; Kotschenreuther et al. Reference Kotschenreuther, Hatch, Mahajan, Valanju, Zheng and Liu2017, Reference Kotschenreuther2019). Namely, these are MTMs (see Drake & Lee Reference Drake and Lee1977; Drake et al. Reference Drake, Gladd, Liu and Chang1980; Applegate et al. Reference Applegate, Roach, Connor, Cowley, Dorland, Hastie and Joiner2007; Guttenfelder et al. Reference Guttenfelder2012; Zocco et al. Reference Zocco, Loureiro, Dickinson, Numata and Roach2015; Larakers et al. Reference Larakers, Hazeltine and Mahajan2020, Reference Larakers, Curie, Hatch, Hazeltine and Mahajan2021; Giacomin et al. Reference Giacomin, Dickinson, Kennedy, Patel and Roach2023), which are electromagnetic instabilities relying on finite resistivity, and ETG modes (see Horton, Hong & Tang Reference Horton, Hong and Tang1988; Farengo, Guzdar & Lee Reference Farengo, Guzdar and Lee1989; Jenko et al. Reference Jenko, Dorland, Kotschenreuther and Rogers2000; Adkins et al. Reference Adkins, Schekochihin, Ivanov and Roach2022, Reference Adkins, Ivanov and Schekochihin2023), which are electrostatic modes. Both of these are driven by the equilibrium electron-temperature gradient.

While gyrokinetic simulations of pedestal-regime plasmas recover various contributions to heat transport from MTMs, ETG modes, or other instabilities (e.g., Guttenfelder et al. Reference Guttenfelder, Groebner, Canik, Grierson, Belli and Candy2021; Chapman-Oplopoiou et al. Reference Chapman-Oplopoiou2022; Leppin et al. Reference Leppin, Gorler, Cavedon, Dunne, Wolfrum and Jenko2023), the consensus appears to be that ETG modes are often responsible for an experimentally significant proportion of the turbulent heat transport in the pedestal (see Ren et al. Reference Ren, Guttenfelder, Kaye and Wang2024 and references therein). A recent study of two JET pulses (one JET-C and one JET-ILW) confirming the dominance of ETG modes was given by Hatch et al. (Reference Hatch2019). Therefore, it makes sense to discuss the implications of ETG turbulence for the pulses in our database.

The ETG instability has two branches, one mediated by the magnetic-field curvature, called toroidal ETG, and one independent of the curvature and mediated by fluctuating parallel electron flow (current), called slab ETG. The slab-ETG instability is driven by the electron-temperature gradient but stabilised by the electron-density gradient. A measure of their competition is the gradient ratio

(5.1) \begin{equation} \eta _e(R) = \frac {{L_{n_e}}}{{L_{T_e}}}, \end{equation}

already introduced in § 1 as a key (local) physical parameter.

Fluid models (Adkins et al. Reference Adkins, Schekochihin, Ivanov and Roach2022) and linear GENE simulations (Jenko et al. Reference Jenko, Dorland and Hammett2001) all result in linear stability thresholds characterised by a near-unity $\eta _{e,\mathrm{crit}}$ such that, for $\eta _e\gt {\eta _{e,\mathrm{crit}}}$ , slab-ETG modes are unstable. The numerical stability scans performed by Jenko et al. (Reference Jenko, Dorland and Hammett2001) yielded

(5.2) \begin{align} \left (\frac {R}{{L_{T_e}}}\right )_{\mathrm{crit}}= \mathrm{max}\left \{ 0.8 \ \frac {R}{{L_{n_e}}} , (1+\tau )(1-1.5\varepsilon )\left (1.33 + 1.91 \frac {\hat {s}}{q}\right ) \left ( 1+0.3 \varepsilon {\frac {\mathrm{d}\kappa }{\mathrm{d}\varepsilon }} \right )\right \} ,\nonumber\\[5pt] \end{align}

where $\tau = Z_{\mathrm{eff}} T_e/T_i$ is the temperature ratio, $\varepsilon = a/R$ is the aspect ratio, $q$ is the magnetic safety factor, $\hat {s}$ is the magnetic shear, and $\kappa$ is the plasma elongation. This formula contains the linear thresholds for the slab-ETG branch (left) and the toroidal-ETG branch (right). Typically, the second term in (5.2) has order-unity values and is therefore much smaller than $0.8 {R/{L_{n_e}}}$ .

The linear gyrokinetic stability analyses of pedestal plasmas by Jenko et al. (Reference Jenko, Told, Xanthopoulos, Merz and Horton2009) and Guttenfelder et al. (Reference Guttenfelder, Groebner, Canik, Grierson, Belli and Candy2021) find a value of $\eta _{e,\mathrm{crit}}$ in the range $0.9{-}1.2$ . This variation in the critical threshold is attributed to the effect of magnetic geometry and toroidal modes on the instability (Jenko et al. Reference Jenko, Told, Xanthopoulos, Merz and Horton2009). The region of ${\eta _{e,\mathrm{crit}}} \in [0.8,1.2]$ is marked in figure 9, bounded by the two blue dash-dotted lines. These order-unity values of $\eta _{e,\mathrm{crit}}$ are situated well below the experimental values of the gradients; only near-separatrix values considerably overlap with this region, albeit they are subject to large uncertainties. This means that, throughout the pedestal, the slab-ETG modes are linearly unstable, to the best of existing theoretical knowledge.

This discussion prepares the analysis of the gradient ratio $\eta _e$ and the outcomes of reconstructing $T_e$ profiles based on the assumption that the pedestal gradients are ‘pinned’ to a specific value of $\eta _e$ in § 6. This is based on the possibility that the system simply pins itself locally to a marginal value of $\eta _e$ (although whether that marginality is with respect to the traditional linear stability threshold is to be discussed). In contrast to this approach, in § 8, we will present the results of reconstructing pedestals using scalings of turbulent heat transport: we will relate the separatrix loss power $ {P_{\mathrm{Sep}}}$ to the background gradients of temperature and density, allowing $\eta _e$ to vary.

6.1. Distributions of the gradient ratio $\eta _e$

Figure 10.

Distributions of the gradient ratio $\eta _e$ , defined in (5.1), across the database conditioned on (a) $R/{L_{T_e}}$ , (b) $R/{L_{n_e}}$ , (c) $Q/ {Q_{e,\mathrm{gB}}}$ . All data are obtained from the pedestal region, between $\psi _{{T_e,}\mathrm{Top}}$ and $\psi _{Sep}$ . Panel (b) suggests that ${R/{L_{n_e}}}\propto \eta _e ^\zeta$ describes the pedestal data well, as will be discussed in § 7. The orange lines in (a) and (b) represent three fit values of $\zeta$ to the maxima of the $R/{L_{n_e}}$ distribution (dashed), the full $R/{L_{n_e}}$ distribution (dotted) and the steep-gradient-region prediction across the database discussed in § 7.2 (solid).

In figures 10(a) and 10(b), the data from figure 9 is recast to show an important property of the pedestals: the steep-gradient region ( ${R/{L_{n_e}}} \gtrsim 20$ ) displays a strong tendency of $\eta _e$ to maintain values of around $2$ across this region. It is also apparent that there is a power-law relationship between $\eta _e$ and $R/{L_{n_e}}$ , which will be discussed in § 7. There is no such relationship between $\eta _e$ and $R/{L_{T_e}}$ evident in figure 10(a), as there is far more variance in $R/{L_{T_e}}$ for any given $\eta _e$ . Furthermore, the gyro-Bohm-normalised heat flux $Q/ {Q_{e,\mathrm{gB}}}$ appears to increase as $\eta _e$ approaches the stability threshold (or a different $\mathcal{O}(1)$ value). This confirms the intuition of Field et al. (Reference Field, Chapman-Oplopoiou, Connor, Frassinetti, Hatch, Roach and Saarelma2023) regarding the role of the linear threshold, although with the notable difference that the heat flux must decrease further away from it.

In (2.8), we defined a radial coordinate $X_{\mathrm{Ren}}$ in order to exploit the geometry of each pulse and to match $\psi _{\mathrm{Sep}}$ , $\psi _{{n_e,}\mathrm{Top}}$ , and $\psi _{{T_e,}\mathrm{Top}}$ across the database. In figure 11 we provide the explicit dependence of $\eta _e$ on the radial location relative to these pedestal landmarks. An important pedestal property highlighted by figure 11 is that the most probable values of $\eta _e$ range between $1.5$ and $3.5$ across the steep-gradient region, increasing from the separatrix (figure 11 e) towards $\psi _{{n_e,}\mathrm{Top}}$ (figure 11 c). Since the stability threshold of ETG modes is expected to be at $\eta _e \approx 1$ (see § 5), this implies a measurable deviation of the turbulence from the linear threshold. Consequently, the turbulence resides in an order-unity supercritical regime where $\eta _e$ is closer to $2$ (which has also been reported in several ASDEX experiments, e.g., Neuhauser et al. Reference Neuhauser2002). The distribution of $\eta _e$ does overlap with the linear threshold in the vicinity of the separatrix, although only just.

Figure 11.

(a) Distribution of $\eta _e$ (as defined in (5.1)) conditioned on the renormalised profile position $X_{\mathrm{Ren}}$ , defined in (2.8). The red lines represent the mean and variance of $\eta _e$ at each $X_{\mathrm{Ren}}$ . (b--e) Cross-sections of the histogram at (b) $({\psi _{{T_e,}\mathrm{Top}}}+{\psi _{{n_e,}\mathrm{Top}}}) / 2$ , (c) $\psi _{{n_e,}\mathrm{Top}}$ , (d) $\psi _{\mathrm{Steep}}$ , and (e) $\psi _{\mathrm{Sep}}$ , corresponding to the white dash-dotted lines in (a). In each of these cross-sections, the continuous red lines represent the mean value of $\eta _e$ , and the dotted red lines represent the values one standard deviation away from the mean. Note that these lines are outside of the range of $\eta _e$ captured in (b).

We note that $\eta _e$ increases rapidly once the profiles cross ${X_{\mathrm{Ren}}} = 1$ , which is a consequence of the inward shift of the electron-temperature pedestal: the temperature pedestal maintains large gradients irrespective of the reduction in the density gradient. This could be indicative of a difference between the saturated turbulent states in the steep-gradient and density-top regions. This qualitative difference between the two regions can be interpreted to imply that, in the steep-gradient region, the density gradient plays an essential role in characterising the turbulence, and the turbulent state resides in a critical regime where the temperature gradient is (almost) uniquely determined by the density gradient. In contrast, inside the density-pedestal top, the density gradient does not uniquely determine the temperature gradient, and $R/{L_{T_e}}$ depends more strongly on other plasma parameters, as can be inferred from the comparison of figures 10(a) and 10(b). This difference could be attributed to the physically expected prevalence of slab-ETG modes in the steep-gradient region, with the density gradient being the dominant stabilising effect on these modes. At small $R/{L_{n_e}}$ , toroidal-ETG modes become unstable before the slab-ETG ones (Jenko et al. Reference Jenko, Dorland and Hammett2001), and the addition of such subdominant modes can alter the heat transport (Parisi et al. Reference Parisi2022). This might be the relevant physics in the density-top region. The inclusion of electromagnetic effects into this picture complicates matters further, as we briefly discussed in § 5.1.

6.2. Electron-temperature reconstruction using constant gradient ratio

Figure 12.

Database predictions obtained by integrating $1 / {L_{T_e}} = \eta _e / {L_{n_e}}$ inwards from the separatrix, (a–d) with the best-fit $\eta _e$ for each pulse over the pedestal region, (e–h) with the best-fit $\eta _e=1.90$ across the database over the pedestal region, and (i–l) with the best-fit $\eta _e=2.19$ across the database over the region $\psi _N \in [0.9,1.0]$ . The colour bars represent the ratio between the separatrix density $n_{e}^{\mathrm{Sep}}$ and the density-pedestal height $n_{e}^{\mathrm{Ped}}$ . The plots follow the same layout as in figure 4.

Using the tendency of $\eta _e$ to have order-unity values in the pedestal region, we will proceed to predict the electron-temperature profiles assuming there is a simple functional relationship between the normalised gradients $R/{L_{T_e}}$ and $R/{L_{n_e}}$ . The simplest such relationship, proposed by Field et al. (Reference Field2020), would be a set value of $\eta _e$ describing the turbulence across the whole profile. This is, in itself, an assumption of infinitely ‘stiff’ transport limited by a critical $\eta _e$ , maintained across the pedestal: the heat flux diverges as soon as this value of $\eta _e$ is crossed, reducing the local temperature gradient and consequently bringing the heat flux back to a finite value. Based on the results of § 6.1, we expect this imposed value of $\eta _e$ to be in the vicinity of $2$ . We note that the experimental value of $\eta _e \approx 2$ across the pedestal was initially measured by Neuhauser et al. (Reference Neuhauser2002) on ASDEX, but has been observed to characterise H-mode pedestals correctly on other devices as well (Ren et al. Reference Ren, Guttenfelder, Kaye and Wang2024). Horton et al. (Reference Horton2005) used the assumption of $\eta _e=2$ to reproduce correctly an electron-temperature-pedestal profile on ASDEX Upgrade.

In order to find the values of $\eta _e$ that best describe the pulses in our database, we reconstruct the electron-temperature profiles by first assuming that there is a constant $\eta _e$ in (5.1), and then integrating inwards from the boundary at $R^{\mathrm{Sep}}$ (as detailed in Appendix C). We then identify the best ‘nominal’ $\eta _e$ such that the difference between the $\texttt {fit}$ and the reconstructed pedestal is minimised (as explained in Appendix C.1).

In figure 12, we present three sets of results obtained using this method of integration and parameter optimisation over different domains. These three database reconstructions are calculated by maximising the accuracy of the $T_e$ profile across:

1. (a-d) the pedestal region, $\psi _N \in [{\psi _{{T_e,}\mathrm{Top}}}, 1.0]$ , with a different $\eta _e$ for each pulse;

2. (e-h) the pedestal region, $\psi _N \in [{\psi _{{T_e,}\mathrm{Top}}}, 1.0]$ , with the best-fit $\eta _e = 1.90$ across the database;

3. (i-l) the region including a part of the core, $\psi _N \in [0.9, 1.0]$ , with the best-fit $\eta _e = 2.19$ across the database.

From the pedestal parameter comparison in figure 12, it is apparent that reconstructed temperature-pedestal profiles have temperature-top locations pushed further outwards than the experimental location. The inflexibility of mandating a constant value of $\eta _e$ yields predicted values of $\psi _{{T_e,}\mathrm{Top}}$ that follow $\psi _{{n_e,}\mathrm{Top}}$ , irrespective of whether $\eta _e$ is fixed across the database or allowed a different value for each pulse. Furthermore, for the database-wide $\eta _e$ predictions in figure 12(e--h), we find that the temperature-pedestal height, width, and top locations are strongly correlated to the ratio between the separatrix density $n_{e}^{\mathrm{Sep}}$ and the pedestal density $n_{e}^{\mathrm{Ped}}$ , supporting this observation. Overall, while it is reassuring that we recover $\eta _e \approx 2$ in our database, setting a constant value of $\eta _e\approx 2$ produces poor predictions of the pedestal-top regions. Furthermore, the negative values of $r^2$ in the case of $T_e({\psi _{{n_e,}\mathrm{Top}}})$ indicate that a fixed value of $\eta _e$ is unable to reproduce the steep-gradient region, as it gives a poorer reconstruction than asserting that the predicted $T_e$ pedestal is the average of the pedestals across our database.

The performance of the models using database-wide $\eta _e$ are presented in entries 31–34 of table 3. In particular, those presented in figures 12(e–h) and 12(i–l) correspond to entries 32 and 34, respectively.

Figure 13.

Values of $\eta _e$ that best predict the $T_e$ profiles across the pedestal region. (a) The distribution of values of fit $\eta _e$ . (b) Comparison between the fit values of $\eta _e$ and the database values of $\eta _e$ at $\psi _{\mathrm{Steep}}$ (as shown in figure 11 d).

Thus, allowing the value of $\eta _e$ to vary across the database, i.e., assuming that the turbulence in each individual pedestal can be described by a single parameter that is constant across the profile, yields reasonably good predictions for the steep-gradient region, while over-predicting $T_e$ at $\psi _{{n_e,}\mathrm{Top}}$ and under-predicting $T_{e}^{\mathrm{Ped}}$ within a factor of $\sqrt {2}$ of the experimental value. The distribution of the best-fit $\eta _e$ across the steep-gradient region is shown in figure 13. We find a considerable overlap between the best-fit values of $\eta _e$ from the pedestal predictions of figure 12(a–d) and the experimental values of $\eta _e({\psi _{\mathrm{Steep}}})$ at the steep-gradient point (shown in figure 11 d). The largest discrepancies between the experimental $\eta _e({\psi _{\mathrm{Steep}}})$ and the fit $\eta _e$ are correlated with the ratio ${n_{e}^{\mathrm{Sep}}}/{n_{e}^{\mathrm{Ped}}}$ : outwards shifted $n_e$ pedestals require a higher optimal $\eta _e$ than the experimental $\eta _e$ in order for the $T_e$ pedestal to rise enough. In general, pedestals with a significant shift have a poor prediction accuracy because of the inability of the constant- $\eta _e$ models to reproduce large $R/{L_{T_e}}$ beyond the density-top point.

Applying symbolic regression models (Cranmer Reference Cranmer2023) to estimate the optimised $\eta _e$ for each pulse as a function of database parameters shows that the engineering and magnetic-field parameters lack any correlation with the obtained $\eta _e$ . The inclusion of ${n_{e}^{\mathrm{Sep}}}/{n_{e}^{\mathrm{Ped}}}$ alongside these parameters is essential for the correct estimation of $\eta _e$ , but the scalings fit to the optimised $\eta _e$ are extremely complex and unlikely to have physical meaning. The importance of ${n_{e}^{\mathrm{Sep}}}/{n_{e}^{\mathrm{Ped}}}$ could not have been elicited in § 4 as the location of the separatrix (and, hence, the value of $n_{e}^{\mathrm{Sep}}$ ) was omitted from the training information of the neural networks. This importance of ${n_{e}^{\mathrm{Sep}}}/{n_{e}^{\mathrm{Ped}}}$ must complement the relationship this ratio bears with the pedestal pressure (Frassinetti et al. Reference Frassinetti2021).

6.3. Physical implications of $\eta _e$ results

The distribution of experimental values of $\eta _e$ (figure 10) over the steep-gradient region and their agreement with the best-fit $\eta _e$ values over the pedestal region in figure 13(b) represent an important finding. These best-fit values of $\eta _e$ are nominally higher than $2$ , with a considerable spread.

One can interpret the results of this section as providing physical insight into the steep-gradient-region turbulence in several alternative ways.

1. (i) The turbulence can be described by a quasilinear model strongly dominated by the linear picture of electrostatic and electromagnetic modes, but the true linear threshold $\eta _{e,\mathrm{crit}}$ differs from the solution of Jenko et al. (Reference Jenko, Dorland and Hammett2001) by an order-unity value.

2. (ii) The turbulence saturates in a nonlinear ‘Dimits’-like state (Dimits et al. Reference Dimits, Cohen, Mattor, Nevins, Shumaker, Parker and Kim2000; Colyer et al. Reference Colyer, Schekochihin, Parra, Roach, Barnes, Ghim and Dorland2017; Ivanov, Schekochihin & Dorland Reference Ivanov, Schekochihin and Dorland2022), and the transport level increases abruptly once some nonlinear threshold $\eta _{e,\mathrm{Dimits}}$ is surpassed; based on our results, we expect that $\eta _{e,\mathrm{Dimits}}$ would be higher than the linear $\eta _{e,\mathrm{crit}}$ by an $\mathcal{O}(1)$ amount.

3. (iii) The turbulence causes a moderately (rather than near-infinitely) ‘stiff’ heat transport (Field et al. Reference Field, Chapman-Oplopoiou, Connor, Frassinetti, Hatch, Roach and Saarelma2023) and, given the separatrix loss power $ {P_{\mathrm{Sep}}}$ in JET-ILW pulses, it results in a value of $\eta _e$ that is greater than $\eta _{e,\mathrm{crit}}$ by an $\mathcal{O}(1)$ amount.

In § 6.1, we also identified and discussed the stark difference between the steep-gradient and density-top regions. Explaining this difference demands a theoretical model that accounts for qualitative changes in the turbulent behaviour with the change of the density gradient $R/{L_{n_e}}$ . Such changes could be the result of the contribution of toroidal-ETG modes to the turbulent transport, as recently found by Chapman-Oplopoiou et al. (Reference Chapman-Oplopoiou, Walker, Hatch, Gorler and Contributors2025) and Krutkin et al. (Reference Krutkin, Chapman-Oplopoiou, Frassinetti, Brunner, Coda and Labit2025). We will discuss gyrokinetics-inspired models (Guttenfelder et al. Reference Guttenfelder, Groebner, Canik, Grierson, Belli and Candy2021; Chapman-Oplopoiou et al. Reference Chapman-Oplopoiou2022; Hatch et al. Reference Hatch2022) that use the turbulent heat flux to reconstruct the $T_e$ profiles in § 8, but before that, in § 7, we will construct and assess a simpler scaling prescription that allows for $\eta _e$ to adjust over the pedestal region.

7.1. Distributions of the scaling exponent $\alpha$

Figure 14.

Distribution of the exponent $\alpha$ , defined in (7.2), across the database as a function of (a) $R/{L_{T_e}}$ , (b) $R/{L_{n_e}}$ , (c) $\eta _e$ . All data are obtained from the pedestal region, between $\psi _{{T_e,}\mathrm{Top}}$ and $\psi _{Sep}$ .

Figure 15.

(a) Distribution of $\alpha$ , as defined in (7.2), conditioned on $X_{\mathrm{Ren}}$ . The red lines represent the mean and variance of $\alpha$ at each $X_{\mathrm{Ren}}$ . (b–e) Cross-sections of the histogram at (b) $({\psi _{{T_e,}\mathrm{Top}}}+{\psi _{{n_e,}\mathrm{Top}}}) / 2$ , (c) $\psi _{{n_e,}\mathrm{Top}}$ , (d) $\psi _{\mathrm{Steep}}$ , and (e) $\psi _{\mathrm{Sep}}$ , corresponding to white dash-dotted lines in (a). In each of these cross-sections, the continuous red lines represent the mean value of $\alpha$ , and the dotted red lines represent the values one standard deviation away from the mean.

In figure 14, we show the distribution of the exponent $\alpha$ conditioned on the gradients $R/{L_{T_e}}$ , $R/{L_{n_e}}$ , and their ratio $\eta _e$ . The values of $\alpha$ at the separatrix are clustered around $\approx 0.33$ (as seen in figure 14 b at large $R/{L_{n_e}}$ ). Overall, the range of values taken by $\alpha$ is narrow, indicating that a power-law relationship between $R/{L_{T_e}}$ and $R/{L_{n_e}}$ with a constant exponent might indeed represent the profiles in the steep-gradient region correctly. The relationship between $\alpha$ and $\eta _e$ is shown in figure 14(c), and it implies a clear correlation between increasing $\eta _e$ and increasing $\alpha$ .

Rearranging the values of $\alpha$ using the renormalised radial coordinate $X_{\mathrm{Ren}}$ given by (2.8) gives us figure 15. Its most important feature is the convergence of all profiles to $\alpha \approx 0.33$ at the separatrix ( $X_{\mathrm{Ren}}$ approaching $0$ in figure 15 a). This coincides with the highest values of the normalised gradients $R/{L_{T_e}}$ (figure 14 a) and $R/{L_{n_e}}$ (figure 14 b). The exponent $\alpha$ commonly lies in the range of $0.3{-} 0.5$ in the steep-gradient region, with a large spread towards $\psi _{{n_e,}\mathrm{Top}}$ . The values of $\alpha$ remain order unity in the density-top region.

7.2. Electron-temperature reconstructions using constant exponent $\alpha$

Because of the small variance of $\alpha$ in the pedestal region, it makes sense to explore the consequences of asserting that $\alpha$ is constant across the pedestal. This is conceptually similar to the analysis carried out in § 6.2, where $\alpha =1$ and $A\equiv \eta _e$ , but now $\alpha \neq 1$ allows the physically relevant parameter $\eta _e$ to adjust over the pedestal (as a function of either of the gradients). The case for such an adjustment of $\eta _e$ across the pedestal is clear in figure 10(a,b), where the orange lines represent fits of $\alpha$ to the distributions of $\eta _e$ vs $R/{L_{n_e}}$ – in the figure caption, these fits are presented as ${R/{L_{n_e}}} \propto \eta _e^\zeta$ , where $ \zeta = (\alpha -1)^{-1}$ . The dotted line with $\alpha \approx 0.1$ is the fit to the whole distribution of $\eta _e$ in figure 10(b), while the dashed line with $\alpha = 0.41$ (obstructed by the continuous orange line with $\alpha = 0.39$ obtained below as a best fit for database reconstructions) is the fit to the highest-likelihood $\eta _e$ values in the histogram. It is clear that $\alpha \approx 0.1$ fails to reproduce the trend of pedestal profiles, whereas $\alpha = 0.41$ meaningfully matches the relationship between $\eta _e$ and $R/{L_{n_e}}$ . Hence, using a constant $\alpha$ to describe a pedestal should lend enough flexibility to the relationship between $R/{L_{T_e}}$ and $R/{L_{n_e}}$ .

Figure 16.

Database predictions obtained by integrating ${R/{L_{T_e}}} = A({R/{L_{n_e}}})^\alpha$ inwards from the separatrix, with (a–d) the best-fit $A$ and $\alpha$ for each pulse, and (e–h) the best-fit $A = 50.6$ and $\alpha =0.39$ across the database. The fits are done over the pedestal region. The colour bar represents the separatrix loss power $ {P_{\mathrm{Sep}}}$ . The layout of the figure is the same as in figure 4.

Figure 17.

Pulse $\#90339$ $\texttt {fit}$ profiles compared with predictions using the scaling (7.1). The layout is the same as in figure 1. The $\texttt {fit}$ profiles are shown as solid lines, while the predictions using the $A$ and $\alpha$ optimised individually for pulse $\#90339$ are shown as dashed lines, and predictions using the $A$ and $\alpha$ optimised for the whole database are shown as dotted lines. Both fits are done over the interval $[{\psi _{{T_e,}\mathrm{Top}}}, {\psi _{\mathrm{Sep}}}]$ . The vertical lines and dots represent the landmarks of the $\texttt {fit}$ profiles (see § 2.4 and figure 1), while the crosses represent the pedestal top of the $T_e$ predictions.

Thus, similar to our approach in § 6.2, we attempt to reproduce the pedestals using (7.1) as a fitting formula, seeking the best values of $A$ and $\alpha$ . Figure 16 shows two database reconstructions that were calculated using the scaling (7.1). These reconstructions maximise the accuracy of the $T_e$ profile across the pedestal region, $\psi _N \in [{\psi _{{T_e,}\mathrm{Top}}}, {\psi _{\mathrm{Sep}}}=1.0]$ , with:

1. (a-d) $A$ and $\alpha$ fit individually for each pulse;

2. (e-h) $A=50.6$ and $\alpha =0.39$ fit across the database.

For these reconstructions in figure 16, the values of $A$ and $\alpha$ are the best-fit values obtained using the optimisation method described in Appendix C.1. Note that the value of $\alpha = 0.39$ in this method agrees with the fit to the highest-likelihood values of $\eta _e$ of figure 10(b).

Figure 18.

Values of $\alpha$ and $A$ that best predict the $T_e$ profiles individually across the pedestal region: (a) the distribution of values of fit $\alpha$ , (b) the fit values of $\alpha$ vs $A$ . The black cross represents the $A$ and $\alpha$ fit for the whole database (shown in figure 16 e–h). The linear fit between $\ln A$ and $\alpha$ is shown as a blue line.

The predictions obtained by allowing $\alpha$ and $A$ to be fit individually for each pulse (figure 16 a–d) are very accurate in reproducing the pedestal region overall, as seen in figure 16(a). Furthermore, the pedestal shapes are reconstructed well for a large proportion of the database, although for some pulses, the $T_e$ profiles continue increasing exponentially towards the core, so $\texttt {mtanh}$ parameters do not describe these profiles well. This can be seen in figure 16(b–d), where a subset of the $\texttt {mtanh}$ parameters is very far from the correct prediction.

This issue of unbounded $T_e$ profiles is more pronounced for the reproductions using a single combination of $A = 50.6$ and $\alpha = 0.39$ across the database. Since many reconstructed profiles do not resemble an $\texttt {mtanh}$ , the corresponding pedestal parameters are obtained poorly (figures 16 f–h). However, the accuracy of these fits across the steep-gradient region (figure 16 e) is much better than the best-fitting $\eta _e$ predictions of § 6.2 (figure 12 e,i), meaning that such an approach is more effective for the steep-gradient region than the methods in § 6. This combination of $A$ and $\alpha$ is shown in figure 10(a,b) as the continuous orange lines, recovering the fit performed directly to the distribution of $R/{L_{n_e}}$ and $\eta _e$ shown by the dashed line.

If $A$ and $\alpha$ are instead optimised over the interval $\psi _N \in [0.9,1.0]$ for a database-wide fit, we obtain $A = 31.7$ and $\alpha = 0.46$ (not shown). This combination produces slightly degraded pedestals over the $\psi _N \in [{\psi _{{T_e,}\mathrm{Top}}}, 1.0]$ interval when compared with the combination in figure 16(e–h), and it does not solve the general issue of the exponential divergence of $T_e$ profiles. The combinations of $A$ and $\alpha$ optimised over the intervals $[{\psi _{{T_e,}\mathrm{Top}}}, 1.0]$ and $[0.9, 1.0]$ correspond to entries 36 and 38 of table 3 in Appendix A, where their accuracies over different ranges are compared with our other attempts at $T_e$ -pedestal reconstructions in this study.

Examples of the profiles that constitute figure 16 are given in figure 17 for the pulse $\#90339$ . The database-optimised fit (dotted lines) has a lower value of $\alpha$ than the optimal value for pulse $\#90339$ , causing $R/{L_{T_e}}$ to vary less with $R/{L_{n_e}}$ . Because of this, the database-optimised values give a large $R/{L_{T_e}}$ beyond the density-pedestal top and a diverging $T_e$ profile without a $\psi _{{T_e,}\mathrm{Top}}$ within the bounds shown in figure 17. The optimal combination of $A$ and $\alpha$ for this individual pulse (dashed lines) results in a good $T_e$ pedestal and reproduces $\eta _e$ correctly between $\psi _{{T_e,}\mathrm{Top}}$ and $\psi _{\mathrm{Sep}}$ .

The distributions of $A$ and $\alpha$ fit for each pedestal (corresponding to figure 16 a–d) are shown in figure 18. The key finding, manifest in figure 18(b), is that, while we found that a two-parameter fit (7.1) is much better than the one-parameter fit ( $\alpha = 1$ , $A = \eta _e$ in § 6), the two parameters appear to be highly correlated. Their relationship is given by $\ln A = 5.64 \pm 0.05 - (4.42 \pm 0.08) \alpha$ , which allows for (7.1) to be recast as

(7.3) \begin{align} \frac {R}{{L_{T_e}}} = (281 \pm 15) \left ( \frac {1}{84 \pm 7} \frac {R}{{L_{n_e}}} \right ) ^ \alpha . \end{align}

The ranges for the numerical constants represent a $95\,\%$ confidence interval. This means that the fits performed in this section in fact intrinsically have only one degree of freedom. We have found that a similar locus of fit parameters $A$ and $\alpha$ , given by $ \ln A = 5.73 \pm 0.09 - (5.18 \pm 0.12) \alpha$ , describes a subset of $273$ JET-C pulses from the EUROfusion pedestal database, selected based on criteria similar to those described in § 3.1. These JET-C pulses are not otherwise included in this study.

Finding a relationship between $\alpha$ and physical parameters would represent a major success in modelling the pedestal steep-gradient region. However, we anticipate that a model like (7.3) might be much harder to justify physically than other alternatives used in §§ 6 and 8. We also note that we have failed to find any correlation between the distribution of $\alpha$ and database parameters.

8.1. Electron-scale turbulence

The nonlinear dynamics of ETG-driven turbulence in the steep-gradient region of pedestals are strongly influenced by the complex interactions between microinstabilities that emerge as a result of the underlying plasma equilibria. Here, we briefly review the key findings from nonlinear gyrokinetic simulations and their implications for pedestal turbulence.

Chapman-Oplopoiou et al. (Reference Chapman-Oplopoiou2022) carried out gyrokinetic simulations using the nominal gradients $R/{L_{T_e}}$ and $R/{L_{n_e}}$ evaluated at $\psi _{\mathrm{Steep}}$ for four JET-ILW pedestals, which are also present in this study. Varying these gradients within the experimental confidence intervals showed that the electron-channel turbulent heat flux $ {Q_{e,\mathrm{Turb}}}$ increased with increasing $R/{L_{T_e}}$ and decreased with increasing $R/{L_{n_e}}$ . This is consistent with the linear picture of the slab-ETG instability, where electron-temperature gradients drive the instability, while electron-density gradients suppress it. Chapman-Oplopoiou et al. (Reference Chapman-Oplopoiou2022) showed that a good fit to their results was the relationship

(8.1) \begin{equation} \frac { {Q_{e,\mathrm{Turb}}}}{ {Q_{e,\mathrm{gB}}}} = A \left (\frac {R}{{L_{T_e}}}\right )^2 \left (\eta _e -{\eta _{e,\mathrm{crit}}}\right )^{\beta }, \end{equation}

where, as defined in § 2.3, $ {Q_{e,\mathrm{gB}}} = n_e T_e {v_{\mathrm{th},e}} \rho _e^2 / R^2$ , and the best-fit parameters were $A=0.85$ , ${\eta _{e,\mathrm{crit}}} = 1.28$ , and $\beta = 1.32$ . The simulated heat fluxes were also consistent with the fit found in simulations of DIII-D pedestals by Guttenfelder et al. (Reference Guttenfelder, Groebner, Canik, Grierson, Belli and Candy2021): $A=1.5$ , ${\eta _{e,\mathrm{crit}}} = 1.4$ , and $\beta =1$ . Models of this form can also reproduce results such as those from the $R/{L_{T_e}}$ scans in Parisi et al. (Reference Parisi2022): keeping $\eta _e$ constant yields $ {Q_{e,\mathrm{Turb}}}/ {Q_{e,\mathrm{gB}}} \propto ({R/{L_{T_e}}})^2$ , whereas keeping $R/{L_{n_e}}$ constant gives $ {Q_{e,\mathrm{Turb}}}/ {Q_{e,\mathrm{gB}}} \propto ({R/{L_{T_e}}})^4$ . This behaviour would be matched by (8.1) with $\beta =2$ when $\eta _e \gg {\eta _{e,\mathrm{crit}}}$ .

The behaviour of the turbulent heat flux $ {Q_{e,\mathrm{Turb}}}$ parametrised by (8.1) can be justified by the physical picture proposed by Field et al. (Reference Field, Chapman-Oplopoiou, Connor, Frassinetti, Hatch, Roach and Saarelma2023). In their model, the $({R/{L_{T_e}}})^2$ term is justified by replacing the radial size $R$ of the tokamak with the locally relevant length scale $L_{T_e}$ in the gyro-Bohm normalisation of the heat flux:

(8.2) \begin{equation} {Q_{e,\mathrm{gB}}} = n_e T_e {v_{\mathrm{th},e}} \frac {\rho _e^2}{R^2} \;\;\; \rightarrow \;\;\; Q_{e,\mathrm{gB}}^* = n_e T_e {v_{\mathrm{th},e}} \frac {\rho _e^2}{{L_{T_e}}^2}, \end{equation}

i.e., the relevant scale associated with free-energy injection is $L_{T_e}$ , rather than $R$ . This is a natural choice if turbulent transport is controlled by the slab-ETG instability. With the redefinition (8.2), (8.1) becomes

(8.3) \begin{equation} \frac { {Q_{e,\mathrm{Turb}}}}{ Q_{e,\mathrm{gB}}^*} =A \left (\eta _e - {\eta _{e,\mathrm{crit}}}\right )^{\beta }, \end{equation}

and, therefore, for $\beta \gt 0$ , the normalised turbulent heat flux is a function of how much the slab-ETG drive (expressed through $\eta _e$ ) exceeds the (nonlinear) threshold $\eta _{e,\mathrm{crit}}$ . This is an essential characteristic that sets this model of turbulent transport apart from the models using a fixed $\eta _e$ (§ 6) or a power law for the relationship between $R/{L_{T_e}}$ and $R/{L_{n_e}}$ (§ 7). Those models amounted to assuming that the turbulence was marginal and, therefore, ‘pinned’ to a specific value of $\eta _{e,\mathrm{crit}}$ , constant across the pedestal or varying in a prescribed manner. Model (8.1) allows the profiles to reach beyond this marginal regime.

This type of modelling of turbulent transport is common in the context of ion-temperature-gradient mode (ITG) turbulence in core plasmas and has been reported to reproduce measurements correctly (e.g., Garbet et al. Reference Garbet2004; Mantica et al. Reference Mantica2009). In the case of ITG turbulence, the critical value of the ion-temperature gradient is usually expected to be larger than the linear stability threshold because the nonlinear saturation just above the linear threshold is in the ‘Dimits’ regime, in which strong, nonlinearly saturated zonal flows suppress heat transport (Dimits et al. Reference Dimits, Cohen, Mattor, Nevins, Shumaker, Parker and Kim2000; Ivanov et al. Reference Ivanov, Schekochihin and Dorland2022). A similar behaviour is sometimes found in simulations of ETG turbulence, where the long-term onset of zonal flows suppresses the heat transport caused by ETG (Colyer et al. Reference Colyer, Schekochihin, Parra, Roach, Barnes, Ghim and Dorland2017). Thus, it is conceivable (although, at this stage, far from certain) that $\eta _{e,\mathrm{crit}}$ is a nonlinear threshold.

Note also that in the limit of strong drive ( $\eta _e \gg {\eta _{e,\mathrm{crit}}}$ ), (8.1) with $\beta = 1$ leads to the scaling $Q / {Q_{e,\mathrm{gB}}} \propto (R/{L_{T_e}})^3$ , which is in accord with a simple scaling theory (Barnes, Parra & Schekochihin Reference Barnes, Parra and Schekochihin2011; Adkins, Ivanov & Schekochihin Reference Adkins, Ivanov and Schekochihin2023).Footnote ¹

Electron-temperature-gradient turbulence in the pedestal is subject to further complexity (see the review by Ren et al. Reference Ren, Guttenfelder, Kaye and Wang2024). The structure of toroidal-ETG modes along the field lines (Jenko et al. Reference Jenko, Told, Xanthopoulos, Merz and Horton2009; Parisi et al. Reference Parisi2020) leads to nonlinear coupling between ion-scale ( $k_y \rho _i \sim 1$ ) and electron-scale ( $k_y \rho _e \sim 1$ ) modes. This can lead to quenching of ETG turbulence under certain conditions (Parisi et al. Reference Parisi2022). Other studies of the effects of toroidal-ETG modes on pedestal turbulence suggest that the contribution of toroidal-ETG modes amplifies transport (Chapman-Oplopoiou et al. Reference Chapman-Oplopoiou, Walker, Hatch, Gorler and Contributors2025; Krutkin et al. Reference Krutkin, Chapman-Oplopoiou, Frassinetti, Brunner, Coda and Labit2025).

The global study by Leppin et al. (Reference Leppin, Gorler, Cavedon, Dunne, Wolfrum and Jenko2023) of pedestal transport over a radial interval wider than the steep-gradient region suggests that the modes that constitute the turbulence are different between the pedestal-top and the steep-gradient region. At the top, ion-scale trapped-electron modes dominate the transport, whereas in the steep-gradient region, they are suppressed as a result of high $\boldsymbol{E} \times \boldsymbol{B}$ shear (Hahm & Burrell Reference Hahm and Burrell1995). Consistent with this, the study of a JET pedestal by Leppin et al. (Reference Leppin, Görler, Frassinetti, Saarelma, Hobirk, Jenko and contributors2024) concluded that the steep-gradient region was dominated by transport through the electron channel, with a large ion-scale heat flux at the top of the pedestal.

In spite of the above, ETG-centric models are unlikely to be a complete description of the turbulence in the pedestal. It has been shown by Hatch et al. (Reference Hatch, Kotschenreuther, Mahajan, Valanju, Jenko, Told, Görler and Saarelma2016, Reference Hatch2021) that MTMs are responsible for magnetic fluctuations that are measured in pedestals in JET-C and JET-ILW. Large transport through MTMs was also found in the multi-machine study by Kotschenreuther et al. (Reference Kotschenreuther2019), and these modes were recovered as non-negligible contributors to the turbulent heat flux in cases dominated by ETG (Chapman-Oplopoiou et al. Reference Chapman-Oplopoiou2022). Electromagnetic effects were also found to be important for the saturation of pedestal turbulence in the comparison carried out by Dicorato et al. (Reference Dicorato2024). Still, they found that turbulent transport had a much higher contribution from ETG modes for the type-I ELMy pedestal when compared with the small-ELM pedestals.

Aside from electron-scale turbulent transport, ion-scale effects can be relevant transport contributors through ion-scale turbulence and ion neoclassical transport (Trinczek et al. Reference Trinczek, Parra, Catto, Calvo and Landreman2023). Predebon et al. (Reference Predebon, Hatch, Frassinetti, Horvath, Saarelma, Chapman-Oplopoiou, Görler and Maggi2023) found, in a study of the isotope effect, the electron-scale heat transport to be smaller than the ion neoclassical heat flux and the ion-scale turbulent heat flux close to the SOL, although the comparison was reversed close to the pedestal top. Furthermore, earlier work by Hatch et al. (Reference Hatch2019) showed that, for a discharge in JET-C and one in JET-ILW, ion-scale and ion neoclassical transport could be larger than that originating from ETG turbulence.

In yet another body of work, the pedestal is considered to be limited by the instability of the kinetic ballooning mode (KBM; see Tang, Connor & Hastie Reference Tang, Connor and Hastie1980; Snyder Reference Snyder1999; Snyder & Hammett Reference Snyder and Hammett2001; Pueschel, Kammerer & Jenko Reference Pueschel, Kammerer and Jenko2008; Waltz Reference Waltz2010). This mode is a source of high particle transport and effectively acts as a stiff transport threshold that limits the pressure gradient of the pedestal. This threshold is used in predictive models such as EPED (Snyder et al. Reference Snyder, Groebner, Leonard, Osborne and Wilson2009; Saarelma et al. Reference Saarelma, Frassinetti, Bilkova, Challis, Chankin, Fridström, Garzotti, Horvath, Maggi and Contributors2019), and pedestals in various devices were consistently found by, e.g., Kotschenreuther et al. (Reference Kotschenreuther2019) to be marginally stable to KBMs, with other forms of instabilities causing transport, such as MTMs.

Keeping in mind the foregoing discussion and acknowledging the potentially diverse nature of pedestal transport, we will make the simplifying assumption that the heat transport in the steep-gradient region is dominated by electron-scale turbulence. It is arguably not entirely unreasonable to claim that the ETG-centric model (8.1) can crudely represent pedestal turbulence, but we also extend our considerations to other heat-flux models that have been obtained by analysing a large dataset of gyrokinetic simulations by Hatch et al. (Reference Hatch2022). Their table 1 contains several symbolic-regression fits to the heat transport in those simulations. These relationships are explicitly stated (see D2 and D3) and tested in Appendix D.1. Optimised versions of these relationships, including (8.1), will also be used in our analysis below in the forms of (8.12) and (8.13).

Further extrapolations obtained from simulation datasets are those found by Farcaş et al. (Reference Farcaş, Merlo and Jenko2024),

(8.4) \begin{equation} \frac { {Q_{e,\mathrm{Turb}}}}{ {Q_{e,\mathrm{gB}}} } \propto \sqrt \frac {m_e}{m_i} \left (\frac {R}{{L_{T_e}}}\right ) ^{1.40} \left (\eta _e - 1\right )^{0.79} q^{-0.51} \beta _e^{-0.87} \lambda _D^{-0.51}, \end{equation}

and by Hatch et al. (Reference Hatch, Kotschenreuther, Li, Chapman-Oplopoiou, Parisi, Mahajan and Groebner2024),

(8.5) \begin{equation} \frac { {Q_{e,\mathrm{Turb}}}}{ {Q_{e,\mathrm{gB}}} } \propto \sqrt \frac {m_e}{m_i} \left ({\frac {\mathrm{d}\ln T_e}{\mathrm{d}\psi }}\right ) ^{2} \left (\eta _e - 1\right ) \eta _e ^{1.57} \tau _e ^ {-0.5} \lambda _D^{-0.4}. \end{equation}

Here $\lambda _D$ is the electron Debye length normalised with respect to the electron gyroradius $\rho _e$ in Farcaş et al. (Reference Farcaş, Merlo and Jenko2024) and to the ion sound radius $\rho _s = \sqrt {m_i T_e }/eB$ in Hatch et al. (Reference Hatch, Kotschenreuther, Li, Chapman-Oplopoiou, Parisi, Mahajan and Groebner2024). These are relationships where the properties of the magnetic field appear via the safety factor $q$ , the electron beta $\beta _e$ , and the normalised temperature gradient with respect to the poloidal flux, $1/\tilde {L}_{T_e} \equiv {\mathrm{d}\ln T_e/\mathrm{d}\psi }$ . Because we do not have a good magnetic-equilibrium reconstruction for our pedestals, we omit these models from our study.

8.2. Assumptions for the turbulent heat flux

The nonlinear dynamics of ETG turbulence provide a pathway to connect the separatrix loss power $ {P_{\mathrm{Sep}}}$ with pedestal profiles. Using the scalings derived from gyrokinetic simulations, $T_e$ -pedestal profiles will be reconstructed in § 8.3 by assuming that the heat flux is dominated by the electron-channel turbulent transport, $Q(R) = {Q_{e,\mathrm{Turb}}}$ , and that $Q$ matches the experimentally determined separatrix loss power $ {P_{\mathrm{Sep}}}$ , i.e.,

(8.6) \begin{equation} Q({R^{\mathrm{Sep}}}) = \frac { {P_{\mathrm{Sep}}} }{ S_{\mathrm{Sep} }} , \end{equation}

where $S_{\mathrm{Sep}} \approx 135\, \mathrm{m}^2$ is the surface area of the separatrix. Further to this, we argue that, given the small width of the pedestal $w_{\mathrm{Ped}} = \mathcal{O}(\mathrm{cm})$ compared with the device length scales $R$ or $a = \mathcal{O}(\mathrm{m})$ , the variation in the area of the flux surface as a function of $R$ is negligible. We also assume that the plasma profiles are in steady state (see § 2 and Frassinetti et al. Reference Frassinetti2020), so the rate of change of the thermal energy over the small extent of the pedestal is small. Therefore, we approximate the turbulent heat flux as constant over the pedestal:

(8.7) \begin{equation} {Q_{e,\mathrm{Turb}}}(R) \approx \frac { {P_{\mathrm{Sep}}} }{ S_{\mathrm{Sep} }} . \end{equation}

To calculate the gyro-Bohm normalisation $ {Q_{e,\mathrm{gB}}}$ of the turbulent heat flux (see 8.2), we use a locally approximated value of the magnetic field, i.e.,

(8.8) \begin{equation} B(R) = \sqrt {B_\theta (R)^2 + B_\phi (R)^2} \approx B_{\mathrm{Vac}} \frac {R_0}{R} \sqrt {1 - \frac {1}{q_{95}^2}} \approx 0.66 B_{\mathrm{Vac}}, \end{equation}

where $B_\theta$ and $B_\phi$ are, respectively, the poloidal and toroidal components of the magnetic field, $B_{\mathrm{Vac}}$ is the vacuum magnetic field on the axis of JET-ILW (shown as $B$ in figure 2 and table 2), $q_{95}$ is the safety factor evaluated at $\psi _N = 0.95$ , and $R_0 = 0.92 \; \mathrm{m}$ is the radius of the axis of JET-ILW. This should remain a good approximation over the extent of the pedestal, irrespective of the divergence of the safety factor at the separatrix, $q \rightarrow \infty$ as $R \rightarrow {R^{\mathrm{Sep}}}$ . This approximation also agrees well with what one would obtain by using $ {I_{\mathrm{Plasma}}}$ and Ampere’s law to estimate $B_\theta (R)^2$ . These approximations fall short of the precision of the magnetic field calculated via the MHD equilibrium, but they are good enough for the purpose of gyro-Bohm normalisation.

Because of the approximations that we use, uncertainties remain as to the exact contribution of additional transport mechanisms such as neoclassical transport (Guttenfelder et al. Reference Guttenfelder, Groebner, Canik, Grierson, Belli and Candy2021; Trinczek et al. Reference Trinczek, Parra, Catto, Calvo and Landreman2023) and ELM-induced losses (Field et al. Reference Field2020). Order-unity errors of the heat flux do not significantly affect the reconstruction of pedestals because of the ‘stiffness’ of pedestal transport described in § 8.1. This property means that small changes in $R/{L_{T_e}}$ in the models introduced below, (8.9)–(8.13), can lead to large changes in heat flux. Conversely, a large inaccuracy in the heat flux results in a comparatively small change in $R/{L_{T_e}}$ , meaning that pedestal reconstructions are robust. This is illustrated in figure 25 (Appendix D.3), where an order-unity change in $ {Q_{e,\mathrm{Turb}}} / {Q_{e,\mathrm{gB}}}$ (represented in figure 25 d by a change in $A$ ) has only a small effect on $R/{L_{T_e}}$ and the reconstructed $T_e$ profile.

8.3. Electron-temperature reconstruction using heat-flux models

Using the methods outlined in Appendix C, we optimise the free parameters in (8.1) and the general forms of the heat-flux models from Hatch et al. (Reference Hatch2022) given in (D2) and (D3). With these, we then reconstruct the $T_e$ profiles.

Figure 19.

Database predictions obtained by inverting heat-flux models for $R/{L_{T_e}}$ and integrating inwards from the separatrix. The models use best-fit parameters over the pedestal region (a–p) (models (8.9)--(8.12)) and over the steep-gradient region (q–t) (model (8.13)). The colour bar represents the separatrix loss power $ {P_{\mathrm{Sep}}}$ . The plots follow the same layout as in figure 4.

Figure 19 shows the mtanh parameters that describe such reconstructions using models optimised over the pedestal region $\psi _N \in [{\psi _{{T_e,}\mathrm{Top}}},1]$ (figure 19 a–p) and the steep-gradient region $\psi _N \in [{\psi _{{n_e,}\mathrm{Top}}},1]$ (figure 19 q–t). The exact models that we used correspond to the panels of figure 19 as follows:

1. (a-d) model (8.1) with fixed $\beta =1$ and ${\eta _{e,\mathrm{crit}}}=1.4$ , and the value of $A=0.95$ optimised over the pedestal region $\psi _N \in [{\psi _{{T_e,}\mathrm{Top}}},1]$ ; this corresponds to entry 8 in table 3:

   (8.9) \begin{equation} \mathrm{Model \; 1}: \;\; \frac { {Q_{e,\mathrm{Turb}}}}{ {Q_{e,\mathrm{gB}}}} = 0.95 \left (\frac {R}{{L_{T_e}}}\right ) ^ 2 \left ( \eta _e - 1.4 \right ) ; \end{equation}

2. (e-h) model (8.1) with fixed $\beta =1$ and the values of ‘ $\eta _{e,\mathrm{crit}}$ ’ $=-0.55$ and $A=0.29$ optimised over the pedestal region $\psi _N \in [{\psi _{{T_e,}\mathrm{Top}}},1]$ ; this corresponds to entry 17 in table 3:

   (8.10) \begin{equation} \mathrm{Model \; 2}: \;\; \frac { {Q_{e,\mathrm{Turb}}}}{ {Q_{e,\mathrm{gB}}}} = 0.29 \left (\frac {R}{{L_{T_e}}}\right ) ^ 2 \left ( \eta _e + 0.55 \right ) ; \end{equation}

3. (i-l) model (8.1) with the values of ‘ $\eta _{e,\mathrm{crit}}$ ’ $=-8.6$ , $\beta =3.75$ , and $A=10^{-3}$ optimised over the pedestal region $\psi _N \in [{\psi _{{T_e,}\mathrm{Top}}},1]$ ; this corresponds to entry 14 in table 3:

   (8.11) \begin{equation} \mathrm{Model \; 3}: \;\; \frac { {Q_{e,\mathrm{Turb}}}}{ {Q_{e,\mathrm{gB}}}} = 10^{-3} \left (\frac {R}{{L_{T_e}}}\right ) ^ 2 \left ( \eta _e + 8.6 \right ) ^ {3.75} ; \end{equation}

4. (m-p) model (D2) from Hatch et al. (Reference Hatch2022) with the values of the numerical constants optimised over the pedestal region $\psi _N \in [{\psi _{{T_e,}\mathrm{Top}}},1]$ ; this corresponds to entry 22 in table 3:

   (8.12) \begin{equation} \mathrm{Model \; 4}: \;\; \frac { {Q_{e,\mathrm{Turb}}}}{ {Q_{e,\mathrm{gB}}}} = 9.64 \frac {R}{{L_{T_e}}} \left ( \eta _e ^ {2.92} + 16.02 \right ) ; \end{equation}

5. (q-t) model (D3) from Hatch et al. (Reference Hatch2022) with the values of the numerical constants optimised over the steep-gradient region $\psi _N \in [{\psi _{{n_e,}\mathrm{Top}}},1]$ ; this corresponds to entry 30 in table 3:

   (8.13) \begin{equation} \mathrm{Model \; 5}: \;\; \frac { {Q_{e,\mathrm{Turb}}}}{ {Q_{e,\mathrm{gB}}}} = 0.01 \frac {R}{{L_{T_e}}} \left ( \eta _e + 8.86 \right ) ^{4.77} . \end{equation}

A more comprehensive list of accuracy metrics for various heat-flux models is given in table 3 of Appendix D. There, we present the performance of the exact models from Chapman-Oplopoiou et al. (Reference Chapman-Oplopoiou2022) and Hatch et al. (Reference Hatch2022): those given by (8.9)--(8.13) and their variations that minimise the mean-squared difference between the reconstructed and the fit $T_e$ profiles. We also compare them with the results of §§ 4, 6, and 7.

We note that the optimised prefactor $A = 0.95$ of (8.1) over the steep-gradient region is in good agreement with the value of $A = 0.85$ proposed by Chapman-Oplopoiou et al. (Reference Chapman-Oplopoiou2022).

Model (8.9) maintains the physical intuition of ETG-based turbulent transport outlined in § 8.1 and manages to capture the shapes of pedestals much better than the simplistic models of §§ 6 and 7. Model (8.10) is a further improvement to all metrics of pedestal shape, although the optimisation of $\eta _{e,\mathrm{crit}}$ yields a negative value over the ranges of the pedestal or steep-gradient region (see entries 12, 14, 16, and 17 in table 3). The improvement in reconstruction accuracy by setting this value to be negative casts doubt on the idea of a turbulent threshold $\eta _{e,\mathrm{crit}}$ . Models (8.9) and (8.10) share two systematic limitations: $R/{L_{T_e}}$ is over-predicted close to the separatrix and the reconstructed $\psi _{{T_e,}\mathrm{Top}}$ tracks the experimental $\psi _{{n_e,}\mathrm{Top}}$ . The inaccuracy in $R/{L_{T_e}}$ comes as a result of the following process, illustrated in figure 20: the large value of $ {Q_{e,\mathrm{Turb}}} / {Q_{e,\mathrm{gB}}}$ near the separatrix yields a large $R/{L_{T_e}}$ (panel d), and this then quickly increases the value of the $T_e$ profile as the integration goes on, reducing the value of $ {Q_{e,\mathrm{Turb}}} / {Q_{e,\mathrm{gB}}}$ faster than the experimental profile suggests (panel c). This means that in the steep-gradient region of the $T_e$ pedestal, the temperature gradient is always over-predicted, with an over-estimated $R/{L_{T_e}}$ at the separatrix. Furthermore, the increase of $n_e$ away from the separatrix increases $ {Q_{e,\mathrm{gB}}}$ , which reduces $ {Q_{e,\mathrm{Turb}}} / {Q_{e,\mathrm{gB}}}$ . This produces the shape of the $T_e$ pedestal, by switching to low values of $R/{L_{T_e}}$ away from the separatrix, and, in the process, causing $\psi _{{T_e,}\mathrm{Top}}$ to match $\psi _{{n_e,}\mathrm{Top}}$ (see figure 19 d,h,l and 20 a).

Figure 20.

Pulse $\#90339$ $\texttt {fit}$ profiles compared with predictions using heat-flux models 1 (8.9) and 2 (8.10). The layout is the same as in figure 1. The $\;\texttt {fit}$ profiles are shown as solid lines, with the predictions using the heat-flux models shown as dashed (model 8.11), and dotted (model 8.9) lines. The fit is done over the interval $[{\psi _{{T_e,}\mathrm{Top}}}$ , ${\psi _{\mathrm{Sep}}}]$ . The vertical lines and dots represent the landmarks of the $\texttt {fit}$ profiles (see § 2.4 and figure 1), while the crosses represent the pedestal top of the $T_e$ predictions.

Seeking optimal values of the exponent $\beta$ in (8.1) yields large exponents for the term containing the gradient ratio $\eta _e$ (entries 13 and 14 in table 3). In spite of this, the parameter scan shown in figure 25 (Appendix D.3) suggests that reconstructions using (8.1) are not very sensitive to the exponent $\beta$ , and that the corresponding reduction of the predicted values of $R/{L_{T_e}}$ are compensated by the decrease in the prefactor $A$ to $10^{-3}$ (effectively increasing the turbulent heat flux). This insensitivity of the qualitative shape of predictions to the exponent $\beta$ is also present for models (8.12) and (8.13).

The predictions using models (8.12) and (8.13) (derived from Hatch et al. Reference Hatch2022) give improved $r^2$ values of the predictions of $T_{e}({\psi _{{n_e,}\mathrm{Top}}})$ and $T_{e}^{\mathrm{Ped}}$ (see figure 19 m,n,q,r) when compared with those obtained from model (8.9). However, the shape of pedestals suggests that there is little variation in $w_{{T_e,}\mathrm{Ped}}$ and $\psi _{{T_e,}\mathrm{Top}}$ (see figure 19 o,p,s,t), and that the reconstructed pedestals are very thin and pushed towards the separatrix. This signals the importance of a larger exponent of the temperature gradient in such models (such as $R^2/L_{T_e}^2$ in 8.1), since the use of $({R/{L_{T_e}}})^1$ in (8.12) and (8.13) (corresponding to models (D2) and (D3) further discussed in Appendix D) demands a large value of the temperature gradient in order to balance the magnitude of $ {Q_{e,\mathrm{gB}}} / {Q_{e,\mathrm{Turb}}}$ near the separatrix. This temperature gradient produces outwards shifted $T_e$ pedestals, as illustrated in figure 21. The sheer magnitude of $R/{L_{T_e}}$ near the separatrix also means that there is little sensitivity of the mtanh parameters to the $n_e$ profile for models (8.12) and (8.13).

Figure 21.

Same as figure 20, but for predictions using heat-flux models 3 (8.11), 4 (8.12), and 5 (8.13). The $\;\texttt {fit}$ profiles are shown as solid lines, while the predictions using different heat-flux models are shown as dashed (model (8.10)), dotted (model (8.12)), and dash-dotted (model (8.13)) lines. The fit is done over the intervals $[{\psi _{{T_e,}\mathrm{Top}}}$ , ${\psi _{\mathrm{Sep}}}]$ for models (8.11) and (8.12), and $[{\psi _{{n_e,}\mathrm{Top}}}$ , ${\psi _{\mathrm{Sep}}}]$ for model (8.13).

Another important common characteristic of our fit parameters is the embarrassing tendency of $\eta _{e,\mathrm{crit}}$ to become negative. This is manifest in (8.10)--(8.13) and in the majority of optimisations in table 3, and we discuss this at length in Appendix D. While the sign of $\eta _{e,\mathrm{crit}}$ does not have a dramatic effect on the reconstructed $T_e$ profiles (see figures 19 a–h and 20), we interpret this result as an indication that either the forms of the models for $ {Q_{e,\mathrm{Turb}}}$ are inaccurate for the data, or that the physical argument for $ {Q_{e,\mathrm{Turb}}} \propto \eta _e - {\eta _{e,\mathrm{crit}}}$ for some physical threshold $\eta _{e,\mathrm{crit}}$ is not supported by the data.

8.4. Discussion of heat-flux model results

In this section, we have obtained proof that the models of Guttenfelder et al. (Reference Guttenfelder, Groebner, Canik, Grierson, Belli and Candy2021), Chapman-Oplopoiou et al. (Reference Chapman-Oplopoiou2022), and Hatch et al. (Reference Hatch2022) perform well in reproducing the $\texttt {mtanh}$ parameters of pedestals (figure 19), alongside matching the experimental profile well over the regions with steep gradients (table 3) when compared with our simple database-wide models from §§ 6 and 7.

A thorough analysis of these gyrokinetics-inspired turbulent transport models has confirmed the necessity of $R/{L_{T_e}}$ appearing with an exponent larger than unity in the expression for $ {Q_{e,\mathrm{Turb}}}/ {Q_{e,\mathrm{gB}}}$ , as also recognised by models (8.4) and (8.5). We have also found a unanimously negative sign of the constant $\eta _{e,\mathrm{crit}}$ , however, this physically dubious feature is not demanded strongly by the predictions: forcing $\eta _{e,\mathrm{crit}}$ to be positive does not severely impact prediction performance.

We note that all of our pedestal reconstructions using optimised heat-flux models generally produce similar $\texttt {mtanh}$ parameters. The minimisation of the sum of squared discrepancies between prediction and experiment also yields comparable reconstruction accuracy between fits done across the same range of the pedestal (entries 7–30 in table 3). These two outcomes mean that a significant part of our assessment of which transport models are best comes from the qualitative shape of the profiles such as those in figures 20 and 21 (as we discuss at the end of § 8.3).

Overall, the inclusion of the turbulent heat flux into our pedestal-reconstruction methods addresses to a noticeable degree the shortcomings of the ‘marginal’ models that used database-wide parameters in §§ 6 and 7. This said, the fact that the reconstructed $T_e$ pedestals inherit the shape of the $n_e$ pedestals is a sign that such models require more complexity in order to capture the density-top region correctly, and hence, the shift between the $T_e$ and $n_e$ pedestals (Frassinetti et al. Reference Frassinetti2020). This is also supported by the significantly better performance of our neural-network models in § 4.2, which successfully learned this difference in turbulent particle and energy transport.

It is also worth noting that the absence in the database of the exact profiles of the magnetic field, e.g., $q(R), \; B(R), \; \hat s (R)$ , limits the range of models that can be tested without including the results of numerical magnetic-field reconstruction.