Complex structure of turbulence across the ASDEX Upgrade pedestal

The theoretical investigation of relevant turbulent transport mechanisms in H-mode pedestals is a great scientific and numerical challenge. In this study, we address this challenge by global, nonlinear gyrokinetic simulations of a full pedestal up to the separatrix, supported by a detailed characterisation of gyrokinetic instabilities from just inside the pedestal top to the pedestal centre and foot. We present ASDEX Upgrade pedestal simulations using an upgraded version of the gyrokinetic, Eulerian, delta-f code GENE (genecode.org) that enables stable global simulations at experimental plasma $\beta$ values. The turbulent transport is found to exhibit a multi-channel, multi-scale character throughout the pedestal with the dominant contribution transitioning from ion-scale trapped electron modes/micro-tearing modes at the pedestal top to electron-scale electron temperature gradient modes in the steep gradient region. Consequently, the turbulent electron heat flux changes from ion to electron scales and the ion heat flux reduces to almost neoclassic values in the pedestal centre. $E\times B$ shear is found to strongly reduce heat flux levels in all channels (electron, ion, electrostatic, electromagnetic) and the interplay of magnetic shear and pressure gradient is found to locally stabilise ion-scale instabilities.


Introduction
The High-connement mode (H-mode) of plasma operation in a tokamak fusion experiment is characterised by the presence of a radially narrow region just inside the Last Closed Flux Surface (LCFS, also called separatrix) in which turbulent transport is strongly suppressed.In this edge transport barrier the proles of electron and ion temperature as well as density are steep and their gradients large, forming a pedestal that lifts the rest of the proles, ultimately improving fusion conditions in the core of the experiment or future reactor.For this reason, H-mode is the baseline scenario of ITER, the world's largest fusion experiment, currently under construction in Cadarache, France.
In this study, we investigate small-scale turbulence in a pedestal of a standard Type-I ELMy H-mode from ASDEX Upgrade (AUG), which has been experimentally well diagnosed and analysed (Cavedon et al. 2017;Viezzer et al. 2020).Standard H-modes are unstable against Edge Localized Modes (ELMs), which expel large amounts of heat and particles (∼ 10% of plasma energy (Cathey et al. 2020)) tens of times per second, each time crashing the pedestal and causing unacceptably high heat loads on the divertor in future reactors.Therefore, special H-modes (EDA (Stimmel et al. 2022;Greenwald et al. 1999;Gil et al. 2020), QH (Burrell et al. 2005), small ELM/ QCE (Harrer et al. 2022)) and techniques (RMPs (Wade et al. 2015;Suttrop et al. 2011Suttrop et al. , 2018)), vertical kicks (de la Luna et al. 2016), pellets (Baylor et al. 2015)) for the suppression or mitigation of ELMs are an area of very active research.Nonetheless, well-developed standard H-mode pedestals remain useful as a prototype to study what turbulent transport mechanisms persist in this region of strong turbulence suppression.
Turbulent transport in the pedestal is a key ingredient in multiple important aspects of edge physics: It inuences the pedestal height and width as well as the dynamics of ELM cycles and its suppression is the basis for the formation of the edge transport barrier.
The simulation and quantitative prediction of turbulent pedestal transport, however, is challenging.The steep gradients in temperature and density, which characterise the pedestal centre, drive many of the classic core instabilities at once, causing the parallel presence and interaction of dierent instability types (e.g.Ion Temperature Gradient (ITG) modes, Electron Temperature Gradient (ETG) modes and Trapped Electron Modes (TEMs)).Furthermore, the strong localisation of the drive poses fundamental challenges for heat ux calculations of ion scale turbulence with usual local gyrokinetic simulations in which the driving gradients are assumed to be constant over the simulation domain: The strong drive causes large eddies, which require wide simulation domains to prevent the eddy from "biting its own tail" across the periodic boundary condition, which would yield unphysically large heat uxes.Since the gradients are assumed to be constant in the local approach, the total drive of the system in the local simulation ends up much higher (strong drive in the whole simulation domain possibly hundreds of gyroradii wide) compared to the very localised steep gradient region (∼ten gyroradii) and may prevent convergence at all.Additionally, the steep gradients make the inclusion of electromagnetic uctuations important, not only in order to include transport caused by electromagnetic instabilities (Kinetic Ballooning Mode (KBM), Micro Tearing Mode (MTM)) but also because they can signicantly inuence the level of electrostatic heat ux.
In recent years the following picture of pedestal micro-instabilities and transport has emerged: MTM and ETG have been identied to be strong candidates for the most relevant instabilities in the steep gradient region of the pedestal (Kotschenreuther et al. 2019;Hatch et al. 2019Hatch et al. , 2016;;Hassan et al. 2022;Halfmoon et al. 2022;Parisi et al. 2020;Chapman-Oplopoiou et al. 2022).In ASDEX Upgrade also TEMs and KBMs have been found to be relevant (Hatch et al. 2015) and in JET-ILW (ITER-like Wall) ITG has been found to be present (Hatch et al. 2017;Predebon et al. 2023) In this paper, we present a radially resolved analysis of heat transport from pedestal top to foot covering ion and electron scales and including electrostatic as well as electromagnetic uctuations.At the centre of our study are global, nonlinear, electromagnetic gyrokinetic simulations of an AUG pedestal, which have been enabled by an upgrade of the GENE code.They are supported by a detailed local, linear analysis of instabilities and dedicated local, nonlinear simulations of heat ux due to electron scale uctuations.We nd that turbulent ion heat ux is present on the pedestal top and vanishes towards the steep gradient region.This is observed even without including the eect of E × B shear due to a radial electric eld E r but is strongly enhanced by it.A combination of magnetic shear and pressure gradient is identied to locally stabilise modes and suppress heat ux.Our ndings are in agreement with experimental observations that the ion heat ux reduces to neoclassic levels in the pedestal centre.Turbulent electron heat ux remains approximately constant across the pedestal but changes scale from dominantly ion scale TEMs at the pedestal top to small-scale electron temperature gradient driven turbulence in the steep gradient region.
The paper is structured as follows: Sec. 2 introduces the GENE code and its new upgrade for nonlinear, electromagnetic, global simulations, Sec. 3 explains the experimental scenario investigated, Sec. 4 shows the results of linear, local scans to characterise instabilities, Sec. 5 shows the results of global and local nonlinear simulations and nally in Sec.6 conclusions are drawn.

Upgrade of the global electromagnetic GENE code
All simulations presented in this paper have been performed with the GENE code (genecode.org)(Jenko et al. 2000;Görler et al. 2011), which is a gyrokinetic, Eulerian, δf Vlasov code that can perform a variety of simulation types, including: linear, nonlinear, electrostatic, electromagnetic, global, local, neoclassic.
It was found that a certain type, namely nonlinear, electromagnetic, global simulations tended to be prone to numerical instabilities that could be avoided by articially reducing plasma β in simulations e.g.(Hatch et al. 2019).In eect, until now, nonlinear, global simulations could in most cases only be run with reduced β and not with experimental values.In this section the upgrade that overcomes this numerical instability is presented, following the proof-of-principle in (Crandall 2019) based on (Reynders 1993).The same electromagnetic model has recently also been implemented in GENE-3D (Wilms et al. 2021), the stellarator version of the GENE code.Related approaches are also being followed in gyrokinetic particle-in-cell (PIC) codes (Mishchenko et al. 2017).
GENE solves the Vlasov-Maxwell system of equations to calculate the time evolution of a gyrocenter distribution F in 5D phase space.The Vlasov equation for one of the species reads: where ∂X/∂t contains the parallel and perpendicular (gradient-B, generalised E × B, curvature) drifts ∂X/∂t = v ∥ b+B/B * ∥ (v ∇B +v χ +v c ), ∂v ∥ /∂t is the parallel acceleration and C symbolises a collision operator.The parallel acceleration depends on the time derivative of the uctuating part of the magnetic vector potential Ā1∥ (the overbar denotes a gyroaverage): where c is the speed of light, q the charge and m the mass of a given particle species.
In the unmodied GENE code the time derivative of the uctuating part F 1 and Ā1∥ are combined and the distribution function which is evolved in time is g 1 = F 1 − (q/mc) Ā1∥ ∂F 0 /∂v ∥ .This approach exhibits numerical instabilities in global, nonlinear, electromagnetic simulations.The numerical instabilities can be solved by retaining F 1 as the main distribution function and solving for Ā1∥ with an additional eld equation derived from Ampère's law ∇ 2 ⊥ A 1∥ = (−4π/c)j.By applying a time derivative to Ampère's law, using E ind ∥ = (−1/c)∂A 1∥ /∂t and writing the time derivative of F 1 as where the current j has been expressed as a velocity space integral over the gyrocenter distribution function F 1 , R denotes all remaining terms, and the sum goes over all species i. Collecting all terms containing E ind ∥ on one side of the equation the nal eld equation becomes: which is solved numerically.Next to the additional eld equation for the plasma induction this approach requires an additional nonlinear term between the elds.In total these changes increase the computational time per time step by approximately 30%.The new electromagnetic model is furthermore compatible with the use of block-structured velocity grids (Jarema et al. 2017).
Before we discuss the results obtained with the presented upgrade in global, nonlinear simulations in Sec. 5, the following two sections introduce the pedestal under consideration and its linear instabilities.

Experimental scenario: H-mode pedestal of ASDEX Upgrade #31529
The particularly well diagnosed and studied shot AUG #31529 (Cavedon et al. 2017;Viezzer et al. 2020) serves as the basis for our investigation.AUG #31529 has NBI (Neutral Beam Injection) and ECRH (Electron Cyclotron Resonance Heating) heating, with a total heating power of P tot = 8.7 MW, an on-axis B-eld of -2.5 T and a plasma current of 0.8 MA.From this shot, we employ ELM-synchronised proles from (Cavedon et al. 2017) and pressure-constrained magnetic equilibria.Fig. 1 shows the proles and corresponding gradient scale length used in this study.We focus on the time point 6 ms after the ELM crash, where the pedestal is mostly recovered and proles are almost pre-ELM, with a slightly (≈ 7%) reduced electron temperature at the pedestal top.The dashed lines indicate representative positions for pedestal top/shoulder, an intermediate region, pedestal centre, and pedestal foot, where linear, local instability scans have been performed (see Sec. 4).With pedestal top/shoulder we refer to the radial position (ρ tor = 0.86) just before the increase of temperature and density gradients, where the growth rate spectrum is still clearly distinct from the pedestal centre.Fig. 2 shows the E × B rotation due to the radial electric eld E r and the corresponding shear.In Fig. 3 the radial proles of further quantities determining edge physics and microinstabilities are shown: plasma β (strongly falls o from pedestal top to foot), collisionality (strongly increases), the gyroradius (decreases), and the safety factor q (increases but exhibits an intermediate at region).
The microinstabilities that dominate under these physical conditions are characterised with linear, local simulations in the next section.

Local, linear scans
To characterise the instabilities present in the given pedestal we have performed scans with linear local simulations in the binormal wavenumber k y from 0.05ρ i −1 to 350ρ i  and pedestal foot) maximised for each ρ tor and k y over the ballooning angle.At selected wavenumbers, ballooning angles and radial positions we have additionally performed scans in temperature and density gradients (±30%), collisionality (0 -18ν * e ) and plasma β (0% -1%).In total about 7000 simulations were performed for the characterisation of linear instabilities.
We use the following criteria to distinguish between the dierent instabilities: Parity of the parallel mode structure in ballooning representation (tearing or ballooning), size (ion or electron gyroradius), drift direction (ion or electron diamagnetic), sensitivity on gradients (T e , T i , n), dependence on collisionality and plasma β as well as diusivity and heat ux ratios following the ngerprint approach (Kotschenreuther et al. 2019).An electromagnetic mode on scales > ρ i with tearing parity is called MTM.An electromagnetic mode on scales > ρ i with ballooning parity and drift in ion diamagnetic direction is called KBM.An electrostatic mode on scales ≈ ρ i , which is stabilised by collisionality and is not destabilised by ∇T i is called TEM.An electrostatic mode on scales ≈ ρ i , which is destabilised by ∇T i and propagates in ion diamagnetic direction is called ITG.An electrostatic mode on scales ⪅ ρ i that is destabilised by ∇T e is called ETG.However, it should be emphasised that a clear categorisation of modes on ion scales in the region of very strong drive close to the separatrix is particularly challenging.Dierent drive mechanisms interact and fuel instabilities with characteristics that do not fall neatly in the mode prototypes developed in the study of core turbulence.
At the pedestal shoulder (ρ tor = 0.86, violet triangles in Fig. 4) we nd MTMs on largest scales, TEMs at ion scales, a region of stable wavenumbers and then ETGs on electron scales.At the intermediate position (ρ tor = 0.94, blue squares in Fig. 4), where the magnetic shear has a minimum, a growth rate gap without any clear mode on ion scales exists.On intermediate scales, more ETG modes occur, that were not present at the pedestal shoulder.In the steep gradient region (ρ tor = 0.97, green crosses in Fig. 4) on ion scales we nd ITG modes that are close to a KBM transition.At smaller scales, ETG driven modes are present that extend to larger scales towards pedestal centre and foot.At the pedestal foot (ρ tor = 0.99, yellow circles in Fig. 4) on ion scales modes show an ETG/TEM character but tend to be destabilised with increasing collision frequency.
The ETG modes at pedestal centre and foot tend to peak increasingly at nite ballooning angle, indicating the presence of toroidal ETG modes (Parisi et al. 2020;Told et al. 2008).
Overall growth rates increase from pedestal top to foot.Scans over plasma β at dierent radial positions show that the pedestal in these linear local simulations sits close to a KBM threshold -being closer at the pedestal foot than at the pedestal top (cf.Fig. 5).The closeness to the KBM threshold in the pedestal centre resembles the use of a KBM constraint in the EPED model (Snyder et al. 2009) for the prediction of the pedestal width.It has, however, been reported that the radial structure of KBMs may not be compatible with the narrow pedestal region (Hatch et al. 2019;Predebon et al. 2023), suggesting that the details of the KBM threshold may be skewed by the local approximation.

Pressure and magnetic shear eect: 2nd stability region
The low growth rates on ion scales in the intermediate region (ρ tor = 0.94) between pedestal top and steep gradient are caused by the interplay of low magnetic shear and already increased pressure gradient in this region.Their interplay locally stabilises ballooning modes (2nd stability region).This can be shown by a scan over the magnetic shear at this position (see Fig. 6, right plot).At nominal parameters (black crosses) no  clear ion scale mode is present, but with increasing magnetic shear ŝ (green triangles and yellow circles) the system leaves the 2nd stability region and an ion scale mode becomes unstable.At the pedestal shoulder, where a lot of transport is driven on ion scales, lowering the magnetic shear, unfortunately, does not decrease growth rates signicantly, since the pressure gradient is too low to access the 2nd stability region (Fig. 6, left plot green triangles).In contrast, a high magnetic shear lets the pedestal shoulder enter the 1st stability region, where TEMs are suppressed (yellow circles).Further scans with e.g.modied proles are not within the scope of this paper, but these results illustrate the potential of reducing microturbulence instabilities through careful tailoring of safety factor and pressure proles.
In the next section, we analyse the nonlinear turbulent system that is fuelled by the presented instabilities and compare to what extent linear mode signatures prevail in the nonlinear state.To reduce computational cost ion scales and electron scales are treated separately.

Global, ion scale simulations
To calculate heat uxes we have performed gradient-driven, global, nonlinear, electromagnetic simulations on ion scales at experimental β values, which have been enabled by the code upgrade presented in Sec. 2. These simulations cover ρ tor = 0.85 − 0.995 in radius (almost the full width shown in Fig. 1) and k y ρ i = 0.05 − 1.6 in binormal wavenumber (corresponding to a toroidal wavenumber N tor ≈ 4 − 124), indicated by the left shaded region in Fig. 4.They are two species simulations (Deuterium and electrons) with correct mass ratio (m e /m D = 1/3670), collisions (Landau collision operator), perpendicular magnetic uctuations Ā1∥ , but without compressional magnetic perturbations B 1,∥ .When indicated, the simulations include the eect of E × B rotation due to the radial electric eld E r (cf.Fig. 2).For numerical reasons the background ow in GENE is restricted to the toroidal direction.As an approximation to the E × B shear eect we retain the magnitude of v E×B , but rotate it to be purely toroidal.More details on e.g.grids are specied in the appendix.
The obtained heat uxes are shown in Fig. 7 as a function of time averaged over real space (ux surface and radius, including the radial buer zones (see appendix)).In both, ion and electron channel, the electrostatic heat ux dominates.But while the ions show vanishing electromagnetic heat ux, the electron heat ux is about 1/4 electromagnetic.When including E × B shear in the simulations the heat ux is strongly damped in all channels.The electrostatic by a factor of three and the electromagnetic even stronger.

Connecting linear and nonlinear results: Frequencies and cross phases
To identify the dominant turbulent transport mechanisms, results from linear and nonlinear simulations have to be connected, to test if the linearly fastest growing modes remain important in the nonlinear saturated state.This can be achieved by comparing frequencies and cross phases.
Fig. 8 shows the mode frequencies of the linear and nonlinear simulations.The left plot shows the frequency spectrum of the local, linear simulations for three radial positions (pedestal shoulder at ρ tor = 0.86, centre at ρ tor = 0.97 and foot at ρ tor = 0.99), in the co-moving frame of reference.In the pedestal shoulder spectrum (violet triangles) the transition from MTM to TEM is visible and at the pedestal centre (yellow circles) the transition from ITG to TEM/ETG.The right plot shows the frequency spectrum of the global, nonlinear simulation analysed at two positions (pedestal shoulder and centre) overlaid by the local, linear spectra.Usually frequencies from local simulations are specied in a frame co-moving with the E ×B rotation.Since E ×B rotation depends on radial position, frequencies of global simulations are specied in the lab frame and for the comparison of both we transform the local frequencies also to the lab frame.The comparison shows that at the pedestal shoulder and even centre the linear frequencies persist in the nonlinear simulations.This indicates that the linearly fastest growing modes remain important in the saturated turbulent state.However, one important dierence appears: At the pedestal shoulder (ρ tor = 0.86) and k y ρ i ≈ 0.3 the nonlinear simulation does not show the mode transition that linear, local simulations present, indicating, that MTMs are suppressed or at least restricted to the very largest scales in global, nonlinear simulations compared to the local, linear ones.
The cross-phases of electric potential and electron density uctuations (see Fig. 9) support this picture: Linear mode structures survive in the nonlinear simulations, but on largest scales at the pedestal shoulder dierences are visible, corroborating the suppression of MTM in global, nonlinear simulations observed in the frequency comparison.Overall, the remarkable agreement seen in the frequency and cross phase comparison between local/linear and global/nonlinear simulations at pedestal shoulder and centre encourages the extension of quasi-linear models to the pedestal region.

Heat ux prole
To study how turbulent transport changes across the pedestal we consider the heat ux averaged over time and ux surface as a function of the radius.Fig. 10 shows important aspects of the turbulent heat ux prole in the studied ASDEX Upgrade pedestal.Particular care has to be taken with regard to the normalisation of the heat ux.Due to the strong temperature changes across the pedestal, the commonly used gyro-Bohm heat ux based on mixing-length estimates Q gB = c s p i ρ * 2 = (T i /m i ) 1/2 n i T i ρ * 2 changes by two orders of magnitude across the pedestal (top left plot).A modied gyro-Bohm heat ux Q gb,mod = Q gb × max (a/L T i , a/L T e ) 2 , in which the minor radius is replaced by the gradient length as the macroscopic length scale, exhibits strong variations across the pedestal as well.We will therefore focus the analysis on the heat ux in SI units.
Fig. 10 (top right) shows the proles of the dierent heat ux components (electrostatic, electromagnetic, electron, ion).The dominant component over most of the pedestal is electrostatic electron heat ux (blue solid line), with the exception of ρ tor ≈ 0.97, where the electrostatic ion heat ux has a local peak.This peak corresponds to the ITG/KBM mode identied in the linear analysis, which occurs at the peak of the ion temperature gradient.Interestingly, the turbulent ion scale heat ux is strongly reduced in all channels from pedestal top to pedestal centre and foot -even without E × B shear.The onset of this reduction coincides with the region of linear stabilisation, discussed in Sec.4.2.A comparison to power balance and neoclassical calculations shows qualitative agreement in heat ux structure and trends (bottom right plot).The electron channel is dominant throughout the pedestal and roughly constant.Our gyrokinetic simulations reveal that while the total electron heat ux remains constant, it transitions in scale.At the pedestal top/shoulder it is driven by ion scale TEM/MTM turbulence while at the pedestal centre and foot it is driven by small-scale ETG turbulence.The region around ρ tor = 0.92 where the total gyrokinetic electron heat ux reaches a minimum likely indicates a limitation of our separate scale ansatz.At this location heat ux is likely driven by scales that are neither resolved in our global simulations nor in the electron scale simulations (cf.Fig. 4) and would require multi-scale simulations to be properly resolved.The ion channel contributes substantially to the total heat ux at the pedestal top/shoulder and reduces to neoclassic values towards the pedestal centre.At the pedestal foot our gyrokinetic simulations do not show the increase of ion heat ux suggested by power balance.The dierences in the grey-shaded region are possibly due to increased measurement uncertainties in the ion prole that aect both power balance (see (Viezzer et al. 2020)) and gyrokinetic simulations.
For the power balance comparison, the turbulent ion heat ux component was estimated by subtracting neoclassic heat ux calculated with NEOART from the ASTRA result (both from (Viezzer et al. 2020)).The total ion heat ux due to power balance is constant (Viezzer et al. 2020) and the minimum in the turbulent heat ux is compensated by an increased neoclassical heat ux, which has a roughly constant diusivity across the pedestal so that its heat ux follows the increasing ion temperature gradient.
The following picture emerges for the investigated scenario: Turbulent heat ux at the pedestal top is dominated by electrostatic TEMs with electromagnetic MTM contributions.The interplay of low magnetic shear and increasing pressure gradient stabilises modes locally before the steep gradient region (ρ tor ≈ 0.94) and reduces heat ux.E × B shear suppresses heat ux strongly in all channels and widens the region of vanishing ion scale heat ux.Turbulent electron heat ux changes scale across the pedestal and turbulent ion heat ux strongly reduces towards the pedestal centre.

Discussion & Conclusions
We have presented a gyrokinetic analysis of an ASDEX Upgrade ELMy H-mode pedestal.The most unstable microinstabilities from just inside the pedestal top to foot were characterised with extensive linear, local scans.At the pedestal top/shoulder MTM, TEM, and ETG were found.In the intermediate region before the pedestal centre modes on ion scales are stabilised by the interplay of magnetic shear and pressure gradient, while on intermediate electron scales additional ETG modes become unstable.In the pedestal centre ITG modes close to the threshold to KBMs were observed and at the pedestal foot modes with TEM/ETG character that are destabilised with increasing collision frequency are present.With nonlinear, global, electromagnetic ion scale simulations and nonlinear, local ETG simulations we have analysed the heat ux in the pedestal -resolved in radius and scale.The ion scale simulations are enabled by an upgrade of the GENE code (cf.Sec. 2).We nd TEM-driven turbulence with electromagnetic components due to MTMs to be dominant at the pedestal top/shoulder.A combination of linear stabilisation and E × B shear suppresses ion scale turbulence towards the steep gradient region.While the turbulent electron heat ux is picked up by small-scale ETG modes, the ion channel reduces to neoclassic heat ux levels.
The global electromagnetic simulations presented in this study are among the most realistic pedestal turbulence simulations performed to date.Together with dedicated local ETG simulations and the extensive linear instability characterisation, they help to conrm the important role of E × B shear stabilisation for pedestal turbulence and demonstrate the transition in scale of electron turbulence from ion scales at pedestal top to electron scales in the steep gradient region.This well-resolved characterisation of turbulence across a pedestal supports the development of reduced models for edge turbulence and the understanding of the L-H transition.
Two limitations of our current approach should be addressed in future work: The separation of ion and electron scale simulations prohibits any mutual scale interaction and excludes a range of wavenumbers in our setup.Multi-scale simulations would promise new insights into how turbulent transport transforms across the pedestal.Furthermore, the use of eld-aligned coordinates strictly restricts our simulation domain to the region of closed ux surfaces.Approaches that can cross the separatrix would be well suited to expand the study of pedestal foot turbulence.

Figure 1 .
Figure 1.Proles (top) and gradient scale length 1/L X = −∂ r X(r)/X (bottom) of ion temperature (orange), electron temperature (blue) and density (green) of AUG #31529 6ms after the ELM crash.It is assumed that n e = n i .The dashed lines indicate positions where instabilities have been characterised in detail (see Sec. 4).

Figure 2 .
Figure 2. E × B rotation velocity (left) and corresponding shear (right) caused by the edge radial electric eld E r .

Figure 3 .
Figure 3. Proles of further relevant quantities inuencing microinstabilities and edge turbulence: Plasma β (top left), collisionality (top right), ρ * (bottom left) and safety factor q combined with magnetic shear ŝ (bottom right).

Figure 4 .
Figure 4. Growth rate spectra for four radial positions.Shaded regions indicate the wavenumber ranges used in nonlinear heat ux simulations.

Figure 5 .
Figure 5. Growth rate scan over plasma β at k y ρ i = 0.075 and k x,center = 0 at four radial positions.Nominal β value indicated in orange.

Figure 6 .
Figure 6.Linear, local growth rates as a function of the toroidal mode number N tor for three magnetic shear (ŝ) values at two dierent radial positions (left: pedestal shoulder ρ tor = 0.86, right: low shear region ρ tor = 0.94).Black crosses are nominal growth rates.

Figure 7 .
Figure 7. Heat uxes (electrostatic Q es and electromagnetic Q em ) for electrons (blue, left) and ions (red, right) as a function of time in MW.The green continuations are performed including an external background velocity shear corresponding to experimentally measured E × B shear.

Figure 8 .
Figure 8. Frequency spectrum on ion scales for linear, local simulations in the co-moving frame (left) and global, nonlinear results compared to the linear, local scans in the lab system (right).The dashed line (left) indicates the transition between ion and electron diamagnetic drift direction.

Figure 9 .
Figure 9. Cross-phases of electric potential ϕ and electron density uctuations n from nonlinear simulations (blue background) and linear simulations (orange circles).
Fig. 10 (bottom left) illustrates the inuence of E × B shear on the heat ux proles: It reduces the ion scale heat ux strongly and widens the region of almost vanishing turbulent heat ux.Fig. 10 (bottom right) shows that the turbulent heat ux on electron scales behaves oppositely across the pedestal compared to the ion scale turbulent heat ux: It vanishes at the pedestal top and strongly increases down the pedestal (stars).

Figure 10 .
Figure 10.Turbulent heat ux prole in an ASDEX Upgrade pedestal from pedestal top to foot.Top left: Gyro-Bohm heat ux prole.Top right: Components of the ion scale heat ux prole without E × B shear.Bottom left: Total heat ux (Q es +Q em ) due to ion scale turbulence with and without E × B shear.Bottom right: Total heat ux due to ion scale turbulence from global simulations (red and blue solid line) as well as ETG heat uxes from local simulations at nominal values (light blue stars) and increased electron temperature gradient (dark blue stars) compared with power balance calculations (broad lines).Region of increased measurement uncertainty in grey.
. Most of these studies are based on local/linear, local/nonlinear, global/linear, or reduced β simulations.However, to study which turbulent mechanisms drive transport under real pedestal conditions, global/nonlinear simulations at experimental β values are indispensable.Such simulations are presented in this work.