Stability and transport of gyrokinetic critical pedestals

A gyrokinetic threshold model for pedestal width–height scaling prediction is applied to multiple devices. A shaping and aspect ratio scan is performed on National Spherical Torus Experiment (NSTX) equilibria, finding Δped=0.92A1.04κ−1.240.38δβθ,ped1.05 for the wide-pedestal branch with pedestal width Δped , aspect ratio A, elongation κ, triangularity δ, and normalized pedestal height βθ,ped . The width–transport scaling is found to vary significantly if the pedestal height is varied either with a fixed density or fixed temperature, showing how fueling and heating sources affect the pedestal density and temperature profiles for the kinetic-ballooning-mode (KBM) limited profiles. For an NSTX equilibrium, at fixed density, the wide branch is Δped=0.028(qe/Γe−1.7)1.5∼ηe1.5 and at fixed temperature Δped=0.31(qe/Γe−4.7)0.85 ∼ηe0.85 , where qe and Γe are turbulent electron heat and particle fluxes and ηe=∇ln⁡Te/∇ln⁡ne for an electron temperature Te and density ne . Pedestals close to the KBM limit are shown to have modified turbulent transport coefficients compared to the strongly driven KBMs. The role of flow shear is studied as a width–height scaling constraint and pedestal saturation mechanism for a standard and lithiated wide pedestal discharge. Finally, the stability, transport, and flow shear constraints are combined and examined for an NSTX experiment.


Introduction
The H-mode pedestal is an edge transport barrier that forms when a strongly heated tokamak plasma transitions into a high confinement regime [1,2].Due to a significant improvement in confinement, H-mode is a leading candidate for burning plasma scenarios [3][4][5][6][7][8].The pedestal pressure profile has a characteristic width and height [9], whose values can be found with the EPED stability threshold model [10], often with remarkable success [7,[11][12][13][14] and some exceptions [15,16].EPED combines a local gradient constraint controlled by infinite-n ideal-ballooning-mode (IBM) stability [17] and a macroscopic stability constraint controlled by peeling-ballooning-mode (PBM) stability [18]; the intersection of IBM and PBM constraints gives a pedestal width-height prediction.
However, extra information from pedestal models is required in order to understand current experiments and design future devices [19,20].For example, while IBM and PBM stability might suffice to predict the pedestal pressure's width-height trajectory, the separate evolution of density and temperature pedestal profiles is needed to determine plasma heating and fueling sources that are consistent with such profiles.Given that gyrokinetic instabilities are often sensitive to density and/or temperature gradients and their relative sizes [21,22], non-IBM transport mechanisms are expected to play a significant role in pedestal evolution [23][24][25][26][27][28].Equilibrium flow shear, part of which is generated by the temperature and density profiles themselves, is also important in pedestal formation [29] and the inter-ELM cycle [30], and is hypothesized as necessary at the pedestal top to allow the pedestal to widen [10] by stabilizing ion-temperature-gradient and trapped-electron-mode instabilities.Given the importance of flow shear in the pedestal -and also its pessimistic scaling with ρ i /a to future devices [31], where ρ i is the ion gyroradius and a the minor radius -generating sufficient flow shear in future reactors (or developing alternative methods [32]) is an important task.
The pedestal is also highly coupled to the core and scrape-off layer (SOL) [33][34][35], meaning that the pedestal's evolution cannot be considered in isolation.Solving this coupling in sufficient detail remains one of the biggest challenges in fusion research [36] due in part to the high dimensionality of the problem, the complex interactions between phenomena, and the uncertainties in models and measurements.For example, predicting the pedestal density profile is a very hard problem.The density pedestal in current experiments is sensitive to neutral particles ionizing in the pedestal and to transport processes [35,[37][38][39][40][41].Changes in divertor and SOL physics and the neutral ionization source in the pedestal will change the pedestal pressure and current profile, in turn strongly affecting core physics, which in turns affects the pedestal, and so on.The strong coupling of the pedestal to other regions motivates high-fidelity models to prevent large error propagation.
In this work, we focus on three challenges mentioned above: (1) accuracy of the pedestal width-height prediction, (2) pedestal heat and particle transport constraints, and (3) flow shear at the pedestal top.
For (1), we introduce a linear gyrokinetic threshold model that builds on the EPED Ballooning Critical Pedestal model [10] to predict the width-height pedestal scaling across aspect-ratio, shaping, and devices.We call these widthheight scaling expressions Gyrokinetic Critical Pedestal constraints.We perform a study of how shaping and aspect-ratio enter the gyrokinetic pedestal widthheight scaling, which may explain the differences in width-height experimental measurements between most devices and the National Spherical Torus Experi-ment (NSTX) [10,15,42], with NSTX typically featuring much wider pedestals.Varying the pedestal shaping and aspect-ratio may also provide new opportunities for controlling pedestal width and height in experiments.For (2), we study heat and particle transport around the gyrokinetic width-height scalings and find the dependence of pedestal width on the electron heat to particle transport ratio.For (3), we study flow shear at the pedestal top in a regular NSTX ELMy H-mode and a NSTX ultra-wide enhanced pedestal (EP) H-mode [43].
The paper layout is as follows: in Section 2, we describe the process for varying the pedestal width and height with self-consistent equilibrium reconstruction.We introduce the gyrokinetic and ideal-ballooning framework used for studying these equilibria.A linear threshold model for gyrokinetic and idealballooning stability is then applied to several devices in Section 3 to find widthheight scaling expressions.The shaping and aspect-ratio dependence of the pedestal width-height-shaping scaling is explored in Section 4. In Section 5, we study gyrokinetic turbulent transport properties close to the Gyrokinetic Critical Pedestal boundaries and find scaling expressions that relate the pedestal width and turbulent transport ratios.In Section 6 we explore the role of flow shear as an additional width-height constraint.In Section 7, we combine the stability, flow shear, and transport constraints to determine regions of pedestal width-height space that are accessible.

Width-Height Scaling Workflow
In this section, we describe the workflow to compute the width-height scaling using the new framework gk ped, outlined in Figure 1.Currently, gk ped is implemented as a module in the integrated modeling and data analysis software OMFIT [44].In addition to the OMFIT tools, we use EFIT-AI [45] combined with EFUND [46] for equilibrium reconstruction, GS2 [47,48] and CGYRO [49] for gyrokinetic simulations, and BALOO [50] and ball stab [48,51] for infinite-n ballooning simulations.

Equilibrium Variation and Reconstruction
The first step in gk ped is to input an equilibrium and profiles.The electron temperature is parameterised as with the same functional form for electron density n e [9,10].Here, H is a step function, ψ is the poloidal flux normalized to 0 at the magnetic axis and 1 at the last-closed-flux-surface, T e,c , T e0 , α T1,2 are constants, ∆ Te and ∆ ne are the electron temperature and density pedestal widths.The pedestal heights T e,ped and n e,ped are T e,ped = T e (ψ ped,Te ), n e,ped = n(ψ ped,ne ) where T e,sep and n e,sep are evaluated at ψ = 1, and ψ ped,Te = ψ mid,Te − ∆ Te /2, ψ ped,ne = ψ mid,ne − ∆ ne /2, For the ions, the density profiles are given by quasineutrality and the temperature profiles by enforcing constant T i /T e across the radial profile.The current density J(ψ) is the sum of bootstrap J bs and non-bootstrap J non−bs contributions, We vary the pedestal width and height with constant plasma current I p , constant β N = β T aB T 0 /I p , and self-consistent bootstrap current.Here, β T = 8π⟨p⟩ V /B 2 T 0 , where ⟨p⟩ V is the pressure averaged over the plasma volume and B T 0 is the magnetic field strength at the magnetic axis.In order to keep I p constant due to changing J bs resulting from different profiles, J non−bs (obtained from the original input equilibrium), is multiplied by a constant k.The bootstrap current can be calculated from several formulae in the literature [52,53] that are implemented in OMFIT, but is recommended that it be the same formula that generated the original input equilibrium.The quantity β N is kept constant by an iterative procedure that allows β N to vary from the input equilibrium by at most 1%.
Once the user is satisfied with the profile parameterization, the pedestal width and height are varied.The height can be varied in two ways: at constant T e,ped with varying n e,ped , or at constant n e,ped with varying T e,ped .As a default option, n e,ped /n e,sep is held constant when varying n e,ped and T e,sep is held constant when varying T e,ped .The total pedestal width is the pedestal top location is ψ ped = ψ mid − ∆ ped /2 where ψ mid = (ψ mid,ne + ψ mid,Te )/2, and the normalized pedestal height is where p ped = 2p e (ψ = ψ ped ) and B pol = 4πI p /lc with flux surface circumference l and speed of light c [10,42,54].

Gyrokinetic and Ideal Analysis
In this section, we describe briefly the gyrokinetic and ideal simulations performed in the pedestal, and the gyrokinetic mode identification scheme.We use GS2 [48] and CGYRO [49] to solve the electromagnetic gyrokinetic equation [55][56][57][58] for distribution function and Maxwell's equations.Here, δf s is the total turbulent distribution function, F 0s is a Maxwellian, Ṙ is the guiding-center particle drift, ϕ and χ s are the electrostatic and gyroaveraged gyrokinetic potentials [58], v χ,s = cB × ∇χ s /B 2 , B is the magnetic field, and C s is a collision operator.We also use BALOO [50] and ball stab [48,51] to solve the infinite-n ballooning equation for the ballooning eigenfunction Y and frequency ω [17], ω 2 λY = −d/dθ[g dY /dθ] + uY where g, u, and λ are geometric coefficients [59].We typically solve both ideal and gyrokinetic equations in the pedestal width-height model.Linear gyrokinetic simulations are performed for a range of binormal wavenumbers k y , typically at scales where kinetic-ballooning-modes (KBMs) [60,61] are expected to be most virulent, i.e. k y ρ i ≈ 0.05 − 0.2, and in a radial domain across the pedestal that is evenly spaced in ψ.Here, ρ i is the ion gyroradius.Once simulations are converged, a mode finder routine adopting a fingerprintslike [38] approach is used to identify the mode type.To aid mode identification, each gyrokinetic simulation is launched with a second simulation with a slightly increased β value (where dβ/dψ is unchanged) -this is particularly helpful for distinguishing between KBMs and trapped-electron-modes (TEMs) [62,63], since both KBMs and TEMs often have similar transport characteristics [38,64].In Table 1, we describe the criteria for each gyrokinetic mode.In addition to KBMs and TEMs, we consider micro-tearing modes (MTMs) [65][66][67][68], electron-temperature-gradient (ETG) modes [22,47,69,70], tearing ETG (TETG) modes [71], and ion-temperature-gradient (ITG) modes [21,[72][73][74].In Table 1, χ s and D s are the turbulent heat and particle diffusivities (later defined in Equation ( 9)), [75] is the mode parity in the fluctuating field A ∥ , γ is the linear growth rate, and ω R is the real frequency.
Because of the wide range of devices we study and the often unusual nature of gyrokinetic instabilities in the pedestal, many of the mode criteria quantities in Table 1 are not hard cutoffs, but rather, approximate values that the mode should satisfy -the more criteria that a given instability satisfies, the more confidence in a given mode type identification.In our experience, distinguishing between KBM, TEM, and ITG was often challenging and required using D e /(χ e + χ i ) and ∂γ/∂β.With the exception of the MTM, we ignore the mode's real frequency in automated mode identification.For all gyrokinetic and ideal simulations in this paper, we only study the radial wavenumber k x = 0   because we focus on KBM stability.While k x = 0 is typically the most unstable mode for KBM [76], for other instabilities this may not be the case [69,[77][78][79][80][81].
We also perform ideal infinite-n ballooning simulations at the same radial locations as the gyrokinetic simulations.At each flux-surface, there are three stability states for the ideal mode: first-stable, unstable, and second-stable.

The Ballooning and Gyrokinetic Critical Pedestal
In this section, we describe the calculation of the Gyrokinetic Critical Pedestal (GCP) and Ballooning Critical Pedestal (BCP) [10] using information from gyrokinetic and ideal infinite-n simulations.
Extensive experimental and theoretical work has shown that the pedestal pressure gradient is often limited by ballooning modes [10,[82][83][84][85].In the EPED model, the ideal infinite-n mode is hypothesized to set the steepest pressure gradient a pedestal achieves across the pedestal half-width.If every radial location within the half-width is ideal-ballooning unstable, EPED determines that the pedestal profile is no longer physically accessible, a constraint known as the Ballooning Critical Pedestal (BCP) [10].The EPED model has been applied successfully to multiple experiments [7,[11][12][13][14].However, recent work [42] showed that for ELMy NSTX pedestals, kinetic-ballooning -rather than ideal-ballooning -stability is needed to match width-height scalings with experiment [15].Such a constraint using the KBM stability threshold is called the Gyrokinetic Critical Pedestal (GCP) [42,86,87].The pedestal half-width region used for the BCP and GCP is shown schematically in Figure 2(a).For the GCP calculation, if KBM is unstable at any k y ρ i wavenumber for a given radius, that radius counts as 'unstable' toward the GCP calculation.If all radii within the pedestal half-width are 'unstable,' the pedestal is GCP unstable.A GCP unstable and stable pedestal are shown schematically in Figure 2(b).
Practically, to find the BCP and GCP we start from an input equilibrium, typically calculated from an experiment, and construct a set of equilibria with varied pedestal width and height as outlined in Section 2.1.We then evaluate ideal-ballooning and gyrokinetic stability across the pedestal half-width on all of these equilibria, and find the boundary in ∆ ped , β θ,ped coordinates between equilibria that are accessible and inaccessible according to the BCP and GCP.A width-height scaling is found by fitting this boundary to a function with constants C and G In this paper we describe the two cases of varying β θ,ped at fixed n e,ped and fixed T e,ped , which previous work [42] has shown gave significant differences in the width-height scaling.

Pedestal Bifurcation
Recently, it was shown that a bifurcation in KBM stability caused by aspectratio and shaping might be responsible for the variation in width-height scaling across devices [86].If both first and second KBM stability can be accessed robustly across the pedestal half-width at different widths and heights, this led to two solutions for ∆ ped : a wide and narrow GCP branch.In Figure 3, we show an example of an equilibrium where both the wide and narrow GCP branches exist.The mode fraction in Figure 3 corresponds to the fraction of all modes across the pedestal half-width and simulated k y ρ i wavenumbers that are KBMs.In Figure 3, three binormal wavenumbers are used k y ρ i ∈ [0.06, 0.12, 0.18], so the minimum KBM mode fraction required to trigger the wide or narrow GCP is 1/3.Throughout this paper, we will refer frequently to the wide and narrow GCP branches.It has also been demonstrated that there is a bifurcation in the macroscopic constraint for pedestal prediction, peeling-ballooning-mode (PBM) stability.First and second PBM stability was achieved in DIII-D by higher fueling and strong positive triangularity [7,13] and in TCV by varying the triangularity from negative to positive [85].

Device Scan
In this section, we give examples of the GCP and BCP calculations for NSTX, DIII-D, and STAR devices.The GCP for more devices is given in [86].

NSTX
We study two NSTX discharges: NSTX 139047 is an ELMy NSTX H-mode [15], and NSTX 132588 [88] is a ultra-wide-pedestal lithiated enhanced-pedestal (EP) H-mode.In Figure 4 we plot the GCP with solid lines and the BCP with dash-dotted lines.For both NSTX 139047 and NSTX 132588, the width-height scaling expression is in excellent agreement with the experimental point, strong evidence that KBM is limiting the pedestal width and height.Notably, for NSTX 139047, the GCP gives a less steep pedestal than the BCP, indicating that the pedestal is limited by KBM first-stability, and therefore is the wide GCP.In contrast, for NSTX 132588, the GCP gives a steeper pedestal than the BCP, indicating that the pedestal is in KBM second-stability and is therefore the narrow GCP.

DIII-D
We now find the BCP and GCP for DIII-D 163303 [89], a previously published ELMy H-mode discharge that is used to study wall conditions, the L-H transition power threshold, and outgassing.Shown in Figure 4, the experimental point for this equilibrium is at a β θ,ped value slightly above the ballooning critical pedestal but slightly below the gyrokinetic critical pedestal.However, within 20% uncertainty, this equilibrium is consistent with both the BCP and GCP, indicating that this pedestal is likely ballooning-limited.

STAR
The Spherical Tokamak Advanced Reactor (STAR) is an A = 2, R 0 = 4m compact high-field tokamak [90] targeting 100-500 MWe net electric power.Here, A = R 0 /a is the aspect-ratio for minor radius and major radius R 0 .In Figure 4, the STAR device is shown to achieve high values of β θ,ped while having a narrower pedestal.Similar to NSTX discharge 132588, STAR relies on accessing KBM second-stability to obtain its steeper kinetic-ballooning-constrained pedestals, which shown in Figure 4, causes the GCP to give a steeper pedestal prediction than the GCP.A final β θ,ped , ∆ ped design point has not yet been determined.Notably, STAR achieves a relatively steeper pedestal measured approximately by β θ,ped /∆ ped due to its low magnetic shear values across most of the pedestal, causing the KBM and ideal-ballooning-mode to be second stable.

Aspect-Ratio and Shaping Scan
In this section, we describe briefly aspect-ratio and shaping scans on ELMy NSTX discharge 132543 [91].We choose a Luce parameterization for the lastclosed-flux-surface shape [92].We define the flux surface elongation κ and triangularity δ as the average ⟨. ..⟩L of the Luce parameters κ = ⟨κ⟩ L , δ = ⟨δ⟩ L .The gk ped tools can perform aspect-ratio and shaping scans and find the BCP and GCP scaling expressions from the resulting new equilibria.The shaping and aspect-ratio scaling results for a different NSTX equilibrium are detailed in [86].For aspect-ratio scans, we use EFIT-AI [45] to construct new equilibria for a range of aspect-ratios keeping the minor radius a constant but allowing the major radius R 0 to vary.In Figure 5(a), we plot the last-closed-flux-surfaces for A ∈ [1.5, 1.96, 2.42, 2.88, 3.34, 3.8] -for each aspect-ratio, a new magnetic coilset is generated by modifying the aspect-ratio of a basecase NSTX coilset, which is used to generate the Green's functions needed for free-boundary equilibrium reconstruction by the EFUND code [46].The boundary points are fixed and the equilibrium parameters are rescaled as follows This gives q 95 ∼ R −3 0 and bootstrap fraction, f bs ∼ R −0.9 0 .In Figure 5(b), we show how these quantities vary with aspect-ratio.For the highest major radius R 0 ≃ 2.4m, the flux-function R 0 B T 0 ≃ 2.4[Tm], which is comparable to or lower than JET [93], SPARC [8], and DIII-D [94].Thus, our high-aspect-ratio equilibria respect reasonable engineering requirements for R 0 B T 0 .
For triangularity scans, we fit the last-closed-flux-surface using a Luce parameterization and rescale δ = (δ upper + δ lower )/2 by a scalar factor.When varying triangularity, we keep all other plasma parameters constant.For elongation scans, we choose to vary the plasma current in order to keep N , in order to keep β N constant we vary plasma current as I p ∼ 1 + κ 2 at fixed A, a, B T [95].
Once the equilibria with different aspect-ratio and shaping are generated, we evaluate the Gyrokinetic Critical Pedestal for each equilibrium.

Least-Squares Fit
Based on our shaping and aspect-ratio scans, we perform a least-squares fit for the pedestal width of the form using the GCP from 11 shaping scans for NSTX 132543.In this paper, we fit the GCP width-scaling for only the wide GCP branch, finding with functional dependencies similar to fits performed in [86] for a different NSTX equilibrium.The fitting parameters in Equation ( 8) have the values and standard deviation uncertainty: C = 0.92 ± 0.16, b = 1.04 ± 0.16, d = −1.24±0.26,f = 0.38±0.04,g = 1.05±0.12.In Figure 6, we plot the normalized width versus fitting and shaping parameters A, κ, and δ.The R 2 value for all fitting parameters is R 2 = 0.94, indicating a relatively good fit.Removing any one of A, κ, δ reduced the R 2 value significantly, indicating the importance of all three of these parameters.This analysis can be improved in future work by: (1) increasing sample size, (2) performing fits on multi-discharge and multi-device experimental database, (3) a comprehensive assessment of more shaping and plasma parameters to find the most important parameters determining ∆ ped .

Turbulent Transport Near The Gyrokinetic Critical Pedestal
In this section, we relate the pedestal width scaling to turbulent transport.We also use transport ratios from linear gyrokinetic simulations to study turbulent transport in the vicinity the GCP first and second stable branches.In these regions, the heat to particle transport ratios from KBM stability can vary significantly compared with transport in strongly-driven KBM regions.Given that experimentally the pedestal is often close to the GCP and far from the strongly-driven KBM regions, the changing transport properties of KBM and other modes has implications for the evolution of density and temperature profiles.In this section, we use equilibria based on NSTX discharge 139047 that have a range of aspect-ratio values, detailed in Table 2. Unless mentioned otherwise, the nominal case used is NSTX discharge 139047 with a slightly increased aspect-ratio A = 2.0, referred to as A 2 .

Gyrokinetic Stability and Transport
Linear gyrokinetic simulations provide information about dominant mode type around the GCP.In Figure 7, we plot the fraction of different mode types across the pedestal half-width for binormal wavenumbers k y ρ i ∈ [0.06, 0.12, 0.18] for NSTX discharge 139047 A 2 .In the GCP unstable region, KBM dominates, with mode fractions lying in 0.5 − 1.0.At pressures below the wide GCP branch, MTM is ubiquitous and for pressures above the narrow GCP branch, TEM is the most common mode.We expect TEM instability generally for higher k y ρ i  values than included in these simulations, which explains the relatively low TEM fraction at pressures above the narrow GCP in Figure 7(d).
In addition to mode type, linear gyrokinetic simulations give flux and diffusivity ratios.The gyroBohm-normalized heat and particle fluxes through a flux surface are where ⟨. ..⟩ ψ is a flux-surface average, m s is the particle mass, q s and Γ s are the gyroBohm normalized heat and particle fluxes for a species s, χ s and D s are the normalized heat and particle diffusivities, and q gB = ρ 2 * r n r T r c, Γ gB = ρ 2 * r n r c where r subscripts refer to a reference species, c = T e /m D is the sound speed, and ρ * r = ρ r /a where ρ r is the gyroradius.The ratio of heat to particle flux is where η s = ∇ ln T s /∇ ln n s .Thus, changing η s by varying pedestal height via density or temperature can affect strongly the heat and particle flux ratios, even if χ s /D s is constant.We find the ratio χ e /D e varies significantly around the GCP.In the top row of Figure 8, we plot χ e /D e versus β θ,ped and ∆ ped for KBM, MTM, and TEM instability in NSTX discharge 139047 A 2 .In strongly driven KBM regions of Figure 8(a), χ e /D e ≃ 1.5.However, as the KBM is stabilized near the narrow and wide GCP branches, its transport coefficients change, often satisfying χ e /D e ≃ 2 − 5.For MTM instability in Figure 8  Around the GCP, χ e /D C is much larger than reported in the literature [38] where it was reported χ e /D C ≃ 3/2.Here, D C is the Carbon-12 particle diffusivity.Plotted in Figure 8(d), along the wide GCP χ e /D C ≃ 15 − 20, indicating that KBM produces relatively weak impurity transport in the inter-ELM period.We use the ratio (χ e + χ D )/D e [38] to distinguish between KBM and TEM, where χ D is the heat diffusivity for the main ion deuterium.For KBM along the GCP we find (χ e + χ D )/D e ≃ 3, increasing substantially near marginal KBM stability to (χ e + χ D )/D e ≃ 10.
The transport coefficients in different GCP regions are summarized later in Table 3 and are discussed more in Section 5.3.

Width-Transport Scaling
In this section, we take initial steps in relating pedestal width and transport.While linear stability threshold models [10,42] provide width-height scalings, they omit the sources and transport required to sustain pedestal profiles.Here, we find the dependence of pedestal width on the transport ratio q e /Γ e .To    simplify analysis, we study quantities averaged over the pedestal half-width, where ψ 1/4 = ψ mid −∆ ped /4 and ψ 3/4 = ψ mid +∆ ped /4.Along the GCP, if KBM dominates electron heat and particle transport and χ e,KBM /D e,KBM = 3/2 is constant, radially averaging Equation (10) gives where the 3/2 subscript in ⟨q e /Γ e ⟩ ped,3/2 indicates that we assumed χ e /D e = 3/2.In Figure 9(a), we plot ⟨q e /Γ e ⟩ ped,3/2 along the GCP wide branch using Equation ( 12) for NSTX 139047 A 2 and also plot ⟨q e /Γ e ⟩ ped,GK using data from KBMs in gyrokinetic simulations, where ky is a sum over all k y wavenumbers included in the simulation and q e /Γ e KBM is evaluated only if the fastest growing mode is a KBM.The close agreement between ⟨q e /Γ e ⟩ ped,3/2 and ⟨q e /Γ e ⟩ ped,GK in Figure 9(a) demonstrates that χ e /D e = 3/2 is an excellent approximation for the KBM along this particular wide branch GCP.
In Figure 9(b), using dashed lines we also plot ∆ ped,GCP for an ⟨η e ⟩ ped function that depends inversely with ∆ ped,GCP : ⟨η e ⟩ ped = 0.5 (∆ ped ) −0.6 + 0.8, but using the same width-height scaling relation.Such an inverse relation for ∆ ped and ⟨η e ⟩ ped would result from varying β θ,ped at fixed T e,ped and fixed n e,sep (see Equation (28), Appendix A).The widest pedestals are given by the largest χ e,KBM /D e,KBM values rather than the smallest χ e,KBM /D e,KBM , which happened when ⟨η e ⟩ ped increased with ∆ ped,GCP .By varying pedestal height at fixed T e,ped for NSTX discharge 139047 A 2 , ∆ ped,GCP is much more sensitive to ⟨η e ⟩ ped , making ∆ ped,GCP more sensitive to ⟨q e /Γ e ⟩ ped at fixed T e,ped than at fixed n e,ped .At fixed T e,ped , we find the width-transport scaling, ∆ ped,wide GCP,fixed T ≃ 0.31 q e Γ e ped,3/2 − 4.7 The stronger sensitivity of pedestal width to ⟨q e /Γ e ⟩ ped at fixed T e,ped occurs because we keep n ped /n sep fixed when increasing β ped , making η e only weakly dependent on β ped .In Figure 10(a) we plot ∆ ped,GCP for fixed n e,ped and fixed T e,ped narrow and wide GCP branches using the fitted form of ∆ ped,GCP in Equation ( 15) -notably, the black curves corresponding to fixed T e,ped narrow and wide GCP branches have relatively large derivatives, as shown in Figure 10(b).This suggests that for KBM-limited pedestals with fixed T e,ped , a small increase of ⟨q e /Γ e ⟩ ped,3/2 (e.g.decreased fueling) would increase ∆ ped much more effectively than increasing ⟨q e /Γ e ⟩ ped by increasing the heating.
To summarize, Equations ( 16) and ( 17) are the wide GCP pedestal widthtransport expressions resulting from a simple transport model that assumes along the GCP, turbulent particle and heat transport are dominated by the KBM with a transport ratio χ e /D e = 3/2.Equations ( 16) and ( 17) have noteworthy features: (a) there is a minimum level of transport ⟨q e /Γ e ⟩ ped,3/2 ≃ 3(1 + d)/2 required for ∆ ped > 0, (b) if ⟨η e ⟩ ped is independent of β θ,ped , then a fixed ⟨q e /Γ e ⟩ ped value can support any pedestal width, and (c) ∆ ′ ped is an increasing function of ⟨q e /Γ e ⟩ ped,3/2 at fixed n e,ped but decreasing for fixed T e,ped .
In the next section, we show that in marginally stable regions close to -but not exactly along -the GCP, transport ratios can differ substantially.

Transport Around GCP Marginality
In this section, we describe turbulent transport ratios in the vicinity of the wide and narrow GCP branches for NSTX 139047 A 2 .The wide and narrow GCP branches can be seen as experimental bounds for β θ,ped : KBM-limited pedestals may not fall exactly on the GCP branches in the inter-ELM cycle.Therefore, it is instructive to study transport not just along the GCP, but also in its vicinity.
Transport ratios change at pressures below the wide GCP scaling.Consider a path where rather than all radial locations in the pedestal half-width being unstable to KBM, only half of all radial locations are unstable -this is shown by path P 1 in Figure 11.Along P 1 there is still a substantial KBM mode fraction (roughly 1/3 of modes), so KBM transport may still be expected.Of the remaining KBMs, χ e /D e and χ e /D C both increase, meaning that electron and impurity particle transport are increasing relative to heat transport.In Table 3, we summarize the KBM transport properties of P 1 in the 'Stable, Near  Wide Branch' row.Along a trajectory above the narrow BCP branch, path P 3 in Figure 11, the ratio χ e /D e also increases while χ e /D C decreases.This indicates that KBM impurity particle transport might be more significant in the narrow than the wide branch.Non-KBM transport is likely increasingly important further away from the GCP.Consider the path P 2 in Figure 11 where the pedestal is at a pressure far below the GCP.Here, the KBM is close to marginally stable at all pedestal flux surfaces.Shown in Figure 11(b), the MTM has a significant mode fraction along P 2 for the k y ρ i values we simulate, indicating significant electron heat transport.Given that KBM is subdominant along P 2 , another mechanism such as TEM or neoclassical physics is required for particle transport.

Aspect-Ratio Dependence
In this section, we show how transport near the GCP varies across aspect-ratio.
Along the wide GCP, χ e /D e along the wide GCP branch is larger at is higher-A (χ e /D e ≃ 2 − 2.5) than at lower-A (χ e /D e ≃ 1.5).At lower-A, any remaining  KBM instabilities also have a higher growth rate than at higher-A along the wide GCP branch, shown in Figure 12 where we plot the average growth rate γ of KBM modes for NSTX 139047 A 2 (A = 2.0) and A 4 (A = 2.9).Here, γ is normalized by the minor radius a and sound speed c.This indicates that the more unstable KBMs along the lower-A wide GCP branch have transport properties expected for an MHD mode χ e /D e = 3/2.At lower growth rates close to KBM marginality, the higher-A KBM has modified transport properties where kinetic effects appear to be particularly important.However, along the narrow GCP branch for both lower and higher aspect-ratio in Figure 12, the KBM growth rate decreases sharply around the narrow GCP boundary, indicating that KBM transport changes quickly across the boundary.
The differing KBM transport coefficients across aspect-ratio along the GCP affects the width-transport scaling.In Figure 13(a), we show the difference between a prediction for ∆ ped,GCP using the χ e /D = 3/2 model and the data from gyrokinetic simulations.At low A values, ∆ ped,GCP is the same for both ⟨q e /Γ e ⟩ ped,3/2 and ⟨q e /Γ e ⟩ ped,GK , whereas at higher values there is a very large discrepancy.This difference comes from the increased χ e /D e value of the KBM for larger A values.We also plot the derivative ∆ ′ ped,GCP in Figure 13(b).The quantity ∆ ′ ped,GCP has an aspect-ratio dependence -the smaller A, the more sensitive ∆ ped,GCP is to ⟨q e /Γ e ⟩ ped,GK .Writing the derivative of ∆ ped,GCP in the form of Equation ( 15 , (18) one sees that increased χ e /D e values generally decrease ∆ ped,GCP and ∆ ′ ped,GCP .

Radial Dependence
In this section, we describe briefly the radial dependence of gyrokinetic instabilities in pedestal width and height space for NSTX 139047 A 2 .We study the 'average' radial location of an instability between ψ 1/4 = ψ mid − ∆ ped /4 and ψ 3/4 = ψ mid + ∆ ped /4 using the quantity gives the average Y radial value across the pedestal half-width.Here, mode=1 if the mode is present and mode=0 if not.For example, if KBM were unstable at all radial and k y ρ i values then ⟨Y ⟩ KBM = 0, indicating a uniform radial distribution across the pedestal half-width relative to the pedestal mid-radius.Figure 14(a) shows that moving along the wide and narrow GCP, initially at small widths the KBM is more common at the pedestal top, but as the pedestal width increases, the KBM is more common around the pedestal foot.This is likely due to the magnetic shear profiles changing as the bootstrap current density increases, shifting the relatively higher magnetic shear (and therefore destabilizing) regions from the pedestal top to the pedestal foot at higher ∆ ped , β θ,ped values.For MTMs in Figure 14(b), the instability is most common near the pedestal top, consistent with other works [24,98,99].Figure 14(c) shows that the TEM only appears in KBM second stability and is also more prevalent at the pedestal top, also seen in other works [20,100].The lack of TEM instability below the wide GCP branch at lower gradients might result from the relatively narrow k y ρ i ∈ [0.06, 0.12, 0.18] range we include in simulations for this work -because steeper gradients tend to destabilize modes at lower k y ρ i values [101,102], we likely require higher k y ρ i values for the pedestals below the wide GCP branch to find TEM instability.Despite the ubiquity of ETG instability in the pedestal [24,27,38,69], we find relatively little dominant ETG modes for k y ρ i ∈ [0.06, 0.12, 0.18], likely because we are again only simulating binormal wavenumbers that are too low.Finally, Figure 14(d) shows that in KBM second stability, much of the pedestal foot is stabilized, which is the only location in β θ,ped , ∆ ped space where stabilization occurs for k y ρ i ∈ [0.06, 0.12, 0.18].

Flow Shear Along the Gyrokinetic Critical Pedestal
In this section, we study the efficacy of pedestal flow shear for two NSTX discharges.The radial shear in plasma rotation Ω ζ is important in the L-H transition [29,103,104] and inter-ELM pedestal dynamics [25,30,[105][106][107][108], where Ω ζ is the toroidal angular rotation frequency.One key assumption in the EPED model is that flow shear at the pedestal top suppresses turbulence [10] allowing the pedestal to broaden.We examine this assumption by comparing flow shear in a standard (139047) and wide (132588) NSTX pedestal along the GCP.
Balancing the radial electric field with the diamagnetic flow [24,69], we approximate the flow shear rate γ E = (r/q)dΩ ζ /dr as As a rough estimate of flow shear efficacy, we compare the flow shear frequency with the linear drive frequency ⟨γ E /ω *i ⟩ top  which is typically comparable to the linear instability growth rate.In Figure 15(a), we plot γ E /ω * i averaged over the pedestal top, versus pedestal width and height for NSTX 139047.Here, ψ 1/4 = ψ mid − ∆/4 and ψ ped = ψ mid − ∆/2.At the pedestal top, larger values of ⟨γ E /ω * i ⟩ top suppress turbulence and may facilitate the pedestal's radially inward propagation [10], although too large values may cause other instabilities [109][110][111].
Moving along the wide GCP and BCP starting from small widths and heights, ⟨γ E /ω * i ⟩ top is a rapidly decreasing function.The marked equilibrium point for NSTX 139047 in Figure 15(a) is at a location where ⟨γ E /ω * i ⟩ top approaches a minimum -moving further along the wide GCP by increasing the width-height may give pedestals with insufficient pedestal top flow shear to permit an increase in pedestal width.
For comparison, we consider pedestal flow shear for ELM-free wide pedestal NSTX discharge 132588.In Figure 15(b), we plot ⟨γ E /ω * i ⟩ top .Compared with NSTX 139047 in Figure 15(a), the wide pedestal has very strong pedestal top flow shear.This strong flow shear indicates that this discharge may have sufficient flow shear to allow the pedestal width to grow to very large values.
We emphasize that using the ratio γ E /ω * i is only heuristic, and actual gyrokinetic simulations are required to definitively determine flow shear efficacy across the β θ,ped , ∆ ped space.Additionally, given that flow shear can also affect peeling-ballooning stability [112], determining the effect of pedestal flow shear on both microstability and macrostability is necessary to understand its role in pedestal evolution.Arrows indicate the direction that the constraint acts on in β θ,ped , ∆ ped space.The translucent color blocks indicate the excluded regions for a constraint curve of the same color.The only accessible pedestal regions are those without any constraints.In this example, for the transport constraints ⟨qe/Γe⟩ ped,lower and ⟨qe/Γe⟩ ped,upper , we have assumed that ηe increases with β θ,ped .

Combined Stability, Flow Shear, and Transport Constraints
In this section, we discuss the combination of three pedestal constraints discussed in this paper: KBM stability, flow shear, and transport.Due to its importance as a saturation mechanism, we also include an ELM constraint [10].
For KBM stability, the Gyrokinetic Critical Pedestal gives the wide and narrow branches, indicated by lines labeled wide and narrow in Figure 16.According to kinetic-ballooning-mode stability, the pedestal can only manifest at pressures above the narrow branch and pressures below the wide branch.Arrows in Figure 16 along the wide and narrow branches indicate the direction in which the constraint applies, ruling out pedestal access to these excluded regions of β θ,ped and ∆ ped .
For flow shear, we show a single constraint given by ⟨γ E /ω * i ⟩ top in Figure 16: values below a critical value indicate that the pedestal top flow shear is no longer sufficiently strong to allow the pedestal to widen.From the results in Section 6 we graph the flow shear constraint as β θ,ped ∼ 1/∆ ped .
For transport, we plot two constraints in Figure 16: ⟨q e /Γ e ⟩ ped,lower and ⟨q e /Γ e ⟩ ped,upper , which are the minimum and maximum values of ⟨q e /Γ e ⟩ ped that the particle and heat sources might permit to support a given pedestal profile.In steady state for heat and particle sources P e and S e q e Γ e ped = P e dV half−width S e dV half−width , where the volume integral is taken over the pedestal half-width.For the example in Figure 16, we have assumed that η e increases with β θ,ped and hence ⟨q e /Γ e ⟩ ped also increases with β θ,ped according to Equation (12).If however, η e were to decrease with β θ,ped , the transport constraints in Figure 16 would change significantly.Finally, we plot a peeling-ballooning ELM constraint in Figure 16 assuming ∆ ped ∼ (β θ,ped ) 4/3 [10].Pedestal pressures above the ELM constraint are inaccessible.
Figure 16 shows that after considering stability, pedestal top flow shear, and transport, only a relatively narrow window of β θ,ped , ∆ ped space remains for a viable pedestal.In the heuristic example in Figure 16, moving along the wide GCP branch, the pedestal growth is eventually stopped by the flow shear constraint.Moving along the narrow GCP branch, the pedestal growth is stopped by the upper transport constraint.

Summary
In this work, we described how a linear gyrokinetic threshold model combined with self-consistent equilibrium variation gives a pedestal width-height scaling expression that depends strongly on aspect-ratio and plasma shaping.These shaping and aspect-ratio scans were performed on an NSTX equilibrium with self-consistent equilibrium reconstruction.The Gyrokinetic Critical Pedestal scaling for the wide branch kinetic-ballooning-mode pedestals with shaping and aspect-ratio dependence in this NSTX equilibrium is ∆ ped = 0.92A 1.04 κ −1.24 0.38 δ (β θ,ped ) 1.05 .It is noteworthy that the width-height scaling has a strong dependence on shaping and aspect-ratio for both the wide and barrow GCP branches [86] -this dependence might not have been reported experimentally due to the relatively narrow range of aspect-ratio, triangularity, and elongation values routinely examined compared with those in the parameter scan in this work.In future work, it is important to study the effect of shaping on both kinetic-ballooning [86] and peeling-ballooning [7,13,85] stability to definitively find attractive pedestal scenarios.
We demonstrated that whether pedestal height β θ,ped is varied with fixed n e,ped or T e,ped changes heat and particle transport along the Gyrokinetic Critical Pedestal significantly.In the wide Gyrokinetic Critical Pedestal branch for fixed n e,ped we find that ∆ ped = 0.028 (q e /Γ e − 1.7) 1.5 ∼ (η e ) 1.5 and for fixed T e,ped we find ∆ ped = 0.31 (q e /Γ e − 4.7) 0.85 ∼ (η e ) 0.85 .Thus, during the inter-ELM buildup, not only do relative contributions to β θ,ped from temperature and density have a big effect on the width-height scaling, but also the relative transport through particle and heat channels.While our linear gyrokinetic model has no information about plasma sources, more sophisticated approaches such as transport solvers [113][114][115] will be sensitive to KBM transport around the Gyrokinetic Critical Pedestal.
In the vicinity of the narrow and wide Gyrokinetic Critical Pedestal branches where KBM becomes marginally stable, the KBM turbulent transport ratios such as χ e /D e and χ e /D Z vary significantly.This also has implications for reduced transport models in the pedestal, given that KBM can produce significant particle and heat flux.
We analyzed the role of flow shear for two NSTX pedestal discharges, finding that the width and height of an ultra-wide-pedestal ELM-free NSTX discharge had much stronger flow shear at the pedestal top, possibly permitting its pedestal to grow to a very large width.This indicates that the flow shear can introduce an additional constraint on the pedestal width and height evolution, and might be an important pedestal growth saturation mechanism.

Acknowledgements
We are grateful for conversations with E. A. Belli, J. Candy A.1 Width-Independent Limit The quantities η e and ⟨η e ⟩ ped in Equations ( 29) and ( 30) are independent of the pedestal width.

Figure 2 :
Figure 2: Schematics of (a) gyrokinetic simulation radial locations and (b) Gyrokinetic Critical Pedestal (GCP) calculation with dominant gyrokinetic instability versus binormal wavenumber kyρi and radial location ψ.Top: GCP unstable example since KBM unstable for at least a single kyρi value at each radial location.Bottom: GCP stable location.Only KBM stability status is shown.

Figure 3 :
Figure 3: Fraction of KBM dominance across the pedestal half-width for fixed n e,ped in NSTX 139047, with wide and narrow GCP branches.The mode fraction is the fraction of all linear gyrokinetic simulations across the pedestal half-width and binormal wavenumbers (here kyρi ∈ [0.06, 0.12, 0.18]) where the fastest growing mode is classified as KBM, MTM, etc.

Figure 4 :
Figure 4: Gyrokinetic Critical Pedestal (GCP) and Ballooning Critical Pedestal (BCP) scaling expressions with experimental points indicated by markers and uncertainty bars of 20 %.The design point for STAR has not yet been finalized.

Figure 6 :
Figure 6: Fitting of ∆ ped in Equation (7) to 11 shaping variations of NSTX discharge 132543, where the dependence on A, κ, and δ is shown in each subplot.For simplicity, we only show points where the x-axis parameter has changed from the nominal value.

Figure 7 :
Figure 7: Fraction of gyrokinetic mode types across the pedestal half-width for fixed n e,ped in NSTX 139047 A2.The mode fraction is the fraction of all linear gyrokinetic simulations across the pedestal half-width and binormal wavenumbers (here kyρi ∈ [0.06, 0.12, 0.18]) where the fastest growing mode is classified as KBM, MTM, etc.

2 < l a t e x i t s h a 1 _
Width versus ⟨qe/Γe⟩ ped along wide GCP branch at fixed n e,ped for transport model ⟨qe/Γe⟩ ped,3/2 and simulations ⟨qe/Γe⟩ ped,GK .b a s e 6 4 = " G 5 D Q 9 Y 7 j s n h 9 o k 3 e x Y H F C + S j s f w

7 +0. 2 < l a t e x i t s h a 1 _
H s T R P a F y o v z s y E i k 1 j n x T m S + p 5 r 1 c / M / r p T q 8 6 m d M J K k G Q a e D w p R j H e M 8 G h w w C V T z s S G E S m Z 2 x X R I J K H a B F g x I T j z J y + S 9 m n d u a i f 3 5 3 V G t e z O M r o A B 2 i E + S g S 9 R A t 6 i J W o i i R / S M X t G b 9 W S 9 W O / W x 7 S 0 Z M 1 6 9 t E f W J 8 / v 4 2 c q A = = < / l a t e x i t > h⌘ e i ped < l a t e x i t s h a 1 _ b a s e 6 4 = "h n A h X U T e U 0 Y b f l n 9 j D q a l O / j m U M = " > A A A C H 3 i c b V D L S g N B E J z 1 G e M r 6 t H L Y B A U Y d k N a n I M 6 s F j B G M C 2 R h m J 7 3 J 4 O y D m V 4 h L P s n X v w V L x 4 U E W / 5 G y c x B 1 8 F D U V V N 9 1 d f i K F R s c Z W 3 P z C 4 t L y 4 W V 4 u r a + s Z m a W v 7 R s e p 4 t D k s Y x V 2 2 c a p I i g i Q I l t B M F L P Q l t P y 7 8 4 n f u g e l R R x d 4 y i B b s g G k Q g E Z 2 i k X u n U r d g 1 6 k k I 8 M C 7 A I m s l 3 k h w 6 E K s w T 6 e e 4 p M R j i 4 W 3 m 2 N W c e h 4 9 o o 5 d 6 Z X K j u 1 M Q f 8 S d 0 b K Z I Z G r / T h 9 W O e h h A h l 0 z r j u s k 2 M 2 Y Q s E l 5 E U v 1 Z A w f s c G 0 D E 0 Y i H o b j b 9 L 6 f 7 R u n T I F a m I q R T 9 f t E x k K t R 6 F v O ie n 6 9 / e R P z P 6 6 Q Y 1 L q Z i J I U I e J f i 4 J U U o z p J C z a F w o 4 y p E h j C t h b q V 8 y B T j a C I t m h D c 3 y / / J T c V 2 z 2 1 T 6 6 O y / W z W R w F s k v 2 y A F x S Z X U y S V p k C b h 5 I E 8 k R f y a j 1 a z 9 a b 9 f 7 V O m f N Z n b I D 1 j j T 9 3 z o P 0 = < / l a t e x i t > 12.8 ( ped) 0.b a s e 6 4 = " D j o 3 n /

7 5 f / k t s j 2 6 3 Y
J 9 f H 5 e r 5 N I 4 C 2 S Y 7 Z I + 4 5 J R U y R W p k T r h 5 J E 8 k 1 f y Z j 1 Z L 9 a 7 9 T F p n b G m M 1 v k B 6 z h F 9 4 Q o P k = < / l a t e x i t > Width prediction using two different fits of ⟨ηe⟩ ped for three χe/De ∈ [1, 1.5, 2.0] values for the KBM.

Figure 11 :
Figure 11: Paths P1, P2, and P3 in the vicinity of the wide and narrow GCPs, plotted over KBM and MTM mode dominance fraction versus pedestal width and height for NSTX 139047 A2.The mode fraction is the fraction of all linear gyrokinetic simulations across the pedestal half-width and binormal wavenumbers (here kyρi ∈ [0.06, 0.12, 0.18]) where the fastest growing mode is classified as KBM, MTM, etc.

⟨
qe / Γe ⟩pe d ,u p p e r ⟨ q e / Γ e ⟩ p e d , l o w e r E L M

Figure 16 :
Figure 16: Combination of stability, flow shear, transport, and ELM constraints.Arrows indicate the direction that the constraint acts on in β θ,ped , ∆ ped space.The translucent color blocks indicate the excluded regions for a constraint curve of the same color.The only accessible pedestal regions are those without any constraints.In this example, for the transport constraints ⟨qe/Γe⟩ ped,lower and ⟨qe/Γe⟩ ped,upper , we have assumed that ηe increases with β θ,ped .

Table 1 :
Mode finder criteria in gk ped.A '−' indicates that the quantity is not considered in the mode criterion.Mode χ i /χ e D e /χ e D i /χ i D e /(χ e + χ i )

Table 2 :
Aspect-ratio scan parameter values for NSTX 139047 used in Section 5.

Table 3 :
KBM transport coefficients in NSTX 139047 A2 in different GCP regions.