Kinetic-ballooning-limited pedestals in spherical tokamak plasmas

A theoretical model is presented that for the first time matches experimental measurements of the pedestal width-height Diallo scaling in the low-aspect-ratio high-β tokamak NSTX. Combining linear gyrokinetics with self-consistent pedestal equilibrium variation, kinetic-ballooning, rather than ideal-ballooning plasma instability, is shown to limit achievable confinement in spherical tokamak pedestals. Simulations are used to find the novel Gyrokinetic Critical Pedestal constraint, which determines the steepest pressure profile a pedestal can sustain subject to gyrokinetic instability. Gyrokinetic width-height scaling expressions for NSTX pedestals with varying density and temperature profiles are obtained. These scalings for STs depart significantly from that of conventional aspect ratio tokamaks.

Introduction.-Fusionenergy is a grand challenge of physics and engineering.The tokamak, a prime candidate for magnetic confinement fusion, achieved its fusion power records [1,2] operating in H-mode, a highconfinement regime characterized by much steeper pressure gradients in the pedestal at the plasma edge [3][4][5].Since fusion power scales approximately quadratically with the pedestal pressure height, accurate prediction of pedestal structure is crucial for upcoming burning plasma experiments such as ITER and SPARC [6][7][8][9][10][11].In this work, we make significant advances in pedestal predictive capabilities, allowing accurate predictions across a much wider range of tokamak operating space.In the lowaspect-ratio (A = R/a where R is the plasma major radius and a is the plasma half-diameter) tokamak NSTX, edge pedestals are much wider than in higher aspect-ratio plasmas.The new work presented here yields the first understanding of this difference.These results have wideranging implications for the physics basis, design, and optimization of fusion pilot plants, and motivates further investigation into kinetic effects on ballooning modes.
In this Letter, we present the first theoretical model for the pedestal width ∆ ped and normalized height β θ,ped to agree with experimental measurements for the National Spherical Torus Experiment (NSTX) [12].The Diallo scaling for NSTX, ∆ ped ≃ 0.4β 1.05  θ,ped , gives pedestals that are much wider than conventional-aspect-ratio tokamaks, ∆ ped ≃ 0.08 β θ,ped [13].We calculate gyrokinetic stability in spherical tokamak (ST) [14][15][16] plasmas with self-consistent equilibrium variations to answer the questions: (1) what additional physics is required for pedestal width-height prediction in STs? (2) what are the steepest-possible pedestal pressure profiles subject to kinetic instability?(3) what are the resulting pedestal width-height scalings?While H-mode has a higher power density, it often has unstable edge-localized-modes (ELMs) [17] and therefore unacceptably high divertor fluxes, necessitating robust ELM-free pedestal regimes [18,19].Kinetic-ballooninglimited pedestals might present such a regime.We use a gyrokinetic model to predict pedestal width-height scalings, showing that low-aspect-ratio high-normalizedpressure (β) plasmas in NSTX pedestals are limited by kinetic rather than ideal-ballooning stability.Starting from experimental equilibria, we construct and analyze equilibria with rescaled pedestal width and height.We use gyrokinetic simulations to obtain a profile constraint that matches NSTX width-height measurements.
H-modes with much lower fitting uncertainty, albeit still with much wider pedestals than for conventional-aspectratio [36].Recent work has shown non-ideal effects to be important for peeling-ballooning stability in NSTX [37,38] and for wide pedestals in DIII-D [19].Additionally, while ideal-MHD models constrain pressure, they do not specify the relative density and temperature contributions required to predict kinetic instabilities and transport [39].
Gyrokinetics and ideal-ballooning.-Westudy plasmas using the gyrokinetic codes GS2 and CGYRO [40][41][42], evolving the distribution function h s = q s ϕ tb F M s /T s +f tb s and the electrostatic and gyrokinetic potentials ϕ tb and χ tb s [43] according to the linear gyrokinetic equation, and Maxwell's equations [43][44][45][46].Here, a species s has a total turbulent distribution function f tb s and a Maxwellian F M s with temperature T s , ω is the frequency, ω * s contains equilibrium instability drives [47], Ω s = q s B/m s c is the gyrofrequency, q s is the electric charge, m s is the mass, v ∥ and v M s are the parallel and magnetic drift velocities, and C l s is a pitch angle scattering and energy diffusion collision operator.We evolve all three electromagnetic fields.
We emphasize that this s − α analysis uses realistic geometry and not the shifted-circle approximation in [57].
Here, r is the minor radial coordinate, q is the safety factor, c is the speed of light, V is the volume, R 0 is the major radius, and p is the pressure.For the s − α scan in Figure 1, we vary temperature and density gradients by the same factor.Shown in Figure 1(a)-(c), in order of increasing pedestal radial fraction, we plot the fastest growing gyrokinetic mode for binormal wavenumber k y = 0.18/ρ i where the deuterium gyroradius is ρ i = T e /m i Ω 2 i , ψ N is the normalized poloidal flux, which is 0 at the axis, 1 at the edge, and ψ ped at the pedestal top.Using an automated 'fingerprints' [58] gyrokinetic mode identifier summarized in Table I, we determine that the fastest growing mode at the pedestal top ∆ = 1/12 (Figure 1(a)) is a microtearing mode (MTM) [47,59], typical of ELMy pedestals [60].In the pedestal half-width ∆ ∈ [1/4 − 3/4], Figure 1(b) and (c) show the equilibrium is marginally unstable to KBMs [22,61], with MTMs and impurity-gradient-driven ion-scale tearing modes dominating at lower α values.Strikingly, each flux surface in Figure 1(a)-(c) is experimentally far below the ideal-ballooning stability boundary, evidence that the KBM clamps the pedestal profiles.KBM is a compelling candidate for limiting pressure profiles as it transports through both particle and energy channels, D s /χ s ∼ 1, where D s and χ s are particle and heat turbulent diffusivities.
Equilibrium Variation.-Whileour radially local analysis showed a NSTX discharge to be marginally KBMunstable, we now describe the approach required to correctly find the steepest-possible pedestal profiles subject to kinetic instability.This approach combines pedestal width-height rescaling with self-consistent plasma equilibrium reconstruction.Following previous works [13], we parameterize electron density and temperature profiles as where H is a Heaviside function, n e,core , T e,core , n e0 , and T e0 are constants, ∆ ne and ∆ Te are the pedestal electron density and temperature widths -which usually differand α {n,T,J},{1,2} are exponents.Quantities A n , A T , and S ∆ rescale the pedestal density, temperature, and width.The pedestal heights n e,ped and T e,ped are n e,ped = n(ψ ped,ne ), T e,ped = T (ψ ped,Te ), ( 6) n e,sep and T e,sep are evaluated at ψ N = 1, t 2 = tanh (2), and ψ ped,ne = ψ mid,ne −∆ ne /2.The current density J(ψ) is the sum of an Ohmic term N ) α J 2 with constants J C , α J1 , α J2 and a bootstrap current term J bs [63].
To distinguish between n and T contributions to pedestal pressure, we change the height with two bracketing cases of 1) varying T at fixed n and 2) varying n at fixed T .Ion profiles are calculated using s q s n s = 0 and maintaining T i /T e .The rescaled pedestal heights n e,ped and T e,ped are obtained with coefficients S n , S T , n e,ped = S n n e,ped , T e,ped = S T T e,ped .
We choose width and height scalings S ∆ ∈ [0.While typically ∆ ne ̸ = ∆ Te , when rescaling the pedestal width, both ∆ ne and ∆ Te are rescaled by S ∆ .Motivated by observations [64], when rescaling the pedestal height, we fix T e,sep and n e,ped /n e,sep .Equilibrium reconstruction is performed using EFIT-AI [65], where total plasma current I p and stored energy are conserved.Gyrokinetic flux-tube simulations are performed according to Equation (1), with wavenumbers where we found KBM to be most prevalent, k y ρ i = 0.06, 0.12, 0.18, radial wavenumber k x = 0, using 3 kinetic species including electrons, and realistic up-down asymmetric geometry.The radial grid has 12 points, equally spaced in Figure 2 shows equilibrium quantities for a height scaling of NSTX 139047 at fixed n e,ped and fixed ∆ ped .The Gyrokinetic Critical Pedestal.-Wenow find the steepest pedestal profiles subject to gyrokinetic stability for NSTX discharge 139047, giving the relation between ∆ ped and β θ,ped .Analogous to the EPED Ballooning-Critical-Gradient (BCP) [13], we hypothesise that the steepest pedestal profiles are marginally unstable to the same gyrokinetic instability across the pedestal halfwidth for any k y ρ i ∈ [0.06, 0.12, 0.18], giving the Gyrokinetic Critical Pedestal (GCP).The instability should FIG.3: a): GCP equilibria for NSTX discharge 139047 for height variations with fixed n (X markers) and fixed T (circle markers), compared with the EPED and Diallo scalings.KBM stability and α across all kyρi values for (b) narrow low pedestals and c) wide high pedestals.
satisfy D s /χ s ∼ 1 (although there are exceptions for marginal KBM [68]) and have similar linear and nonlinear critical thresholds [69]: KBM is a natural candidate [70], although a trapped-electron-mode [71,72] may also be suitable.Our NSTX pedestal simulations found KBM dominates, so we here focus on KBM-limited profiles.
We find the GCP for the two bracketing cases of increasing pedestal height at fixed n and T , shown in Figure 3(a).While the fixed n GCP profiles match NSTX measurements, the fixed T GCP profiles under-predict ∆ ped for lower p ped .This largely stems from our choice to fix T e,sep while allowing n e,sep to vary.For narrow low pedestals, fixing T e,sep while varying T e,ped for fixed n height variation gives relatively lower α values than for fixed T variation, shown in Figure 3(b).The higher α values for fixed T make the KBM more unstable, hence giving a wider pedestal than for fixed n.For wide high pedestals, shown in Figure 3(c), fixed n and T have similar α profiles and KBM stability properties near the GCP and hence predict similar pedestal profiles.There is also an effect due to bootstrap current modifying the magnetic shear, but since most NSTX pedestals we analyzed were in ideal first-stability, the change in the α profiles is likely more important.
Averaged Pedestal Scalings.-We now find the GCP averaged over five NSTX discharges.In Figure 4, we plot GCP equilibria points in gold calculated from NSTX discharges 130670, 132543, 139034, 139047, and 141300.Black markers indicate the Diallo scaling NSTX measurements ∆ ped = (0.4 ± 0.1)β 1.05±0.2ped [12], in excellent agreement with our GCP scaling ∆ ped ≃ 0.43β  We also perform ideal-ballooning stability calculations to obtain the BCP.The corresponding BCP -purple markers in Figure 4 -underpredicts pedestal width, ∆ ped ≃ 0.26β 0.96 ped .The dispersion of BCP datapoints is due to variations between NSTX discharges.KBM-limited pedestals are wider than idealballooning-limited pedestals at fixed β θ,ped due to a lower α instability threshold for KBM.To elucidate this, in Figure 5 we show ideal s − α analysis for NSTX 139047 equilibria on the GCP.In Figure 5(a) and (b), we plot the s and α locations of GCP equilibria with different widths for two radial locations, ∆ = 3/12, 5/12.The lines in Figure 5(a) and (b) show the ideal stability boundaries for each equilibrium.The large distance in α between GCP equilibria and ideal-ballooning stability boundaries causes the large width prediction difference for the GCP and BCP in Figure 4.In Figure 5(c)-(e), we plot pressure, α, and T e profiles for three GCP and three BCP equilbria with three different widths.The kinetic-ballooning-limited profiles can only support roughly one half of the pressure gradient than ideal-ballooning-limited profiles.
Conclusion.-Combining self-consistent pedestal equilibrium variation with a gyrokinetic stability threshold model, we obtained a pedestal width-height scaling for kinetic-ballooning-limited NSTX spherical tokamak plasmas, ∆ ped ≃ 0.43β 1.03  θ,ped , which is in excellent agreement with experimental measurements, ∆ ped = (0.4 ± 0.1)β 1.05±0.2θ,ped [12].Notably, using an ideal rather than a kinetic-ballooning threshold for NSTX underpredicts the width by 40%, but the scaling ∆ ped ≃ 0.26β 0.96 θ,ped is still linear.Our gyrokinetic model, applicable in the ρ i /a → 0 limit, is likely to become an even better approximation to future fusion reactors.While kineticballooning-limited pedestals give lower average gradients than for ideal modes, their extra width might permit a larger β θ,ped before an ELM occurs [13,18].For sufficiently wide pedestals, there may even exist ELM-free regimes with other saturation mechanisms [19,73], and thus paradoxically, as has been suggested in the context of other pedestal transport mechanisms [13,18], kineticballooning-degraded pedestal profiles may increase β θ,ped and therefore fusion power.Recent results from ELMfree negative triangularity operation [74] that aims to prevent ideal second stability access [75] may be modified by a larger KBM instability region [76], a phenomenon also seen in ST reactor studies [77,78].Scaling expressions presented in this paper are summarized in Table II.
Code and data availability.-Part of the data analysis was performed using the OMFIT integrated modeling framework [79] using the Github projects gk ped [80] and ideal-ballooning-solver [50].The data that support the findings of this study are openly available in Princeton Data Commons at https://doi.org/10.34770/vj4h-6120.

FIG. 2 :
FIG.2: Quantities versus ψN for equilibria with increasing T e,ped based on NSTX 139047 with fixed n e,ped and ∆ ped : a) Te, b) a/LT e , c) ne, d) a/Ln s , e) J bs /R, f) safety factor q and magnetic shear s, and α with ideal balooning stability (g)) and gyrokinetic stability for kyρi = 0.18 (h)).larger boostrap current, shown in Figure2(e), modifiying q and reducing s, shown in Figure2(f).In Figure 2(g) and (h), we plot gyrokinetic and ideal stability over α profiles.Figure2(g) shows larger α destabilizes the ideal-ballooning mode near the pedestal foot, while the pedestal center and top become second-stable.Figure2(h) shows the fastest growing gyrokinetic mode for k y ρ i = 0.18.Strikingly, the KBM can be the fastest growing gyrokinetic mode at roughly half the critical α for ideal instability.At lower α values, we find MTMs and electron-temperature-gradient (ETG) modes[40,66,67].The Gyrokinetic Critical Pedestal.-Wenow find the steepest pedestal profiles subject to gyrokinetic stability for NSTX discharge 139047, giving the relation between ∆ ped and β θ,ped .Analogous to the EPED Ballooning-Critical-Gradient (BCP)[13], we hypothesise that the steepest pedestal profiles are marginally unstable to the same gyrokinetic instability across the pedestal halfwidth for any k y ρ i ∈ [0.06, 0.12, 0.18], giving the Gyrokinetic Critical Pedestal (GCP).The instability should