Nonlinear gyrokinetic modelling of high confinement negative triangularity plasmas

Nonlinear gyrokinetic simulations correctly predict particle as well as ion and electron energy fluxes of high confinement plasmas with a negative triangularity cross sectional shape, showing that core transport in these plasmas is well described by standard gyrokinetic models. Experimentally inferred power balance fluxes are mostly reproduced within one standard deviation across a wide portion of the minor radius. Experimental conditions are reproduced by ion scale simulations, without the need to include density and temperature profile curvature effects. The experimental case is used as baseline to predict that the non-dimensional confinement scaling in negative triangularity plasmas increases strongly with plasma current while slightly degrading at increasing normalized pressure and decreasing collisionality. Recent experiments showed that low toroidal rotation negatively impacts confinement; consistent with the experiment, simulations predict that low rotational shear significantly affects confinement unless the plasma effective charge is maintained above a minimum level. Core confinement is predicted to significantly degrade in low aspect ratio devices.


Introduction
Tokamak plasmas with a poloidal cross sectional shape featuring Negative Triangularity (NegT) have recently attracted worldwide theoretical and experimental interest thanks * Author to whom any correspondence should be addressed.
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to a number of characteristics that are promising in solving the core-edge integration challenge.Indeed, plasmas sustain reactor grade normalized pressure and confinement times despite robustly maintaining relaxed edge pressure profiles typical of L-mode plasmas [1][2][3][4].The good core confinement is thought to be caused by the stabilization of trapped electron modes (TEM) across a conspicuous part of the plasma volume [5,6], while the inability to develop steep edge pedestals can be attributed to ballooning modes closing access to the second stability region [7][8][9].The absence of an edge pedestal is favorable in several ways, such as providing a passive safety system against wall damaging edge localized modes (ELMs), decreasing the impurity confinement time and widening the reconstructed scrape-off layer heat flux width, λ q , that is seen to exceed the ITPA multi-machine scaling law by up to 50% [3] 7 .Together with technological advantages granted by placing the divertor at larger radii, e.g.larger divertor wetted area or the fact that internal coils are placed in an environment at lower magnetic field, plasmas with a NegT shape are looked at as a promising candidate for reactors.For a review of experimental and theoretical work on this configurations we refer interested readers to [11].Despite the favorable features described above, the viability of NegT configurations as a scenario for future fusion reactors has to be assessed by projecting performance to an operational space compatible with what a future power plant will likely operate in.A way to obtain credible theoretical predictions is to start with an accurate model of an existing experiment and then scale the simulations to parameters that will be obtained in reactors.It is important to underline that the wording reactor relevant might means different things to different people, such as metal plasma facing components, non inductive current drive, divertor detachment etc.In this paper, reactor relevant is to be understood purely from a plasma core transport perspective, i.e. low neutral beam torque, reasonably low effective charge, confinement quality of order unity, normalized pressure equal or larger than that expected in ITER, i.e. β N > 1.8, as well as a plasma discharge long enough to allow perfect ionelectron energy equipartition.In this paper we describe nonlinear gyrokinetic modelling of a NegT discharge obtained on the DIII-D tokamak taken as a starting point to derive nondimensional confinement scaling as well as to predict fusion performance in machines characterized by low and high effective charge, low rotation and aspect ratio less than two.
The paper is organized as follows: section 2 describes the experiment that was modelled along with the numerical workflow; section 3 overviews the main results; conclusions are offered at the end.

Description of the experimental data and modelling workflow
The experimental case under consideration is DIII-D innerwall limited discharge #171421, that was part of the study published in [12].The temporal evolution of a few discharge parameters is displayed in figure 1, where it can be seen how the discharge features a staircase increase of the beam auxiliary power with steps long enough to obtain stationary phases corresponding to varying amounts of coupled power.
Although this study analyzed phases of the discharge at varying auxiliary heating levels resulting in 1.5 < β N < 2.7, most of the non-linear simulations herein reported utilize profiles averaged during time window 2.7-3.0 s, where the discharge sustains a normalized pressure corresponding to β N ≃ 2.0, a normalized thermal energy confinement time corresponding to H 98(y,2) ≃ 0.95, Greenwald fraction F GW ≃ 0.55 and edge safety factor q lim ≃ 4.2.The radial profile of the safety factor is monotonic with central value below unity triggering sawteeth.Relevant dimensional parameters of the discharge in this time window include plasma current I P = 0.9 MA, confining magnetic field B T = 2.0 T, auxiliary beam power P NBI = 5.6 MW, auxiliary direct electron heating P ECH = 3 MW deposited near the r/a = 0.4 surface, where r/a is the half width of a flux surface at the elevation of the midplane normalized to its value at the last closed flux surface.The launching angles of the various gyrotrons were configured in such a way as to provide over 99% absorption almost exclusively as heating, with the overall current drive limited to less than 2 kA.Simulations using the TRANSP code indicate that 65.3% of the coupled beam power is delivered to ions and 34.7% to electrons, with a total of torque equal to 5.6 N m.Given the above, the discharge is dominated by electron heating.The choice of this time window was dictated by the need to analyze a plasma sustaining a relatively high value of normalized pressure while recovering good fits in the kinetic profiles and equilibrium reconstruction.The main diagnostics monitoring the experiment are listed below.The electron temperature was measured by Thomson Scattering [13] (TS) and Electron Cyclotron Emission [14] (ECE); density was gauged by TS, CO2 Interferometer [15] and profile reflectometer; Charge Exchange Recombination [16,17] (CER) was used to obtain the main and impurity ion density and temperatures, along with the toroidal and poloidal velocity components.Detailed radial profile analysis was performed during a 300 ms long time window starting 100 ms after the last increase in auxiliary power to let the fast ions profile relax to the new stationary level given that the full energy beam classical slowing down time is less than 70 ms.Given the lack of the edge pedestal, the position of the separatrix could not be set following the usual approach that relies on the pressure radial profile assuming a hyperbolic tangent shape in the pedestal region; rather, the two-point model [18] was employed to determine the value of the electron temperature at the separatrix, Te-sep.More specifically, the value for Te-sep was obtained by constraining the kinetically corrected electron Spitzer parallel heat flux computed using Thomson electron temperature and density radial profiles to that derived using the power entering the Scrape-Off Layer (SOL).The procedure is iterated until values for Te-sep and for the power fall-off length in the SOL, λ q , are self-consistently obtained.On DIII-D the ion radial profiles do not usually require any significant spatial shift thanks to the fact that the position of the separatrix on the mid-plane, where the lines of sight of the CER diagnostic are located, is determined by the equilibrium reconstruction more accurately than that in the upper poloidal cross section where the electron temperature profiles are measured by the Thomson diagnostic.Once reliable edge density and temperature profiles are obtained, any mismatch between the measured and computed surface loop voltage is to be attributed to errors in the effective charge of the plasma.The value of the effective charge near the plasma edge was determined by employing an iterative procedure using the 1.5 dimensional ONETWO transport code and the NUBEAM Monte-Carlo code.More specifically, the value of the effective charge and that of the anomalous fast ions diffusion coefficient was altered until satisfactory agreement was obtained for stored energy, neutron rate and surface loop voltage.The procedure determined that the effective charge near the plasma edge should be Z eff = 3.5 ± 0.3, with the radial profile approximating a roughly linear behavior.Similar values were obtained using the TRANSP code.It has to be pointed out that this value of Z eff at the edge is not consistent with that contributed by the amount of carbon 6+ measured by the CER diagnostic, for which the radial Z eff profile would be a relatively flat function of the minor radius, assuming values near Z eff = 2.8 ± 0.1 for r/a > 0.6-0.7.The higher Z eff value near the plasma edge is instead consistent with the visible bremsstrahlung diagnostic, which corroborates the transport analysis reported above.The larger edge Z eff values can be attributed to the presence of other impurities near the plasma periphery, as well as to the fact that, in contrast to typical H-mode plasmas in which carbon ions are fully stripped near the pedestal top, L-mode plasmas feature edge electron temperature values in the range 50-200 eV, depending on the amount of auxiliary power coupled and on the radial position considered, for which other carbon charge states not measured by the CER system do contribute to the overall Z eff value.The discrepancy between the edge value of Z eff measured by the CER system and that inferred by a transport analysis is within two population uncertainty, which could appear as negligible; however, the corresponding difference in the radial profile of Z eff leads to implications that will be discussed in section 3.
Gyrokinetic (GK) simulations were performed using the non-linear CGYRO code [19] that solves the GK Vlasov-Maxwell system of equations as an initial value problem in the local, or flux-tube, approximation.The simulations evolve perturbations in the electrostatic and parallel vector potential, retain three kinetic species: electrons, deuterium and fully stripped carbon which was the main impurity.Particle drifts and geometric coefficients are computed in real geometry, meaning that an initial equilibrium reconstruction was performed by the EFIT code constrained by measurements from magnetic loops and the Motional Stark Effect (MSE) diagnostic; the ONETWO transport code was used to derive the steady-state current profile, which was then provided as input to the EFIT code during the calculation of the complete kinetic equilibrium, which is subsequently read directly by CGYRO.This method ensures a correct evaluation of the magnetic geometry.Non-linear simulations typically evolve the perturbed spectrum over 30 poloidal and 400 radial wave numbers in the region k y ρ s < 2 and |k ρ ρ s | < 15, with a box size that is approximately 90 ρ s wide in both directions, where ρ s ≡ c s /Ω D is the ion-sound gyro-radius and where a is the plasma minor radius and c s ≡ √ T e /M D is the deuterium sound speed.Convergence studies were performed by resolving the same maximum wave-numbers while increasing the box size to 200 ρ s in both directions, as well as by using a grid sized at 60 ρ s wide in both directions and evolving up to k y ρ s ≃ 7 and |k ρ ρ s | < 25.The numerical grid is also composed of 32 poloidal points along the flux-tube, 8 energy and 16 pitch angle points.The convergence along these latter parameters was verified only on linear modes across the entire spectrum of linearly unstable modes.The gyrokinetic model includes sonic rotation and the Sugama collision operator which conserves particles, energy and momentum.Converged non-linear simulations typically require a few 10 5 dynamically adjusted time steps to evolve the equations over 600-800 a/c s .Uncertainties in the computed fluxes are estimated as ensemble standard deviations of quantities averaged over segments 100 a/c s long.
Modelling was performed at a number of radial locations in the region 0.45 < r/a < 0.9.The choice of the innermost radial surface is dictated by the saw-tooth inversion radius as well as by the ECH power deposition location, while the outermost radial surface was chosen so that uncertainties in profiles and in the corresponding terms composing the transport equations does not compromise the analysis.

Overview of results
The linear stability analysis of this discharge depends to some extent on the effective charge radial profile that is adopted.Indeed, while most quantities that enter the gyrokinetic equations are measured with good resolution and agreement between various systems, measurements of the effective charge are far less cross validated and discrepancies between various systems lead to large uncertainties in the density profile of the main ion species.This issue is particularly pronounced near the plasma edge, where the largest uncertainties in the experimentally measured density of electrons and ions exist.Indeed, the most unstable mode followed by initial value codes may jump from one unstable branch to another one based on how peaked the density radial profile of the main ion species is.To elucidate this point, we display in figure 2 the real and imaginary parts of the most unstable linear mode as a function of the toroidal mode number and the radial surface under consideration.The three cases considered are for the profile inferred by assuming that the amount of carbon 6+ measured by the CER system is the only impurity present in the plasma, for a flat radial profiles of Z eff = 2.7 and Z eff = 1.3.Although the local carbon concentrations in the cases corresponding to the CER measurement and to the flat Z eff = 2.7 case are similar, it is to be noted that the two dimensional plots displaying the growth rates are not significantly altered while the nature of the most unstable modes is, as evidenced by the direction of propagation which varies from the ion to the electron diamagnetic direction.One could therefore expect the particle flux predicted by the simulations to be a rather sensitive function of the Z eff profile, with a somewhat lesser impact on the energy fluxes.The importance of having a properly reconstructed radial profile of the effective charge is displayed in table 1, which reports the dependence of the saturated ion and electron particle fluxes, as predicted by CGYRO simulations, at varying radial gradient of the effective charge with all other parameters fixed, including the value of Z eff itself.Given the results obtained in the case of a flat Z eff profile, peaked impurity profiles were considered because the corresponding ion particle fluxes would assume unrealistic values.Considering instead the case corresponding to a flat Z eff = 1.3 profile, the reduced main ion dilution appears to enhance the presence and strength of ITG modes at ion scale, but also affecting the growth rates of electron scale modes likely due to finite Larmor radius effects.We note that, while such small modifications to the Z eff radial profiles do not usually entail any major modifications to the instability at play in plasmas that are strongly dominated by one given branch, the impact is much more severe in cases where additional modes are almost equally unstable, such as deuterium ITG, carbon ITG and TEM in the case here considered.We note that a micro-stability analysis of recent NegT experiments on the ASDEX Upgrade tokamak also indicated that the experimentally measured profiles sit near the transition between competing modes [20].Sensitivity studies of the saturated fluxes on the value of Z eff and on its radial gradient were carried out while keeping all other parameters fixed.By varying Z eff by 20% the sum of the ions and electron energy fluxes is seen to respond in the region 5%-20%, depending on the value of ∇Z eff ; particle fluxes respond in the region 10%-30% unless the Z eff variation is large enough to cause the dominant instability to approach, or cause, a transition to a different mode thereby generating much larger variations.Larger variations of Z eff generally lead to proportionally larger variations in the fluxes.The sensitivity to the radial gradient of Z eff is somehow more pronounced than those for Z eff , leading to about twice as large variations for both energy and particle fluxes.As for Z eff , variations in ∇Z eff can also cause mode transitions with corresponding large impacts on the particle fluxes.Dedicated analyses of the main ion temperature [17] radial profiles were carried out and are reported in figure 3.While the temperature profiles of deuterons and carbon 6+ are relatively close to each other over most of the major radius, the differences are actually large enough to make the ion temperature gradient inverse scale length up to 12% larger for the deuterons in the outer half of the minor radius.At inner radial locations, the increased radial gradient is balanced by the increased temperature, so that the scale length is essentially unaffected.This feature was observed at various time slices throughout the discharge evolution.It is remarkable to note that, while non-linear simulations carried out at r/a = 0.75 using carbon data for both ions yield turbulent energy and particle fluxes much below the experimentally inferred values, a good match against power balance is obtained by including the main ion measurements.This study did not evaluate whether the carbon profiles are actually below the non-linear threshold or marginally unstable so that a much longer simulation would have eventually generated fluxes close to the experiment.Additional minor adjustments to the local value of Z eff , up to the order of the experimental uncertainty from the CER measurement, yield better agreement between the simulations and the experiment, especially on the particle flux; other quantities such as the electron density and temperature scale lengths were not altered from the experimentally measured values.Overall, CGYRO does an excellent job in recovering the turbulent fluxes of this plasma, as displayed in figure 4 where the ion and electron energy as well as the electron particle fluxes predicted by non-linear simulations from CGYRO are compared, at four radial locations, to values inferred from power balance analysis minus the neoclassical contribution predicted by the NEO code [21].The only noticeable discrepancy that is clearly outside several statistical uncertainties is in the electron energy flux at the outermost radial location, which is the only location among those considered where electron scale fluctuations are non-linearly unstable but are not included in the present modelling effort.The impact of triangularity on micro-instabilities appears to be correctly captured by CGYRO because, besides recovering the experimental fluxes in the negative triangularity plasma considered, it also predicts larger energy fluxes when the sign of triangularity is reversed, which is in line with the observed improvement in confinement with respect to expected values based on the ITER89P scaling law for plasmas with an L-mode edge.
Electron scale fluctuations, although linearly unstable, are not active in non-linear simulations carried out with experimentally measured parameters.More specifically, electron scale fluctuations are seen to persist in the non-linear phase of the simulation only when the intensity of ion scale fluctuations is weakened beyond a threshold.This was accomplished by artificially increasing collisionality at fixed β as well as density and temperature profile scale lengths.Beyond such a threshold, electron scale fluctuations are seen to exchange energy with the ion scale part of the spectrum, a behavior that is reminiscent of multi-scale interactions, whose strength appears to depend on the intensity of the ion scale part of the spectrum [22].
The impact of Shafranov shift on the non-linearly saturated fluxes is negligible, being less than 9% on both energy and particle fluxes when one order of magnitude reduction of the Shafranov shift is artificially imposed.As such, although NegT equilibria are inherently characterized by a larger Shafranov shift than their positive triangularity counterparts, a quantity that is generally observed to improve confinement, the observed stabilizing effect exerted by negative triangularity on micro-instabilities does not predominantly rest on it, at least for the cases here considered.Rather, it appears that negative triangularity improves confinement by modifying the toroidal precession drift of trapped electrons in such a way as to  weaken the TEMs [5].It is currently an open research question to determine whether negative triangularity has a direct stabilizing impact on ITG modes as well, with predictions showing either moderate to substantial effects [23][24][25].While we do not have experimental or theoretical evidence to point either way, we note that in plasmas where TEM and ITG are both unstable, weakening of either branch would cause the overall fluxes to decrease [26], in agreement with what is experimentally and numerically observed in the case reported in this work.Given the excellent agreement between CGYRO and the experiment, we adopt this case as a test-bed for extrapolations that are detailed in the following subsections.

Impact of rotational shear
The original TCV work that demonstrated H-mode level of confinement in L-mode plasmas with negative triangularity was obtained in a regime of low-density plasmas, subject to pure electron heating and relatively large impurity content whose radial profile was assumed to be flat yielding Z eff = 3.5.Such cases were characterized by a relatively large electronto-ion temperature ratios across most of the core section and were strongly dominated by TEM turbulence, with that driven by ITG modes being severely weakened due to the low ion temperature and the large main ion dilution.Although H-mode grade confinement was routinely obtained using torque-free auxiliary heating, the peculiar turbulent state of such plasmas makes it legitimate to ask whether low rotational shear would induce a significant confinement degradation in case a more reactor relevant regime was obtained.In this respect, the DIII-D experiments also obtained H-mode grade confinement levels, i.e.H 98(y,2) close to unity, in plasmas with the ion and electron temperature close to each other over most of the cross section, thereby nearing one aspect of reactor conditions; however, they did so by using large external beam torque.Since no experimental time was available to study the impact of low external torque, the baseline case described in section 3 was used to predict the impact of rotational shear.The procedure adopted involved maintaining the density and temperature profiles for all kinetic species unchanged as well as the toroidal velocity of the ions near the edge, which was fixed to the experimentally measured value, while the rotation velocity in the core was progressively lowered in a linear way going towards the magnetic axis, where the toroidal rotation was reduced up to a factor of three as compared to the value measured during the experiment.At fixed density and temperature profiles, i.e. at fixed stored energy, the variation in the predicted energy flux for ions and electrons would indicate how much power, or fueling, the discharges are predicted to require in order to sustain that given stored energy, thereby giving indications on the resulting energy and particle confinement times.Figure 5 displays the variation in the electron plus ion energy fluxes that is to be expected when lowering the on-axis rotation by a given factor with respect to the experimentally measured value.The lowest rotation considered corresponds to a reduction of a factor of three in the on-axis toroidal rotation, which approximately corresponds to values obtained on DIII-D when the NBI systems delivers no net external torque.Such values are taken as rough proxies of plasmas where the radial electric field is dominated by the diamagnetic contribution, which is a regime to be expected in future reactors.The low rotation point computed at r/a = 0.45 is characterized by a larger uncertainty, when compared to other points, due to intermittency in the simulation.For all cases, the energy flux carried by electrons and deuterons are very similar, while that carried by carbon ions is about 10% of the total.The larger impact of low rotation at the inner radial location can be understood by considering that the rotational shear can be roughly approximated by the variation in toroidal rotation divided by the safety factor.Given that the safety factor is an increasing function of the minor radius and that the toroidal rotation kept fixed at the plasma edge, the corresponding reduction in the rotational shear is larger near the plasma center.While the actual confinement time at low rotation should be computed by performing several flux tubes simulations at various radii and radially integrating the corresponding plasma profiles, a rough estimate of the variation in confinement time can be obtained by computing the required increase in auxiliary power required to sustain the profiles, weighted by the plasma volume enclosed by the flux surface.By assuming a negligible temporal variation of the stored energy, such a procedure yields where W is the stored energy and the index i spans through the number of flux surfaces N that were computed.When evolving the flux increase at flux surfaces corresponding to r/a = [0.45,0.75, 0.9], we obtain that the confinement time at low rotation should be approximately equal to 80% of that measured experimentally at high rotation.It is instructive to carry out a similar exercise in an ad-hoc case where the ion and electron temperatures were set to be equal to their arithmetic average, i.e. mimicking perfect equipartition across the radial profile, as well as in the simplifying case of a flat Z eff radial profile with values equal to 2.7 and 1.6.When lowering the toroidal rotation by a factor of three in the higher Z eff case, the combined ion-electron energy and particle fluxes increased by ∆Q tot = +15% ± 10% and ∆Γ tot = +20% ± 11%, while at the lowest Z eff value the variation is expected to be ∆Q tot = +37% ± 6% and ∆Γ tot = +45% ± 20%.Slight variations are expected when carrying out computations on more radial locations.The corresponding variations in the respective energy and particle confinement times are obtained by taking the reciprocal of the variations in the fluxes above.As noted earlier, lowering Z eff causes an increase in the expected fluxes because, even at fixed density and temperature profiles, this results in a stronger turbulent drive for both ITG, due to lower main ion dilution, and TEM, due to reduced collisionality.We note that the Z eff dependence on the rotational shear stabilization is not a distinct feature of NegT plasmas as a similar trend was obtained by artificially reversing triangularity in CGYRO.A general comment on the impact of Z eff on confinement in future reactors is warranted.In order ot preserve the lifespan of plasma facing components, future reactors will likely operate in the divertor detached regime, possibly using impurity seeding.Although the fusion community is quickly advancing its ability to build models able to predict the impurity enrichment necessary to detach for given upstream conditions and the resulting impurity contamination in the plasma core, there are still considerable uncertainties [27,28].In this respect, while results in this section provide a first order estimate for the confinement degradation at low rotational shear as a function of core Z eff , it is far beyond the scope of this work to predict how much of a confinement degradation a prospective NegT reactor will experience because we cannot reliably predict the actual core impurity contamination.Additionally, while we estimated the percentage of confinement degradation with respect to the IPB98(y,2) scaling law in NegT plasmas, the dependence of the normalized confinement as a function of ρ * , the ion Larmor radius normalized to the machine size, is still unknown.As a result, any given loss due to low rotational shear, e.g.35%, might cause the reactor to operate near L-mode or H-mode levels of confinement, depending on how the actual confinement time compares to the existing H-mode scaling law at ρ * reactor levels.

Impact of aspect ratio
Decreasing aspect ratio increases the fraction of trapped particles, which is an effective way to increase the bootstrap current fraction [29].A larger amount of bootstrap current is appealing for reactors because it reduces the capital cost and recirculating power in a prospective reactor.Although increasing the trapped particle fraction by decreasing aspect ratio should also increase TEM driven turbulent fluxes and the associated confinement degradation, NegT configurations might suffer from that to a lesser extent because TEM are less unstable than in standard positive triangularity plasmas.If that was the case, then NegT configurations would just see the benefit of more bootstrap current.
Starting from our baseline case at r/a = 0.75, the aspect ratio was artificially reduced in the CGYRO simulations by reducing the major radius of the equilibrium, while maintaining minor radius, plasma density and temperature radial profiles as well as the rest of the equilibrium fixed.As displayed in figure 6, a reduction of the aspect ratio R/a from 2.8 to 1.9 causes the combined ion and electron energy flux to increase by 75%, while a further reduction to R/a = 1.5 yields the energy flux to increase by almost 400%.It is to be noted that, while varying aspect ratio, the energy flux remains approximately balanced between the two ion species and electrons.By increasing the aspect ratio beyond three, i.e. above the experimental value on DIII-D, fluxes are expected to vary by an insignificant amount for any of the kinetic species evolved.This indicates that, in view of the cost associated to building large devices, the optimal value of aspect ratio is indeed close to three, which is the value chosen by most existing machines.This behavior is tracked by linear simulations which, for all wave-numbers evolved, observe increasing growth rates at decreasing aspect ratio.We note that, both in linear and nonlinear simulations, the increase in the fluxes/growth rates is much larger than the corresponding increase in the trapped particle fraction.Further projections of the expected confinement loss in NegT plasmas in small aspect ratio devices, along with an investigation of the physical reason causing it, will be the subject of future work.

Non-dimensional confinement scaling
The fact that a given scenario of operation obtains good confinement levels, which is commonly expressed by displaying confinement quality factors H 98(y,2) reaching or exceeding unity, does not imply, per se, that such a scenario will be applicable to future reactors.Indeed, the scaling of the confinement time to reactors conditions should instead be determined in order to project fusion performance in terms of the triple product.An effective way of achieving this is to apply the scale invariance paradigm proposed by Connor [30] which, in a concise form, states that if the set of equations describing a given system are invariant under a scale transformation, then any quantity derived from such equations must bear the same scale invariance.If the model under consideration is the Fokker-Plank equation at arbitrary large values of beta, then the confinement time τ can be cast in the form where Ω is the ion cyclotron frequency, ρ * the Larmor radius normalized to the machine size, ν * the normalized collisionality, β the normalized plasma pressure and r i are ratios of like quantities, one example of which is the safety factor q. With the exception of the normalized Larmor radius, gyrokinetic codes employing the flux tube formalism, such as CGYRO, can be used to estimate the unknown function F. Numerical scans in q, β and ν with other parameters held fixed were carried out for flux surfaces corresponding to normalized minor radii r/a = 0.75 and r/a = 0.90.Quantities were typically varied by a factor of two and included three points in q − β and three to five points in ν.The overall scan included over twenty non linear simulations.By assuming that such a function is the product of the independent quantities to some power, scans in the safety factor at fixed magnetic shear, collisionality and pressure suggest that the energy confinement is proportional to ν 0.5±0.2* β −0.4±0.1 q −3.8±0.2

95
, thereby significantly improving at large plasma current, with minor improvement with decreasing normalized pressure and increasing collisionality.The uncertainty on the exponent was estimated via a Monte-Carlo analysis based on the uncertainty associated to the nonlinearly saturated fluxes, rounded to the first digit.We stress that, using a flux tube formalism, the calculations are effectively carried out at vanishing ρ * , which approximates large devices at high confining magnetic field.The positive dependence on collisionality can be understood in terms of collisions reducing the turbulent drive from electrons.This behavior is opposite to what is observed in normalized confinement experiments in H-mode plasmas at positive triangularity, where the normalized confinement is observed to degrade with increasing collisionality [31].However, the IPB98(y,2) scaling law, when converted to non dimensional units, translates into a negligible positive dependence of the normalized confinement time on collisionality, or α ν * = 0.08, while that for the ITER97-L law is more robustly positive with α ν * = 0.2.The confinement degradation with q 95 is comparable to that derived for the ITER97-L law, for which α q = −3.6.Finally, the confinement degradation with normalized pressure is weaker than that derived for the IPB98(y,2) and ITER97-L scaling laws, for which the exponents are α β = −0.90 and α β = −1.4,respectively.Based on the numerical values assumed by the exponents, our analysis indicates that NegT plasmas might behave differently from both standard H-mode and L-mode plasmas at positive triangularity, although they share some features.It has to be said that large uncertainties are typically associated to the exponents of non-dimensional confinement scaling laws, when derived via both an experimental and a modelling approach.Moreover, exponents of dimensional scaling laws, when converted to non-dimensional units, do not always agree with their counterparts derived in controlled nondimensional experiments.This discrepancy is possibly due to issues either in the controlled experiments or in the determination of the scaling law, or both.Indeed, when scaling one dimensionless quantity in experiments, it is generally hard to maintain all others fixed across a portion of the minor radius that is large enough as not to bias the determination of the confinement time.Conversely, the determination of scaling laws from data-sets is affected by hidden variables, co-linearities among quantities of interest, or scarcely populated parametric spaces [32].To this extent, in order to confidently predict performance in future devices, it is important to derive scaling laws that, when expressed in dimensionless and engineering units, are consistent with each other as well as with modelling work.

Conclusions
In this paper we performed a linear and non-linear gyrokinetic analysis of a high performance DIII-D inner wall limited discharge at negative triangularity using the CGYRO code.The modelling, coupled to neoclassical predictions from the NEO code, quantitatively reproduces ion and electron particle and energy fluxes inferred from a power balance analysis at a number of radial locations.Apart from the outermost radial location, the experimental conditions are well reproduced by ion scale simulations, without the need to include density and temperature profile curvature effects.Scoping simulations at positive triangularity predict fluxes about twice as large as those computed in the actual equilibrium, thereby corroborating the stabilizing effect of negative triangularity.Simulations on artificially created equilibria conclude that the Shafranov shift induced stabilization is not the cause for the improved confinement experienced by NegT configurations.The linear ion scale turbulent regime of this plasma is composed of competing ITG and TEM modes, with the most unstable mode being determined by the exact shape of the Z eff radial profile assumed in the simulations, with other parameters being fixed.Low values of rotational shear are seen to degrade the energy confinement time in the region 15%-37%, depending on the value of the plasma effective charge and the stabilization role this exerts on ion scale instabilities.A prospective reactor will likely operate with seeded impurities to ease divertor detachment, possibly accompanied by a radiative mantle.The resulting confinement quality at low rotational shear will depend on which seeded impurity will be needed to detach, its enrichment and the resulting contribution to the effective charge in the plasma core.Although the low rotational shear induced confinement degradation is estimated, predictions of the actual confinement quality in a prospective reactor are beyond the scope and the means of this paper.The impact of decreasing aspect ratio is predicted to be strongly destabilizing both linearly and nonlinearly, with turbulent fluxes predicted to double at aspect ratio values slightly less than two and increasing much more rapidly at even smaller values.An attempt at estimating the non dimensional scaling of the energy confinement time yields positive dependence on normalized collisionality and inverse safety factor, while the magnetic component of the overall transport causes weak degradation.

Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States Government.Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Figure 1 .
Figure 1.Temporal evolution of a few parameters for DIII-D discharge #171421.(a) Plasma current.(b) Axuliariay neutral beam, electron cyclotron and Ohmic power.(c) Electron line averaged density.(d) Normalized plasma pressure.The two vertical dashed lines in (b) indicate the time window mostly used for the analysis.Sudden short drops in neutral beam power are used to improve the accuracy of ion data from the charge exchange recombination spectroscopy system.

Figure 2 .
Figure 2. Frequency (top) and growth-rate (bottom) of the most unstable linear mode, normalized to c s,D /a, for a number of radial locations and bi-normal wave-vectors as computed by CGYRO.The three cases refer to the Z eff profile inferred by assuming that C 6+ is the only impurity in the plasma (left), a flat Z eff = 2.7 (centre) and a flat Z eff = 1.3 profile (right).Black contour lines in the plots on the top separate modes propagating in the ion (positive) and electron (negative) diamagnetic drift directions.

Figure 3 .
Figure 3.Comparison between the temperatures measured by the CER system for carbon (black squares) and deuterium (red triangles) ions as a function of the radial coordinate expressed as the squared root of the normalized poloidal flux.Profiles were measured during phases where the plasma was subject to low(left), mid (centre) and high (right) auxiliary power, resulting in the three normalized pressure levels indicated in each plot.

Figure 4 .
Figure 4. Comparison between the electron particle (left), electron energy (centre) and ion energy (right) fluxes, normalized to GyroBohm units, predicted by non-linear gyrokinetic simulations (green triangles) with those inferred by a power balance analysis minus the neoclassical contribution (blue), as a function of the normalized minor radius.

Figure 5 .
Figure 5. Sum of the electron and ion energy fluxes, normalized to values expected for the experimental case, as a function of the plasma rotation on-axis.Computation carried out for flux surface corresponding to r/a = 0.45 (left) and r/a = 0.9 (right).

Figure 6 .
Figure 6.Energy fluxes as a function of aspect ratio for electrons (blue), deuterons (black) and carbon impurities (red) at radial location r/a = 0.75.

Table 1 .
Dependence on the radial gradient of Z eff of the non-linearly saturated particle fluxes, expressed in gyro-Bohm units, for electrons, deuterons and carbon.Inferred values from TRANSP are reported for a quantitative comparison.Simulations are carried out at flux surface r/a = 0.90 with all other profiles fixed.