Full-flux-surface effects on electrostatic turbulence in Wendelstein 7-X-like plasmas

We present the first nonlinear, gyrokinetic, surface-global simulations of a Wendelstein 7-X-like stellarator with kinetic electrons. As a first application, we investigate the interplay between Ion Temperature Gradient (ITG) and Trapped Electron Mode (TEM) driven turbulence in a Full-Flux-Surface (FFS) approach, as well as the effect of a neoclassical radial electric field, something that escapes the capabilities of flux-tube simulations. We find that even in this more complex setup, ITG turbulence is stabilised through a finite density gradient while TEM turbulence remains relatively weak. Furthermore, we show that the effect of the radial electric field itself is small in comparison with the variation of the gradients. Nevertheless, we observe that for some of the cases shown here, there is not only quantitative but also qualitative disagreement between flux-tube and FFS simulations, in contrast to earlier studies with an adiabatic electron model. These results emphasise the potential importance of retaining geometrical variations on the flux-surface when describing stellarator turbulence under realistic conditions.


Introduction
Modern day techniques in numerical optimisation have made it possible to lower the neoclassical transport in stellarators to the point where some of the designs have energy confinement levels comparable to that of tokamaks [1]. Due to this, significant effort is put into finding ways to also reduce plasma turbulence in modern day stellarator research in order to further increase their performance [2,3]. One particular obstacle observed in the Wendelstein 7-X (W7-X) stellarator is the * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. clamping of the ion temperature to around 1.5 keV in ECRHheated plasmas [4]. Unlike in tokamaks, however, using pellet injection together with increased heating power instead of gaspuff fuelling has been proven to transiently decrease turbulent transport significantly [5] in W7-X. The mechanisms behind this improved performance have been explained in [6]: due to the supposedly localised nature of Ion Temperature Gradient (ITG) turbulence in W7-X [7], an increase in the strength of the neoclassical radial electric field causes a dislocation of turbulent fluctuations into regions of better curvature, therefore weakening ITG driven transport. Nevertheless, this effect was considered to be of rather secondary importance. The main driver of turbulence stabilisation was attributed to a large fraction of trapped particles to reside in regions of good curvature. It has been shown in [8] that such a constellation causes trapped electrons to have a stabilising effect on the plasma, therefore entering a regime in which Ion Temperature Gradient and Trapped Electron Mode (TEM) activity are largely suppressed if the background density and temperature gradients approach the same value [9].
The underlying idea was built on flux-tube theory, only considering a change in gradients for a single flux-tube at a time, which neglects the possibility of different types of modes to reside in different field lines on a flux-surface, as well as the effects of geometrical variations on nonlinear mechanisms such as interaction with zonal flows. Moreover, the Full-Flux-Surface (FFS) simulations performed in [6] employed an adiabatic electron model, therefore leaving the effect of a radial electric field on TEM turbulence and ITG-TEM-hybrid scenarios unexplored.
Due to a recent upgrade [10], GENE-3D [11] has become a suitable tool to look at these effects in combination with each other. Therefore, we plan to bridge the gap and provide a first insight into the interplay of ITG and TEM turbulence in surface-global simulations of W7-X-like stellarators in this work. In particular, we confirm the stabilising influence of a finite density gradient under this more general model. Additionally, we investigate the impact of a non-local geometry on ITG and TEM turbulence, as well as the influence of a neoclassical radial electric field and draw comparisons with their flux-tube counterparts. We find that the electric field does not seem to have a significant impact on any of the cases under consideration, which we attribute to the small amount of dislocation of turbulent fluctuations. Furthermore we observe that, while flux-tube simulations seem to reproduce the trends of some of the FFS simulations, there are cases where the two models give distinctly different results from each other, highlighting the importance of using the higher-fidelity model whenever possible.
The rest of the paper is structured as follows: in section 2, we introduce the numerical setup of the simulations presented in this paper. We reproduce the stabilisation of ITG-dominated turbulence through a finite density gradient, as predicted by local theory, in a nonlinear, full-flux-surface setup. We investigate the spatial structure of turbulent fluctuations, finding a rather extended structure of the fluctuations on the surface. In section 3 we investigate the configurational effects of ITG stabilisation through a density gradient in the context of the lowmirror configuration of W7-X. In there, we observe the same qualitative behaviour as for the cases shown in section 2, which we attribute to the fact that the driving background gradients are not large enough to highlight the configurational differences of the two geometries with respect to density gradient driven ITG stabilisation. In section 4, we give an overview over the influence of a neoclassical electric field on different types of turbulence under consideration. In there, we show that its effect on the heat flux is small in comparison with the variation of the background gradients. In section 5 we study the influence of a finite electron temperature gradient on the comparison between flux-tube and FFS simulations. In there, we find substantial disagreement between the two models for the cases under consideration, which cannot even be remedied by increasing the number of flux-tubes used to compare against the higher-fidelity model. Finally, the results are summarised in section 6 and an outlook for future projects is given.

Density-gradient-induced ITG stabilisation in local and surface-global simulations
In this section, we will compare the influence of full-surface effects on the density-gradient-induced stabilisation of ITG turbulence in W7-X by performing FFS simulations using the GENE-3D code. For this, we consider the Standard configuration [12], sometimes also labeled as EIM configuration, at ρ tor = 0.65. Here, ρ tor = √ Φ tor /Φ edge is the surface label based on the toroidal flux, with Φ edge being the toroidal flux at the last closed flux-surface. As this paper will be focusing on electrostatcially driven, collisionless turbulence, we assume a collisionless and electrostatic limit with an electron plasma-β of β e = 8π n ref T ref /B 2 axis = 10 −4 . We assume a reference temperature of T ref = 3.41 keV, which results in a finite-size parameter ρ * = ρ s /a = 1/200. Here we introduced the ion sound Larmor radius ρ s , which we define via the ion sound velocity c s = √ T ref /m p and the ion Larmor frequency Ω s = eB axis /(m p c) as ρ s = c s /Ω s , where m p is the proton mass. Furthermore, we have introduced an effective minor radius a = √ Φ edge /(π B axis ), with B axis being the magnetic field strength on the magnetic axis.
In order to study the turbulence stabilisation that is associated with a finite density gradient, we consider three scenarios: in the first case, we only retain an ion temperature gradient of a/L T i = 2.5, while setting all other background gradients to zero. This will ensure that the resulting turbulence will be purely ITG-driven and can be seen as a reference case. The second scenario is that of a purely density-gradient driven TEM with a/L n = 2.5 and a/L T i = 0, which will serve as a comparison at the opposite extreme. While the first two cases distinctly provide a free-energy source for only ITG and ∇ndriven TEM turbulence, respectively, we also consider a third case with a mixed drive, right between the other two. The background gradients are chosen to be a/L T i = a/L n = 2.5, which according to [9,13] is considered to be a very favourable parameter set for turbulence suppression since η i = L n /L T i = 1. For now, we set the electron temperature gradient to zero for all three cases and postpone studying its influence to section 5. For convenience, the gradients, together with the respective labelling of the cases, are summarised in table 1.
The phase space domain chosen for both flux-tube and FFS simulations is (l x , l y , l z , l v , l µ ) = (225 ρ s , 145.157 ρ s , 2π, 3.0 √ 2T σ /m σ , 9.0 T σ /B axis ) with a resolution of (n x , n y ,n z ,n v || ,n µ ) = (225,198,128,32,9). Here, x, y and z are our radial, binormal and parallel spatial coordinates, whereas v || and µ are the velocity parallel to the magnetic field and the magnetic moment, respectively. Since Dirichlet boundary conditions were employed in the radial direction, numerical heat and particle sources with an amplitude of κ H = κ P = 0.02 were added to the equations in order to retain the background density and temperature profiles [14]. For the flux-tube Table 1. Choice of gradients for the different turbulence scenarios used in this paper. Unless stated otherwise, the electron temperature gradient is assumed to be zero. simulations, we restrict ourselves to the bean-shaped (α = 0) flux-tube, while for the FFS simulations, one-fifth of the entire surface is considered, taking advantage of the discrete symmetry of W7-X. For the spatial direction parallel to the magnetic field, twist-and-shift boundary conditions [15] were used for both, flux-tube and full-flux-surface simulations.

Case label a/L T i a/Ln
Comparing the transport levels of the three scenarios in figure 1, we first confirm the density gradient-driven reduction of turbulence in flux-tube simulations that was suggested by linear theory [8,9] and confirmed for nonlinear flux-tube simulations [16]. We observe the pure ITG having by far the largest ion and total transport. In particular, the density-driven TEM is much more benign, a fact that is understood to be caused by the trapped electrons primarily residing in regions of positive average curvature [8]. On top of that, the mixed case has the lowest transport of all flux-tube simulations, be it in each of the channels or the total transport, which is in line with the linear results presented in [6,9]. On top of that, we show in figure 1 that this stabilisation even extends to surfaceglobal simulations. In particular, we see that the FFS simulations results follow the same trends as those of their flux-tube counterparts, with the ITG case having the highest amount of transport, followed by the TEM case and the Mixed case having the lowest transport overall.
We are particularly interested in how well a flux-tube result agrees with the overall behaviour of the flux-surface simulations, in order to estimate if the latter could be mimicked through multiple local simulations. To this end, the averaged ion and electron heat fluxes obtained by the FFS simulations are displayed as a function of the field line label α in figure 2 for all three cases. The heat flux is rather uniform for the cases under consideration, with the variation always under 20% with respect to the corresponding mean value. Putting the flux obtained by the corresponding flux-tube simulation on top, marked by a cross, we can see that for the ITG case, the local and flux-surface results differ by up to 30%, which indicates that the combination of multiple independent local simulations could still yield different results than the FFS approach. These findings differ from results shown in [7], where under the usage of an adiabatic electron model, a close agreement was found between flux-tube and FFS simulations of ITG. To test the influence of the electron model, we repeated the simulation of the ITG case with adiabatic electrons. A comparison of the heat flux structure can be found in figure 3. In there, one can see that the agreement between flux-tube and fluxsurface simulations is much better for the adiabatic than the kinetic electron model. Beyond that, one can also see a clear maximum of the heat flux levels towards the α = 0−field line for the former, whereas the latter model produces a rather uniform distribution of the flux. This could be an indication that the usage of an adiabatic electron model over-exaggerates the localisation of turbulence in comparison with its kinetic counterpart, something that is left to be studied in more detail in the future. Furthermore, it becomes apparent from this figure that flux-tube simulations produce significantly more transport towards the α = 0−tube for both electron models, which together with the previously mentioned peaked structure of the adiabatic electron model could be one factor explaining the good agreement between flux-tube and FFS simulations with adiabatic electrons that was presented in [7]. Despite these differences, there seems to be a close agreement between the fluxtube and surface simulations for the TEM and mixed cases, indicating that for these cases the former might serve as an adequate proxy for predicting the heat fluxes.
Previous studies of FFS simulations with an adiabatic electron model [6,7,13,17,18], using the surface-global version of the GENE code, propose that turbulent density fluctuations in most stellarators are localised on a thin strip near the outboard midplane, where the fluctuation amplitude was varying up to a factor of around 10 for some parameters. These claims could not be reproduced in global simulations of W7-X with adiabatic electrons [19], where the amplitude variation was found to be rather close to a factor of 2. We wish to investigate this phenomenon using the more accurate model of kinetic electrons. To this end, the density fluctuations on the surface are shown as a function of the poloidal and toroidal PEST angles θ * and ϕ [20], assuming a right-handed system, in figure 4.
Here we can see that the fluctuations seem to be fairly extended in both the poloidal and toroidal direction for all three cases, with the fluctuation level over the surface varying by at most around a factor of two for all three cases. These results agree qualitatively with global simulations presented in [19], which were performed using Euterpe and GENE-3D, although an adiabatic electron model was used there. In order to test the influence of the treatment of the electrons further, we additionally compare the spatial distribution of density fluctuations between the adiabatic and kinetic electron simulation in figure 5. Although having a slightly larger ratio between maximum and minimum amplitude than its kinetic electron counterpart, the results  obtained with adiabatic electrons are still far more extended than what was shown in [6,7,13,17,18] and is again in line with [19].
In summary, we conclude that, while the flux-tube results produce a good qualitative approximation of the more complex model of FFS simulations for the particular cases presented in this section, we still observe differences between the two models for the ITG case, making it necessary to use the higherfidelity model if one is interested in an accurate quantitative prediction of the turbulent fluxes. Furthermore, we showed  that the use of an adiabatic electron model results in a stronger agreement between the two simulation procedures, on top of producing more localised fluctuation amplitudes in general in comparison with a kinetic electron model for the cases under consideration. We will investigate the quantitative differences between the two simulation domains in more detail in section 5.

Comparison between standard and low-mirror configuration
In this section, we want to study the influence of geometry on the stabilisation of turbulence. We repeat the simulations of section 2 using the low-mirror configuration of W7-X, but keeping the other parameters the same. We can see in figure 6 that the trend of ITG stabilisation is also present in this configuration. One can see that the overall transport of all cases seems to be larger than the heat fluxes obtained in the simulations of the standard configuration. This increase can could be linked to the fact that this configuration has a larger overlap between the regions of bad curvature and magnetic wells, something that can act as a destabilising mechanism of trapped-particle dynamics, even if no TEM-drive is available. However, one still observes a substantial decrease of both flux channels when adding a density gradient to the ITG scenario, just as is the case in figure 1.
The reason for the similar behaviour between the two configurations can be understood by the fact that, for the gradients smaller or equal to the ones used here, all configurations considered in [9] seem to benefit from η close to 1 [13], and the difference between the geometries only becomes apparent at significantly larger gradients. This is actually supported by results presented in [16], where flux-tube simulations performed with the gyrokinetic code Stella showed an ITG stabilisation through a finite density gradient even in NCSX, a stellarator whose quasiaxisymmetric nature results in unfavourable electron curvature drifts on the surface. We, therefore, expect similar trends regarding the stabilisation of core turbulence through a finite density gradient between the standard and low-mirror configuration for experimental parameters.

Influence of a radial electric field
One crucial feature that sets apart FFS from flux-tube simulations is that the former can account for the influence of an equilibrium radial electric field E r (x) = −∇ϕ 0 (x). While both models can include the influence of its radial derivative [21], only surface simulations can capture the influence of the electric field itself, as a constant electric field can be eliminated from the equation by transforming into a reference system that rotates with the corresponding ExB-velocity. The first publication investigating this aspect was [6], showing a substantial reduction of ITG turbulence. The mechanism for the stabilisation can be summarised as follows: the electric field causes an additional advective velocity drive, which in turn will dislocate turbulent fluctuations on the surface under consideration. In comparison with the hypothesised strong localisation of turbulence in W7-X [6,7], most of the fluctuations, which reside in a thin strip around the outboard midplane, will be moved into regions of weaker curvature, therefore reducing the effective drive of ITG turbulence.
While it was concluded that its stabilising effect would most likely be minor in comparison with an increase in the density gradient, its implications on turbulence levels in the presence of kinetic electrons or impurities are unknown so far. In this work, we will only focus on the former and leave the latter for future work. In particular, we consider again the three FFS cases in the standard configuration used before, but add a radial electric field with Mach number M ExB = v ExB /c s = ±0.015, where v ExB is the ExB-velocity and c s is the ion sound velocity. The choice of this Mach number corresponds approximately to a radial electric field of ±19 kV m −1 for the given reference temperature. The electric field is accounted for by adding the term ] to the right-hand side of the gyrokinetic equation, where F 1,σ is the first order distribution function of particle species σ and G{A 1,|| } is the gyroaveraged parallel vector potential. For further details, the reader is referred to [21].
A comparison between the heat fluxes with and without the electric field is given in figure 7. Here, one can see that the effect on turbulence is indeed rather small, especially in comparison with the variation of the gradients. This is again in line with results published in [19], where a noticeable impact was only obtained in linear cases, while nonlinear turbulence was also rather unaffected. The reason for that can be understood when considering the explanation mentioned at the beginning of this section, together with the results of section 2: this dislocating effect of the radial electric field can only have a noticeable impact if turbulence is sufficiently localised. This can be easily understood by considering an extreme scenario of completely homogeneously spread turbulence, where any sort of dislocation would go unnoticed. As was already shown in figure 4, the turbulence is spread out rather evenly for all three cases under consideration. Likewise, as can be seen in figure 8, the degree of dislocation is substantially smaller for our cases in comparison with the results shown in [6]. On top of that, we argue that even in the case of strong localisation, the electric field will only cause dislocations on equilibrium length scales if the associated velocities, namely, the ExB-and thermal velocity, are comparable with each other, which would require that M ExB ∼ 1. We will support this claim during the analysis of the cases considered in section 5.
In summary of this section, we conclude that a constant neoclassical electric field will not have a significant influence on the turbulent behaviour of the plasma, due to the rather extended nature of the fluctuations as well as the weak advective drive in comparison with the equilibrium scale. In this work, we do not address the impact of turbulence shearing through the electric field, which might still have a beneficial effect [22], which will be investigated in the future.

Effect of a finite electron temperature gradient
As a final step, we will consider the effect of retaining a finite electron temperature gradient on the turbulent dynamics at hand. We, therefore, repeat the simulations shown in section 2 with an additional electron temperature gradient of a/L Te = 2.5. Doing so leads to a significantly different behaviour than what was observed in section 2. Inspecting the ion and electron heat fluxes of the three scenarios, shown in figure 9, the first thing we observe is a substantial increase of the fluxes in the ITG case in comparison with the ones shown in figure 1. Especially, the increase in the electron heat flux can be interpreted as an indication of the existence of an ITG-TEM hybrid in this case. We further support this hypothesis by looking at the structure of the ion and electron heat fluxes along the magnetic field line for the flux-tube case, shown in figure 10. One can see that, while the ion heat flux still is largely localised around the outboard midplane, similar to the structure of the pure ITG case, the structure of the electron heat flux shows several local maxima, corresponding to the positions of the magnetic wells along the field line, similar to the structure of the density-gradient driven TEM turbulence. In comparison with the ITG-case fluxes presented in section 2 however, the flux-tube simulations produce heat fluxes that are almost twice as large as the ones obtained with the FFS model. One can see that the two models now do not only differ quantitatively but also qualitatively for the mixed  and TEM case: while the flux-tube simulations predict that the mixed case should have similar levels of total transport compared with the TEM case, the FFS simulations predict that the former produces significantly more transport than the latter. This can be seen in figure 11, where we plotted the heat fluxes again like in figure 9, but without the ITG results for clarity. To explore these differences further, we show the variation of the heat flux along the field-line label for all three FFS simulations in figure 12. Furthermore, we added the transport levels of the flux-tube simulations at the position corresponding to the respective tubes with crosses. To highlight the discrepancy, we perform additional simulations of the ITG case, as the difference between the two models is strongest there. On top of the α = 0−flux-tube, we performed local simulations at α = (0.25, 0.5, 0.75) * (2π/5) for this case, as can be seen in figure 12. In doing so, we see that, while the FFS simulation produces heat fluxes that are spread out fairly evenly over the different tubes, the local simulations predict a strong peaking of the fluxes in the bean-shaped tube, with a rapid fall-off towards the other tubes. Comparing the local with the surfaceglobal results, it becomes apparent that even an averaging procedure over the local simulations will not only still give significantly different transport levels, but also produce a different spatial profile of turbulence.
We can observe similar behaviour in the same figure for the mixed and TEM cases, where local simulations either underor over-predict the heat fluxes, respectively. In addition, we can see that for both cases, the FFS simulation seems to indicate a transport slightly increasing towards the α = π/5-flux-tube, in contrast to the ITG case, which has slightly higher transport around α = 0. This implies that one should not only compare a single flux-tube in one's turbulence studies, as there is no guarantee that the transport will always peak in the same binormal position.
As a next step, we again compare the structure of the turbulent density fluctuations between the three cases in figure 13. We can see that the ITG and TEM show a similar structure compared to the cases shown in figure 4, in that the fluctuations are localised on the outboard-midplane in the ITG case and fairly spread out over the surface in the TEM scenario. In contrast to section 2, the mixed case with finite electron temperature gradient produces a highly localised fluctuation profile that peaks mainly on the inboard side of the flux-surface.
While a dedicated investigation of this type of turbulence is postponed to future work, its localised nature still makes it interesting to study the influence of an electric field on its fluctuation and transport levels. Adding an electric field with M ExB = −0.015, the ion heat flux reduces from (1.70 ± 0.10) Q GB to (1.26 ± 0.11) Q GB and the electron flux from (1.98 ± 0.12) Q GB to (1.46 ± 0.13) Q GB . Likewise, the density fluctuations are again only shifted very slightly, as can be seen  in figure 14, which overall implies that even for these highly localised cases, the neoclassical radial electric field seems to have no significant impact on the spatial location of the fluctuations. As was mentioned already in section 2, we attribute this behaviour to the fact that the ExB-velocity is small compared with the thermal velocity and will therefore not produce significant dislocations on equilibrium length scales.
Overall, we showed in this section that for the case considered with an additional electron temperature gradient, fluxtube simulations seem to be insufficient, as they do capture neither the quantitative nor the qualitative behaviour of turbulence in the same way as FFS simulations.

Summary and outlook
In this paper, we investigated the effect of surface-global effects on ITG and TEM-driven turbulence and the interaction between the two in W7-X-like configurations. For this, we performed the first FFS simulations with kinetic electrons. In doing so, we confirmed the stabilising effect of a finite density gradient on ITG-dominated turbulence within the more complex model. Our findings indicate that, in most cases, turbulence is spread out rather evenly over the surface, in contrast to previously published FFS simulations. We also found that, while flux-tube simulations can be a sufficient approximation of transport levels for some cases, there is a significant chance of either over-or under-predicting transport levels, as well as producing an incorrect spatial structure of turbulence on the surface itself. In addition, we showed that an external radial electric field given through neoclassical physics does not have a significant impact on turbulent transport for the cases under consideration, not even if the turbulent fluctuations are highly localised.
While these results can be understood as being representative cases of the most relevant types of ion-scale turbulence in the previous W7-X campaign, we consider it to be beneficial to extend the studies presented here to a larger set of parameters in future work. This includes varying the radial position and the range of gradients and testing the behaviour in a larger set of configurations, as such an endeavour will be of significant help for constructing reduced models of turbulent transport in the future. Furthermore, we plan on performing an extensive comparison between flux-tube, FFS and radially global simulations for an experimental discharge in a future work in order to compare the predictive capabilities of the respective model for realistic parameter setups.