3.1 Simulation details

The gyrokinetic simulations are carried out with the GENE code, which uses input data from the VMEC (Hirshman et al. Reference Hirshman, Merkel and van Rij1986) and geometry interface for stellarators and tokamaks (GIST) (Xanthopoulos et al. Reference Xanthopoulos, Cooper, Jenko, Turkin, Runov and Geiger2009) codes for data concerning the geometry of the magnetic field. The nonlinear gyrokinetic equations as used in the GENE code may be found in Jenko et al. (Reference Jenko, Dorland, Kotschenreuther and Rogers2000) and Goerler et al. (Reference Goerler, Lapillonne, Brunner, Dannert, Jenko, Merz and Told2011). Both ions and electrons are treated kinetically. The code GIST has been designed to provide the nonlinear gyrokinetic solver GENE with the necessary geometric information, in order to perform simulations in an arbitrary toroidal plasma with closed flux surfaces. The simulations are carried out for a range of density ( $a/L_{ni,e}$ ) and temperature ( $a/L_{Ti,e}$ ) gradients, where $a$ is a normalisation length, and as already mentioned, it is important to vary the magnetic equilibrium from simulation to simulation so that it is consistent with the resulting pressure gradient. The magnetic equilibrium is varied in the following way: we keep the shape of the pressure profile constant but vary its absolute magnitude, so that $\unicode[STIX]{x1D6FD}$ becomes proportional to the pressure gradient.

We consider a hydrogen plasma in the collisionless regime, in W7-X and in a generic tokamak. For W7-X, we take two configurations (KJM and TEH). In particular, the TEH configuration has no magnetic well and is therefore MHD-unstable for very low $\unicode[STIX]{x1D6FD}$ . We also consider two distinct flux tubes, one with its centre in the outboard midplane of the so-called bean-shaped plane, the other centred around the outboard midplane of the ‘triangular’ plane, with both tubes extending one poloidal turn around the torus. This choice is guided by the experience gained in previous iMHD stability studies. The bean-shaped cross-section is therefore paradigmatic of the most unstable situation. To render the picture somewhat more complete, we report on the results obtained by using a less unstable triangular cross-section. Electromagnetic effects are always taken to be consistent with equilibrium pressure gradients. Furthermore: $T_{i}/T_{e}=1$ , $m_{i}/m_{e}=1836$ , $r/R=0.26$ , $q=1.3$ , local magnetic shear ${\hat{s}}=0.745$ for the tokamak case and $r/R\approx 0.095$ , $q=1.1$ , ${\hat{s}}\approx -0.1$ for the W7-X case. Here, $T_{i,e}$ are the ion and electron temperature, $R$ is the major radius, $r$ is the minor radius and ${\hat{s}}$ is the global magnetic shear. Here, $\unicode[STIX]{x1D6FD}_{GENE}=\unicode[STIX]{x1D6FD}_{i,e}=\unicode[STIX]{x1D6FD}_{total}/2=(8\unicode[STIX]{x03C0}n_{i0}T_{ref})/(B_{ref}^{2})$ , where $T_{ref}$ is a reference temperature, $B_{ref}$ is a reference magnetic field and $n_{i_{0}}$ is the equilibrium ion density.

We consider $k_{x}\unicode[STIX]{x1D70C}_{s}=0$ , where $k_{x}$ is the radial wavenumber, starting our reference case, further scans in this parameter are discussed later. Parallel magnetic fluctuations $\unicode[STIX]{x1D6FF}B_{\Vert }$ are taken into account and their role will be discussed first.

Figure 1.

Influence of $\unicode[STIX]{x1D6FF}B_{\Vert }$ , spectra for W7-X from GENE simulations. (a) $a/L_{Ti,e}=3$ . (b) $a/L_{Ti,e}=7$ .

3.2 Parallel magnetic fluctuations influence

The role of $\unicode[STIX]{x1D6FF}B_{\Vert }$ in microinstabilities is a subtle one. While in slab geometry $\unicode[STIX]{x1D6FF}B_{\Vert }$ could generate a spurious mode had one missed its importance in maintaining pressure balance (Rogers, Zhu & Francisquez Reference Rogers, Zhu and Francisquez2018), for ITG instabilities in toroidal geometry it provides a non-trivial cancellation of the magnetic drift modification due to equilibrium pressure gradients (Zocco et al. Reference Zocco, Helander and Connor2015). Despite this complication, in our simulations, we always retain $\unicode[STIX]{x1D6FF}B_{\Vert }$ and a full picture of the influence of pressure gradients on magnetic drifts. That is, we always retain a difference between curvature and $\unicode[STIX]{x1D735}B$ drifts. In the language of Tang, Connor and Hastie, this means (Tang et al. Reference Tang, Connor and Hastie1980)

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D705}}-\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D735}B}=\frac{\unicode[STIX]{x1D6FD}_{i}}{2}\unicode[STIX]{x1D714}_{p},\end{eqnarray}$$

which can be easily derived from $\boldsymbol{j}\times \boldsymbol{B}=\unicode[STIX]{x1D735}p$ .

The $\unicode[STIX]{x1D6FF}B_{\Vert }$ effects have been neglected in many numerical studies of microinstabilities and turbulence in the literature. Here, we find that neglecting these effects has a noticeable impact on growth rates even for reasonably small gradients (see figure 1). In figure 1, we show the spectrum of the KBMs obtained by GENE simulations for W7-X with and without $\unicode[STIX]{x1D6FF}B_{\Vert }$ . For these cases, $\unicode[STIX]{x1D6FD}=3\,\%$ , $a/L_{ni,e}=0$ , $a/L_{Ti,e}=3$ and $a/L_{Ti,e}=7$ . The same type of study is then repeated for the tokamak case (see figure 2).

The comparison between figures 1 and 2 shows that in both cases, an absence of $\unicode[STIX]{x1D6FF}B_{\Vert }$ effects leads to a decrease of the KBM growth rate up to a factor of 6. This result is in line with earlier findings that parallel magnetic fluctuations act to destabilise (Belli & Candy Reference Belli and Candy2010; Zocco et al. Reference Zocco, Helander and Connor2015). In the stellarator case, for small gradient $a/L_{Ti,e}=3$ , $\unicode[STIX]{x1D6FF}B_{\Vert }$ is essential for KBMs destabilization (see figure 1 a). We also note that, for W7-X, the growth rate of the high- $k_{y}$ end of the spectrum TEMs are not affected by parallel magnetic fluctuations.

For the stellarator, at large gradients, the mode frequency is $\unicode[STIX]{x1D714}_{r}=\unicode[STIX]{x1D714}_{p_{i}}/2$ only if $\unicode[STIX]{x1D6FF}B_{\Vert }$ is included (see figure 1 b) and the curvature and $\unicode[STIX]{x1D735}B$ drifts are kept consistent with the Grad–Shafranov equation. In the tokamak case, we observe the same trend (see figure 2 b), however, even with $\unicode[STIX]{x1D6FF}B_{\Vert }$ included, the frequency of the mode is not $\unicode[STIX]{x1D714}_{p_{i}}/2$ for finite $k_{y}$ s since the temperature gradient $a/L_{Ti,e}=7$ is not high enough (for a more detailed discussion about the tokamak case see Aleynikova & Zocco Reference Aleynikova and Zocco2017).

We conclude that local GK KBM simulations are consistent with diamagnetic modification of the iMHD only if magnetic compressibility is taken into account.

Figure 2.

Influence of $\unicode[STIX]{x1D6FF}B_{\Vert }$ , spectra for tokamak from GENE simulations. (a) $a/L_{Ti,e}=3$ . (b) $a/L_{Ti,e}=7$ .

3.3 Scan in radial wavenumber

The most unstable KBM in W7-X geometry has a vanishing radial wavenumber (figure 3 a), which corresponds to $\unicode[STIX]{x1D703}_{0}=0$ in the ballooning representation. This is connected with the fact that $k_{x}$ is a function of the global shear, ${\hat{s}}$ , and W7-X has a very small ${\hat{s}}$ . From this it follows the weak dependence of our results on the specific choice of the simulated tube. We remark that the radial wavenumber of the most unstable KBM in W7-X is in contrast with that of LHD (Ishizawa et al. Reference Ishizawa, Maeyama, Watanabe, Sugama and Nakajima2015), which is finite. The ITG mode also has the peak of the growth rate curve located around $k_{x}\unicode[STIX]{x1D70C}_{s}=0$ (figure 3 b). In calculations shown in this figure, the density gradient and $\unicode[STIX]{x1D6FD}$ have both been set to zero, so that all microinstabilities other than the ITG mode are eliminated. Periodicity in both plots is explained by the fact that $k_{x}$ is connected with a ballooning angle and it should be periodic in a flux tube approach (see, for example, Beer, Cowley & Hammett (Reference Beer, Cowley and Hammett1995) or equation (4) of Dickinson et al. (Reference Dickinson, Roach, Skipp and Wilson2014)). In figure 4, a KBM-normalized eigenfunction is presented for two simulation tubes. Peaking of eigenfunction around non-zero $\unicode[STIX]{x1D703}$ (see figure 4 b) is an interesting feature of the simulation tube which is centred around the outboard midplane of the ‘triangular’ plane. Earlier simulations have consistently shown this feature of the eigenfunction as being very small at the ‘helical edge’ on the outboard side of W7-X, presumably because the local magnetic shear is exceedingly large there (Kleiber & Scott Reference Kleiber and Scott2005; Helander et al. Reference Helander, Beidler, Bird, Drevlak, Feng, Hatzky, Jenko, Kleiber, Proll and Turkin2012).

Figure 3.

Growth rate (purple, $+$ ) and real frequency (green, $\times$ ) dependence on $k_{x}\unicode[STIX]{x1D70C}_{s}$ for (a) KBM instability, $a/L_{Ti,e}=2$ , $k_{y}\unicode[STIX]{x1D70C}_{s}=0.05$ , $\unicode[STIX]{x1D6FD}=2.4\,\%$ , and (b) ITG instability, $a/L_{Ti,e}=2$ , $k_{y}\unicode[STIX]{x1D70C}_{s}=0.8$ , $\unicode[STIX]{x1D6FD}=0.0\,\%$ . Simulation tube is centred around the outboard midplane of the so-called (a) ‘bean’ plane and (b) ‘triangular’ plane.

Figure 4.

KBM instability eigenfunction, normalized, logarithmic scale. From (a) to (b): simulation tube centred around the outboard midplane of the so-called ‘bean’ plane and around the outboard midplane of the ‘triangular’ plane.

3.4 The effect of temperature gradients in W7-X

Equation (2.3) implies that a necessary condition for instability is $\text{Re}(\unicode[STIX]{x1D714})=\unicode[STIX]{x1D714}_{p_{i}}/2$ . In Aleynikova & Zocco (Reference Aleynikova and Zocco2017), it was shown that this is indeed observed in numerical simulations for large temperature gradients for a simple tokamak geometry. We found that the same is true in the stellarator W7-X (see figure 5). This simulation corresponds to a MHD mode in the limit $k_{y}\rightarrow 0$ , since on this plot, we can observe a purely growing mode with a maximum of growth rate at $k_{y}=0$ . Exactly the same type of instability in simple tokamak geometry was discussed in Aleynikova & Zocco (Reference Aleynikova and Zocco2017).

Figure 5.

Spectrum for $a/L_{Ti,e}=7$ . Here, $\unicode[STIX]{x1D6FD}=3\,\%$ and $a/L_{ni,e}=0$ . $\unicode[STIX]{x1D6FE}$ : solid curve ( $+$ ) and $\unicode[STIX]{x1D714}_{r}$ : crosses ( $\times$ ). The blue line represents the $\unicode[STIX]{x1D714}_{pi}/2$ frequency.

Figure 6.

Spectra for (a) $a/L_{Ti,e}=5$ and for (b) $a/L_{Ti,e}=3$ . Here, $\unicode[STIX]{x1D6FD}=3\,\%$ and $a/L_{ni,e}=0$ . $\unicode[STIX]{x1D6FE}$ : solid curve ( $+$ ) and $\unicode[STIX]{x1D714}_{r}$ : crosses ( $\times$ ). The blue line represents the $\unicode[STIX]{x1D714}_{pi}/2$ frequency, the orange one represents the $\unicode[STIX]{x1D714}_{pi}$ frequency.

Figure 7.

Dependence on $\unicode[STIX]{x1D6FD}$ for the fastest growing mode with $a/L_{Ti,e}=2$ and $k_{y}\unicode[STIX]{x1D70C}_{s}$ in the range from 0.05 to 0.8, in two different W7-X configurations (KJM and TEH) and two simulation tubes (centred around the outboard midplane of the ‘bean’ and ‘triangular’ plane). (a) Growth rate of the instabilities. (b) Real frequency of the corresponding modes. Large dots specify where the magnetic equilibrium was varied.

In W7-X, we find that KBMs have zero frequency at $k_{y}=0$ for a range of temperature gradients (see figure 6). Another important aspect is that, despite gradients being so small that $\unicode[STIX]{x1D714}_{r}$ deviates substantially from $\unicode[STIX]{x1D714}_{pi}/2$ , the growth rate of the mode peaks at $k_{y}=0$ , as opposed to the tokamak, in which an increase in frequency is accompanied by a peaking of the growth rate at finite wavelengths. However, this behaviour of KBM in the stellarator might be far from general, since we cannot claim that we have investigated all possible configurations and parameters.

W7-X was optimized in such away that most magnetic configurations have a high threshold for ideal-MHD instability (Nührenberg Reference Nührenberg2016). It is, therefore, important to establish whether kinetic effects can destabilize ballooning modes for $\unicode[STIX]{x1D6FD}$ s lower than the ideal ones. In the next subsection, we investigate this question.

3.5 KBM and iMHD thresholds in W7-X

Results of GENE simulations for two W7-X configurations are shown in figure 7. One of these configurations (TEH, purple color, $+$ ) is atypical (non-optimised) in the sense that it has no magnetic well and is, therefore, MHD- unstable for very low $\unicode[STIX]{x1D6FD}$ , whereas the other configuration (KJM, green, $\times$ ) is a typical one and is MHD stable up to $\langle \unicode[STIX]{x1D6FD}\rangle =5\,\%$ (where $\langle \unicode[STIX]{x1D6FD}\rangle$ is a plasma volume average $\unicode[STIX]{x1D6FD}$ , not to be confused with local $\unicode[STIX]{x1D6FD}$ which is used in the present work). Growth rates (a) and real frequencies (b) are displayed as functions of $\unicode[STIX]{x1D6FD}$ . For these calculations, we used zero density gradient, $a/L_{Ti,e}=2$ , and $k_{y}\unicode[STIX]{x1D70C}_{s}$ was varied from 0.05 to 0.8 to identify the maxima of growth rates. The TEH configuration is characterised by a small rotational transform, low iota ${<}5/6$ and very large mirror ratio $=$  24 %, whereas the KJM configuration corresponds to a more typical case with mirror ratio $=$  10 % and standard iota $=$  1. For the sake of completeness, it is important to note that, although the magnetic well of the configuration is taken as the figure of merit for MHD stability, there are experimental results that show stable configurations with a magnetic hill (de Aguilera et al. Reference de Aguilera, Castejón, Ascasíbar, Blanco, De la Cal, Hidalgo, Liu, López-Fraguas, Medina and Ochando2015). We see that the ‘diamagnetically-modified iMHD’ ballooning mode description (based on our $\unicode[STIX]{x1D714}_{r}$ condition, blue line figure 7 b) is valid for all KBMs presented on this plot. The extrapolation of the growth rate to where the KBM branch of the curves in figure 7 would intersect the horizontal axis, $\unicode[STIX]{x1D6FE}=0$ , suggests that the critical beta of KBM destabilization in the MHD non-optimised configuration is $\unicode[STIX]{x1D6FD}\sim 0.65\,\%$ while, for MHD-optimised ones, it is $\unicode[STIX]{x1D6FD}\sim 1.9\,\%$ (‘bean’ tube) and $\unicode[STIX]{x1D6FD}\sim 2.2\,\%$ (‘triangular’ tube). We conclude that the MHD-optimized configuration has a higher threshold for KBMs. This result confirms the expectation that KBM stability should be related to ideal-MHD stability. A device like W7-X that has been optimised to be MHD stable up to very high $\unicode[STIX]{x1D6FD}$ should, therefore, also benefit from enhanced KBM stability. However, in previous simulations, it has typically been seen that the KBM stability threshold lies somewhat below that of iMHD ballooning modes. In figure 8, we show KBM critical $\unicode[STIX]{x1D6FD}$ values (for KJM and TEH configurations) corresponding to the point of marginal KBM stability: for $k_{y}\unicode[STIX]{x1D70C}_{s}$ from 0.005 to 0.4 and a range of $\unicode[STIX]{x1D6FD}$ values, the linear growth rates are obtained and KBM curves are extrapolated to a subdominant region where KBMs are marginally stable. Reference ideal MHD $\unicode[STIX]{x1D6FD}_{crit}$ values (note, $\unicode[STIX]{x1D6FD}_{crit}$ is a standard local one) for both configurations are obtained by using a different code (Nührenberg et al. Reference Nührenberg, Bondeson, Zille, Sindoni and Troyon1988), which solves the iMHD ballooning equation along the field line. Note that the kinetic threshold is determined at the specific $k_{y}$ where the KBM is destabilized first, which, in the TEH case, is close to $k_{y}\approx 0$ , as it should in the iMHD limit. In the TEH case, a higher stability limit of KBMs (as compared with the iMHD limit), comes from kinetic effects, for example, diamagnetic (see (2.3)). However, in the KJM case, we see a KBM threshold significantly lower than the corresponding iMHD limit. This might be connected with further kinetic effects, which can provide additional destabilization close to marginality. One can convince oneself that kinetic effects are important even if $k_{y}$ goes to zero, since the driving terms of the right-hand side of equation (3.40) of Tang et al. (Reference Tang, Connor and Hastie1980) and equation (11) of Aleynikova & Zocco (Reference Aleynikova and Zocco2017) scale similarly with $k_{y}\unicode[STIX]{x1D70C}_{s}$ . Kinetic correction to iMHD then survives in the limit $k_{y}\unicode[STIX]{x1D70C}_{s}\rightarrow 0$ . When one extrapolates to marginality (as is done for figure 8), it is expected that high-order corrections in the magnetic drift can compete with the main drive, even for $k_{y}\rightarrow 0$ , as was discussed by Tang, Connor and Hastie and Aleynikova and Zocco in their works, and result in an instability below the iMHD threshold. One could also perhaps attribute it to the resonant ion-transit destabilization (for more details, see Tsai & Chen Reference Tsai and Chen1993).

Figure 8.

KBM critical $\unicode[STIX]{x1D6FD}$ values corresponding to the point of marginal KBM stability (see the text). For reference: the MHD estimates obtained by an ideal ballooning code are shown as lines.

Thus, we conclude that, even though W7-X has been optimised to be MHD-stable up to very high $\unicode[STIX]{x1D6FD}$ and indeed the MHD-optimized configuration has a relatively high threshold for KBMs, kinetic effects are important and we may expect KBM instability before we reach the iMHD ballooning threshold. In figure 7, the growth rate of ITG instability decreases moderately with $\unicode[STIX]{x1D6FD}$ and the ITG mode is subdominant for $\unicode[STIX]{x1D6FD}>2.4\,\%$ in the MHD-optimized case (KJM) and for $\unicode[STIX]{x1D6FD}>1.2\,\%$ in the MHD-non-optimized one (TEH). However, we do not observe regions where instability is completely suppressed between ITG and KBM. A TEM window is not observed between KBMs and ITGs either (in contrast to the usual picture of the tokamak case, figure 1 of Pueschel et al. Reference Pueschel, Kammerer and Jenko2008) and figure 9 of this article). This might be connected with, first, more unstable ITGs (less stabilized by $\unicode[STIX]{x1D6FD}$ ) and, second – TEMs in W7-X are more stable given the fact that most trapped electrons have favourable bounce-averaged curvature (Proll et al. Reference Proll, Helander, Connor and Plunk2012). The same was observed in Ishizawa et al. (Reference Ishizawa, Maeyama, Watanabe, Sugama and Nakajima2015), where the authors reported that the TEM is not observed in the same type of calculations for the standard LHD case. This is a major difference between electromagnetic microinstabilities in stellarators and tokamaks.

Figure 9.

Local flux-tube GENE simulations of the growth rate dependence on plasma $\unicode[STIX]{x1D6FD}$ for different $a/L_{Ti,e}$ , standard cyclone base case parameters, collisionless regime, ${\hat{s}}-\unicode[STIX]{x1D6FC}$ geometry.

Figure 10.

Growth rates as a function of $\unicode[STIX]{x1D6FD}$ and $k_{y}\unicode[STIX]{x1D70C}_{s}$ (colorcoded), when pressure gradients are consistent with the plasma equilibrium. (a) Simulations for standard tokamak case. (b) W7-X standard configuration (KJM, ‘bean’ tube).

We further proceed with a comparison of the W7-X results with a generic tokamak case, emphasising the consistency of the local pressure gradient and the plasma equilibrium. In figure 10, we show the growth rates obtained from the gyrokinetic calculations performed by GENE code as functions of $\unicode[STIX]{x1D6FD}$ and $k_{y}$ . For these calculations, we include $\unicode[STIX]{x1D6FF}B_{\Vert }$ effects (as we do in all our calculations) and take pressure gradients in GENE to be consistent with the plasma equilibrium obtained by the VMEC code (Hirshman et al. Reference Hirshman, Merkel and van Rij1986). We stress the importance of this type of consistency because the difference with results obtained in a non-consistent way (see figure 11) is evident. In figure 10, we can observe different types of electromagnetic microinstabilities. For the tokamak standard case (figure 10 a), the region with $\unicode[STIX]{x1D6FD}<1.7\,\%$ and $k_{y}\unicode[STIX]{x1D70C}_{s}<0.7$ shows electromagnetic ITG modes, while the region with $\unicode[STIX]{x1D6FD}>2.3\,\%$ and $0.1<k_{y}\unicode[STIX]{x1D70C}_{s}<0.4$ contains KBMs. TEMs can be observed in the region with $k_{y}\unicode[STIX]{x1D70C}_{s}>0.8$ . For the W7-X standard case (figure 10 b), ITG instability exists in a wider region of parameters, and complete electromagnetic stabilization of this instability does not appear. KBMs are more stable than in the tokamak case and the most unstable ones are those with the smallest wavenumbers. TEMs are present for high $k_{y}$ , but cannot be observed in the region between ITG and KBM (as has been previously discussed). We conclude this W7-X configuration appears to be somewhat more stable than the tokamak with respect to low $k_{y}$ modes, including KBMs.

As already mentioned, simulations with consistent plasma equilibria are quite different from those without. In figure 11, we show the result of simulations in a fixed equilibrium (zero $\unicode[STIX]{x1D6FD}$ ) that does not vary with $\unicode[STIX]{x1D6FD}$ and observe that growth rates for ITG and TEM for small values of $\unicode[STIX]{x1D6FD}$ are approximately the same (in comparison with figure 10), but KBMs are significantly more unstable at higher values of $\unicode[STIX]{x1D6FD}$ , both in the tokamak and in W7-X. However, in the stellarator case, they share the same feature of having the most unstable mode around $k_{y}\unicode[STIX]{x1D70C}_{s}=0$ . The most interesting point here is that the TEM for the tokamak case seems to not be affected by $\unicode[STIX]{x1D6FD}$ if the equilibrium is kept constant, but if it is varied consistently with the pressure gradients (figure 10), the TEM is stabilized by $\unicode[STIX]{x1D6FD}$ as originally predicted by Rosenbluth & Sloan (Reference Rosenbluth and Sloan1971). For our W7-X calculations, the TEM is moderately stabilized by $\unicode[STIX]{x1D6FD}$ in both cases, both with consistent and non-consistent equilibria.

Figure 11.

Growth rates as a function of $\unicode[STIX]{x1D6FD}$ and $k_{y}\unicode[STIX]{x1D70C}_{s}$ (colorcoded), when pressure gradients are not (!) consistent with the plasma equilibrium. (a) Simulations for standard tokamak case. (b) W7-X standard configuration (KJM, ‘bean’ tube).