3.1. Neoclassical radial electric field

In order to obtain a radial profile of $v_{E\times B}$, we compare neoclassical simulations and DR measurements of the background radial electric field, $E_r$. In stellarators, it is predominantly determined by the ambipolar neoclassical (known as NC) transport (Helander et al. Reference Helander, Beidler, Bird, Drevlak, Feng, Hatzky, Jenko, Kleiber, Proll and Turkin2012). The monoenergetic diffusion coefficients for electrons and ions scale differently with the collision frequency, $\nu$, and have different magnitudes. The ions usually are in the $\sqrt {\nu }$-regime and the electrons in the $1/\nu$-regime (Helander et al. Reference Helander, Beidler, Bird, Drevlak, Feng, Hatzky, Jenko, Kleiber, Proll and Turkin2012). In order to maintain ambipolarity of the particle fluxes, $\varGamma _i = \varGamma _e$, a self-consistent radial electric field is established (Mynick & Hitchon Reference Mynick and Hitchon1983; Yokoyama et al. Reference Yokoyama, Maaßberg, Beidler, Tribaldos, Ida, Estrada, Castejon, Fujisawa, Minami and Shimozuma2007). For equal ion and electron temperature, the ambipolar electric field is usually negative and compensating for a larger ion particle transport. This situation is called ion root. Due to the strong temperature dependence of the diffusion coefficient in the $1/\nu$-regime, $D_e\propto T_e^{7/2}$, the electron diffusion increases strongly for local electron heating ($T_e>T_i$) and a second stable solution with a positive $E_r$ arises, which is called electron root (Yokoyama et al. Reference Yokoyama, Maaßberg, Beidler, Tribaldos, Ida, Estrada, Castejon, Fujisawa, Minami and Shimozuma2007; Helander et al. Reference Helander, Beidler, Bird, Drevlak, Feng, Hatzky, Jenko, Kleiber, Proll and Turkin2012).

Measurements by CXRS and XICS show the existence of an electron root in the core and an ion root in the outer half of ECR heated plasmas in W7-X (Pablant et al. Reference Pablant, Langenberg, Alonso, Beidler, Bitter, Bozhenkov, Burhenn, Beurskens, Delgado-Aparicio and Dinklage2018; Ford et al. Reference Ford, Vanó, Alonso, Baldzuhn, Beurskens, Biedermann, Bozhenkov, Fuchert, Geiger and Hartmann2020). They have a generally good agreement with numerical neoclassical calculations by codes like the drift kinetic equation solver, DKES (Hirshman et al. Reference Hirshman, Shaing, van Rij, Beasley and Crume1986; Turkin et al. Reference Turkin, Beidler, Maaßberg, Murakami, Tribaldos and Wakasa2011). The DR measurements also show a good agreement between the radial electric field obtained from the poloidal velocity of fluctuations, $E_r^\mathrm {DR}=u_y B$, and neoclassical calculations by DKES (Windisch et al. Reference Windisch, Smith, Alcusón, Lechte, Beurskens, Bozhenkov, Carralero, Damm, Estrada and Fuchert2019; Carralero et al. Reference Carralero, Estrada, Windisch, Velasco, Alonso, Beurskens, Bozhenkov, Damm, Fuchert and Gao2020). Figure 5 shows the neoclassical electric field calculated for the profiles of the reference plasma, given in figure 2. At the plasma edge, ion and electron temperatures are set equal where the ion temperature exceeds the electron temperature (since there is no ion heating) and smoothing is applied to avoid a discontinuity in the gradient. The dashed line is the electron-root solution, which only appears in the core, where $T_e \gg T_i$. The ion root, plotted as a solid line, is relevant in the outer half of the plasma, where it forms an $E_r$-well. Sensitivity studies have previously produced an uncertainty of the simulated $E_r^\mathrm {NC}$ of the order of 10 %, by varying the local density and temperature values and gradients by 10 % (Carralero et al. Reference Carralero, Estrada, Windisch, Velasco, Alonso, Beurskens, Bozhenkov, Damm, Fuchert and Gao2020; Estrada et al. Reference Estrada, Carralero, Windisch, Sánchez, García-Regaña, Martínez-Fernández, de la Peña, Velasco, Alonso and Beurskens2021). The black error bars are measurement points by DR, which agree well with the ion-root solution by DKES in the region of interest (shadowed region around 75 % of the minor radius, where the drive of ITG turbulence should be large). At the last closed flux surface (known as LCFS), the measured $E_r$ changes sign, resulting in a large shear. The corresponding $\boldsymbol {E\times B}$ velocity of the $E_r$-well in the region of interest is $v_{E\times B}\approx 4\ \mathrm {km}\,\mathrm {s}^{-1}$, which is the same order of magnitude as the phase velocity measured in the PCI wavenumber–frequency spectra. The good agreement of the DR measurement and the simulated $E_r$ supports the assumption that the propagation velocity of fluctuations is dominated by the $\boldsymbol {E\times B}$ rotation, $u_y \approx v_{E\times B}$.

Figure 5.

Flux-surface averaged neoclassical radial electric field by DKES (blue) and measurements by the Doppler reflectometer (black) for the reference discharge.

3.2. Comparison of velocities

For a direct comparison of the dominant phase velocity in the PCI wavenumber–frequency spectrum, $v_{\mathrm {ph}}^\mathrm {PCI}$, with the local, neoclassical $\boldsymbol {E\times B}$ velocity, geometrical effects along the PCI line of sight have to be taken into account. The PCI measures wavenumbers and velocities along a fixed measurement direction, $\boldsymbol {\hat {e}}_{m}$, which does not strictly correspond to the propagation direction of turbulent fluctuations. The latter is assumed to be perpendicular to the magnetic field direction, $\boldsymbol {\hat {b}} = \boldsymbol {B} / B$, and the flux surface normal, $\boldsymbol {\hat {n}}_s = \boldsymbol {\nabla } s / |\boldsymbol {\nabla } s|$

(3.2)\begin{equation} \boldsymbol{u}_y = u_y \boldsymbol{\hat{n}}_s \times \boldsymbol{\hat{b}}, \end{equation}

where $s$ is the normalised toroidal flux. The phase velocity measured by PCI is a projection of $\boldsymbol {u}_y$ onto the measurement direction,

(3.3)\begin{equation} v_{\mathrm{ph}}^\mathrm{PCI} = \boldsymbol{u}_y\boldsymbol{\cdot}\boldsymbol{\hat{e}}_{m}. \end{equation}

This projection changes along the line of sight, since both the angle between the magnetic field and the measurement direction as well as the angle between the laser beam and the flux surface normal changes. Measured phase velocities therefore correspond to different $u_y$ depending on where the fluctuations occur along the line of sight. The local neoclassical electric field is calculated by multiplying the flux surface averaged value of the radial electric field given by DKES, $\langle E_r \rangle ^{\textrm {NC}}$, with $|\boldsymbol {\nabla } r_{\mathrm {eff}}|$, accounting for the local flux compression. The magnitude of the corresponding $\boldsymbol {E\times B}$ velocity is given by

(3.4)\begin{equation} v_{E\times B} = \frac{\langle E_r \rangle^{\textrm{NC}} |\boldsymbol{\nabla} r_{\mathrm{eff}}|}{B}. \end{equation}

Besides the profile of $\langle E_r \rangle ^{\textrm {NC}}$ that was shown in figure 5, the magnetic field magnitude, $B$, as well as $|\boldsymbol {\nabla } r_{\mathrm {eff}}|$ vary along the PCI line of sight. The geometrical effects described above depend on the exact position of the line of sight and therefore vary slightly across the detector channels. The 32 channels of the detector correspond to 32 parallel lines of sight distributed along $\boldsymbol {\hat {e}}_{m}$ within the width of the laser beam. Since the intensity of the laser across its width has a Gaussian profile, not all channels contribute equally to the wavenumber–frequency spectrum. To illustrate this effect, the local $v_{E\times B}$ and $v_{\mathrm {ph}}$ were calculated for all channel lines of sight and opacity weighted by the respective laser amplitude.

Figure 6 shows the direct comparison of the local $\boldsymbol {E\times B}$ velocity from neoclassical calculations and the local binormal velocity, which corresponds to the measured dominant phase velocity in the PCI wavenumber–frequency spectrum along the PCI line of sight. The $v_{E\times B}$ in the ion diamagnetic direction stems from the electron-root solution for $E_r$, the $v_{E\times B}$ in the electron diamagnetic direction from the ion-root solution. Since it has been shown experimentally that an electron-root radial electric field exists in the core, the corresponding electron-root velocity should be preferred where both solutions exist. A typical phase velocity of the ITG modes, $v_{\mathrm {ph}}=500\ \mathrm {m}\,\mathrm {s}^{-1}$ in the ion diamagnetic direction, which is later confirmed by linear GENE simulations, is added to $v_{E\times B}$ at $r_{\mathrm {eff}}/a=0.75$ for reference. The negative phase velocity in the PCI wavenumber–frequency spectrum was used for the outboard side mapping and the positive phase velocity for the inboard side (solid lines). The dotted lines correspond to the inverse attribution for the central line of sight. The difference between the 32 lines of sight is very small on the inboard side but relevant on the outboard side. The velocities match best at $r_{\mathrm {eff}}/a\approx 0.75$, which is also the position of the well in $\langle E_r \rangle ^{\textrm {NC}}$. With the inverse attribution, the phase velocity measured by PCI would agree with the neoclassical results only quite close to the core, near the electron-root solution. Since we are looking for the region of strongest fluctuations, we concentrate on the attribution of wavenumber branches to the inboard and outboard side for which the velocities match in the outer half of the plasma, where we expect a larger drive for ion scale turbulence. The turbulent contribution to $u_y$ is just a small correction and of the same order as the geometrical effects of the finite beam width and the uncertainty of $v_{E\times B}$ due to the uncertainty of the input profile parameters ($\approx 10\,\%$, not shown here). The observed match suggests that strong fluctuations exist at $r_{\mathrm {eff}}/a\approx 0.75$ which cause a dominant linear phase velocity in the PCI wavenumber–frequency spectrum determined by $v_{E\times B}$ at that position.

Figure 6.

Local $\boldsymbol {E\times B}$ rotation velocity from neoclassical calculations and the binormal velocity corresponding to the dominant phase velocity in PCI the spectrum along the line of sight of PCI. A linear estimate of the phase velocity of ITG modes at $r_{\mathrm {eff}}/a=0.75$ (see § 4.1) is added to $v_{E\times B}$ for reference. The dotted lines show the alternative attribution of wavenumber branch to inboard/outboard side for the central line of sight. The faded black and blue lines illustrate the geometric variation of the mapping across the 32 lines of sight weighted by the respective laser power, as described in the text.

A simple test for the hypothesis that $v_{\mathrm {ph}}^\mathrm {PCI}$ in each wavenumber branch is determined by $v_{E\times B}$, is an experiment with reversed magnetic field direction but otherwise similar plasma parameters. The $\boldsymbol {E\times B}$ rotation changes direction while everything else stays the same, if the discharges are similar enough. We compare two almost identical experiments in a magnetic configuration with lower rotational transform. In both discharges, $5\ \mathrm {MW}$ of ECRH were applied at a line-integrated density of $4\times 10^{19}\ \mathrm {m}^{-2}$. Figure 7 shows the wavenumber–frequency spectra and the simulated neoclassical radial electric field of the respective discharges. The radial electric field profiles are very similar. Since $v_{E\times B}$ changes direction, the spectrum is mirrored at the $k=0$ axis when going from forward to reversed field, but looks otherwise identical. The measured dominant phase velocities for the negative (positive) wavenumber branch are $v_{\mathrm {ph}}^\mathrm {PCI} = -2.3\,(2.9) \ \mathrm {km}\,\mathrm {s}^{-1}$ for the forward field case and $v_{\mathrm {ph}}^\mathrm {PCI} = -2.9\,(2.3) \ \mathrm {km}\,\mathrm {s}^{-1}$ for the reversed field case. The asymmetry in the spectral power also switches sides. The velocity comparison yields a similarly good match as in the reference plasma for both cases (see Appendix A), only if the attribution of each $k$-branch to the inboard or outboard side is reversed. There is no electron-root solution in this case, which excludes an inverse attribution. The match is best at the location of the $E_r$-well, which is not at $r_{\mathrm {eff}}/a=0.75$ but farther inwards. This poses the question, whether the location of strongest fluctuations is different to the reference case and which role the $E_r$-well itself plays for the dominant phase velocity in PCI spectra.

Figure 7.

(a) Wavenumber–frequency spectrum and dominant phase velocity of discharge with forward and reversed field direction. The spectra are normalised to the mean along the $k$-axis for a better visualisation of the dominant phase velocity. (b) The DKES $E_r$ for the respective discharges.

We repeated the analysis for discharges with varying input power or density. Both variations have an effect on the radial location and the depth of the well in the ion root $E_r$. For the scan in ECRH power, three time windows within the discharge presented in figure 1 (shaded regions) were analysed, where the ECRH power was stepped down from $4\ \mathrm {MW}$ over $3.5\ \mathrm {MW}$ to $2\ \mathrm {MW}$, while keeping the line-integrated density constant at $4\times 10^{19}\ \mathrm {m}^{-2}$. For the scan of plasma density, three discharges with heating power $P_{\textrm {ECRH}}=4.7\ \mathrm {MW}$ were chosen. The line-integrated density was increased between the discharges from $4\times 10^{19}\ \mathrm {m}^{-2}$ over $6\times 10^{19}\ \mathrm {m}^{-2}$ to $8\times 10^{19}\ \mathrm {m}^{-2}$. Details of the velocity comparison for each case can be found in Appendix A. In all situations, the measured dominant phase velocity follows the trend of increasing or decreasing $E_r$-well depth. The match is usually best around the position of the $E_r$-well, which also varies but only within the outer half of the plasma. In many cases, the electron-root solution is very small or does not exist at all, which excludes the possibility of a significant contribution of core density fluctuations to the dominant phase velocity. We conclude that strong density fluctuations must be present in the outer half of the plasma and that the well in the $v_{E\times B}$-profile, which appears in the same radial region, determines the observed dominant phase velocity in PCI wavenumber–frequency spectra.

While there must be strong fluctuations at the position of the $E_r$-well, the form of the $v_{E\times B}$-profile itself must have an impact on the observed wavenumber–frequency spectrum as well. The spectral power originating from fluctuations at a radial location with a strong radial velocity gradient will be spread out in $(kf)$-space. If the fluctuations arise at a radial location with small velocity gradient, the spectral power is focused along a single angle in the wavenumber–frequency spectrum. Fluctuations will produce clearer dominant phase velocities, if they appear in a region in which the velocity profile is flat. While this does not disagree with the interpretation of the strongest density fluctuations causing the dominant phase velocity, both mechanisms have to be taken into account to understand the PCI spectra. In the following subsection, we examine the effect of the velocity gradient on PCI spectra with a synthetic PCI diagnostic (Coda Reference Coda1997; Lin et al. Reference Lin, Porkolab, Edlund, Rost, Fiore, Greenwald, Lin, Mikkelsen, Tsujii and Wukitch2009; Rost, Lin & Porkolab Reference Rost, Lin and Porkolab2010; Tsujii et al. Reference Tsujii, Porkolab, Bonoli, Edlund, Ennever, Lin, Wright, Wukitch, Jaeger and Green2015).

3.3. Synthetic PCI diagnostic

A synthetic PCI diagnostic is a useful tool for investigating simplified situations such as different radial gradients of the background velocity field and for comparing simulated three-dimensional density fluctuations with measurement data. It reproduces the phase contrast imaging signal numerically by integrating the $\tilde {n}_e$ profile along the PCI line of sight based on the model found in chapter 2 of Coda (Reference Coda1997). This model includes known instrument functions that affect the measurement. The details of the synthetic diagnostic will be presented elsewhere. The primary application in this work is investigating the effect of a radial velocity gradient on the dominant phase velocity in PCI wavenumber–frequency spectra. We implement a simplified model which captures the basic features of the $v_{E\times B}$-profile around the PCI line of sight without the need for a fully self-consistent solution of the problem, since global nonlinear gyrokinetic simulations including the effects of a radial electric field have not been performed at the time of writing. A model density fluctuation background, which is derived from a simulation for W7-X with GENE-3D (Bañón Navarro et al. Reference Bañón Navarro, Merlo, Plunk, Xanthopoulos, von Stechow, Di Siena, Maurer, Hindenlang, Wilms and Jenko2020), is rotated along the poloidal angle according to a predefined rotation velocity. The model is described in more detail in Appendix B. It is important to point out that the investigated effect is different from turbulence suppression by $\boldsymbol {E\times B}$ shear (Burrell Reference Burrell1997). Neither the linear stabilisation of turbulent modes nor the nonlinear decorrelation through $\boldsymbol {E\times B}$ shear can be simulated with the simplified model of the $v_{E\times B}$ rotation. However, a shift of spectral power to higher wavenumbers is possible in the present model through shearing of structures spanning several flux surfaces. In order to test the effect of a velocity gradient, fluctuations were restricted to a narrow radial interval by a top-hat function of width $0.2\,a$ centred around $r_{\mathrm {eff}}/a=0.7$. The velocity profiles have the same value at the centre with a constant gradient over the interval as shown in Appendix B. In principle, one would expect the resulting wavenumber–frequency spectra to show the same dominant phase velocity for each case with a varying concentration of spectral power. Figure 8 shows the wavenumber–frequency spectra for three different gradients of the model velocity profile. The radial velocity gradient increases from left to right. In the left-hand case, the gradient corresponds roughly to a rigid plasma rotation and shows a very distinct dominant phase velocity with little spectral power in other parts of the spectrum. Model profiles with even flatter gradient produced very similar results and are therefore not shown here. For increasing gradients, the spectral power is clearly spread out over a larger area in $(kf)$-space. The gradient of the right-hand case corresponds roughly to the maximum gradient that is observed experimentally at the edge of the plasma in discharges without pellet injection, NBI or impurity injection. In order to measure this effect quantitatively, we compare the integrated spectral power in a ${\pm }50\ \mathrm {kHz}$ band around the dominant phase velocity, $s_v$, with the total power in the spectrum, $s_\mathrm {total}$. The velocity band is marked by white dashes in figure 8. In the left-hand case, 68 % of the power is concentrated along the negative $v_{\mathrm {ph}}$ and 31 % along the positive $v_{\mathrm {ph}}$. Only 1 % of spectral power appears outside the velocity bands. For the most severe gradient, the right-hand case, these fractions go down to 50 % and 8 % for the negative and positive velocity band, respectively. Additionally, the dominant phase velocity increases for the cases with higher gradient, even though the average velocity over the top-hat function stays the same. This is because the maximum velocity on the domain of the $\tilde {n}_e$ step function increases. Due to the nonlinear relation between phase velocity and the slope in the wavenumber–frequency spectrum, more power is concentrated at a larger slope. In other words, the dominant phase velocity is prone to show the higher end of phase velocities in the fluctuations. These results have important consequences: if the velocity profile has a well, as it does in the experiment, the concentration of spectral power at the slope of the velocity in the well is not only supported by the low gradient at that position but also by the fact that fluctuations with a high velocity appear on the same small area in the $(kf)$-space.

Figure 8.

Wavenumber–frequency spectra of synthetic PCI measurements corresponding to model velocity profiles with increasing radial gradient from (a–c). The colour scale is in dB. The white lines mark the dominant linear phase velocities, the white dashes mark a ${\pm }50\ \textrm {kHz}$ interval around the dominant $v_{\mathrm {ph}}$.

Finally, for a direct comparison with the experimental wavenumber–frequency spectrum, the simulated neoclassical $v_{E\times B}$-profile, taken from figure 5, and a simulated fluctuation amplitude profile, taken from Bañón Navarro et al. (Reference Bañón Navarro, Merlo, Plunk, Xanthopoulos, von Stechow, Di Siena, Maurer, Hindenlang, Wilms and Jenko2020), are used as input for the model. Figure 9 shows the resulting wavenumber–frequency spectrum, which has very similar features as the experimental spectrum (figure 4). The dominant phase velocity in the spectrum matches $v_{E\times B}$ at the position of the well on both the inboard and outboard side, which confirms that the analysis is consistent. A more detailed or even quantitative comparison of experimental and simulated spectra is not reasonable, since the simulation in (Bañón Navarro et al. Reference Bañón Navarro, Merlo, Plunk, Xanthopoulos, von Stechow, Di Siena, Maurer, Hindenlang, Wilms and Jenko2020) does not closely match the experimental situation of the reference discharge. Nonlinear global simulations with GENE-3D including the background radial electric field are planned, which can then be directly compared with measurement data via the synthetic diagnostic.

Figure 9.

Wavenumber–frequency spectrum of synthetic PCI measurement with the simplified model for the $\boldsymbol {E\times B}$ rotation. Density fluctuations from global gyrokinetic simulation (Bañón Navarro et al. Reference Bañón Navarro, Merlo, Plunk, Xanthopoulos, von Stechow, Di Siena, Maurer, Hindenlang, Wilms and Jenko2020) are rotated according to the simulated neoclassical $v_{E\times B}$-profile in figure 5.

4.1. Linear gyrokinetic flux-tube simulations

Linear gyrokinetic simulations are used to study the linear drive and dominant mode type in order to identify how turbulence reacts to changes in the density and temperature profiles. We first investigate the dependence of the fundamental modes on the normalised ion-temperature-gradient, $a/L_{T_{i}}$, and the temperature ratio, $\tau =T_i/T_e$, for W7-X with a scan over those parameters, similar to Zocco et al. (Reference Zocco, Xanthopoulos, Doerk, Connor and Helander2018). It has been shown that ITG modes in W7-X are most unstable in a narrow toroidal band located at the outboard midplane of the bean shape cross-section, where the curvature is the most unfavourable (Xanthopoulos et al. Reference Xanthopoulos, Mynick, Helander, Turkin, Plunk, Jenko, Görler, Told, Bird and Proll2014). A flux-tube at a minor radius of $r_{\mathrm {eff}}/a=0.75$ passing through that region was chosen for the collisionless simulations with kinetic electrons and without electromagnetic effects. The density gradient was chosen according to the value of the experimental profile at this radius ($a/L_n=1.5$, see figure 2) and a resolution of $(n_x,n_y,n_z,n_{v_\parallel },n_\mu )=(31,1,128,64,8)$. By including kinetic electrons and a finite density gradient, TEMs are simulated as well and the situation is closer to the experiment. The ETG modes are expected to be unstable for the given experimental electron-temperature-gradient. However, they grow at smaller scales than TEMs and ITG modes and are likely not detected by PCI. They are, consequently, not relevant for the interpretation of the PCI density fluctuation measurements. Additionally, it has been shown that at the chosen radius, where $\tau =1$ in the experiment, the nonlinear contribution of ETG to turbulent transport is negligible compared with ITG and TEM, since the inherently small parallel connection lengths in modular stellarators prevent ETG from forming radially extended streamers (Plunk et al. Reference Plunk, Xanthopoulos, Weir, Bozhenkov, Dinklage, Fuchert, Geiger, Hirsch, Hoefel and Jakubowski2019). We focus on ITG and TEM and neglect ETG modes by assuming $a/L_{T_{e}}=0$. The normalised ion-temperature-gradient length is varied between 1 and 8 for three different temperature ratios. Figure 10 shows the linear growth rates of the most unstable modes along the simulated wavenumbers that include ion and intermediate scales ($k_y\rho _s\lesssim 10$) for the described scan. There is a clear positive scaling of the growth rates with $a/L_{T_{i}}$ and a negative scaling with $\tau$, which is typical for ITG modes. In the reference discharge, $a/L_{T_{i}}=5.4$ at this radial location, which is marked by a vertical line in the figure. The experiment situation is far from classical marginal stability, which appears at $a/L_{T_{i}} \approx 1.5$. Figure 11 shows the growth rates and real frequencies versus the binormal wavenumber, $k_y$, for three temperature ratios and $a/L_{T_{i}}=5.4$. In order to facilitate a direct comparison with the experimental data of PCI, the quantities in figures 10 and 11 are shown in real units, which are obtained from the normalised simulation output with the temperature and density at $r_{\mathrm {eff}}/a=0.75$ according to the experimental profiles (figure 2) and the local magnetic field at that position. The most unstable wavenumber is around $k_y=6\ \mathrm {cm}^{-1}$ for all considered temperature ratios, which is slightly below $k_y\rho _{s}=1$, with the ion drift scale, $\rho _{s}$, as expected for ITG and TEM modes. At this wavenumber range, the real frequencies show a linear scaling with $k_y$. Together with the clear scaling with $a/L_{T_{i}}$, which can also be observed for the real frequencies over most of the wavenumber range, this suggests that the underlying modes are resonant toroidal ITG modes. The TEMs appear to play a minor role, since the presented characteristics are clearly ITG-like. Pure TEMs propagate in the electron diamagnetic direction, which is expressed by negative frequencies in GENE and not seen in figure 11. Additional flux-tube simulations of pure ITG modes ($a/L_n=0$) and pure TEMs ($a/L_{T_{i}}=0$) confirms that the TEM growth rates are much smaller than those of pure ITG or the combined case for ion- and intermediate scales. On the one hand, this can be attributed to the comparably small density gradient, on the other hand, it has been shown that TEMs are inherently suppressed in quasi-isodynamic magnetic configurations such as W7-X (Proll et al. Reference Proll, Helander, Connor and Plunk2012, Reference Proll, Mynick, Xanthopoulos, Lazerson and Faber2016). A moderate increase of $a/L_n$ is likely to have an overall stabilising effect as it suppresses ITG before TEMs become relevant (Alcusón et al. Reference Alcusón, Xanthopoulos, Plunk, Helander, Wilms, Turkin, von Stechow and Grulke2020). We conclude that the turbulence at the chosen location in the reference discharge is determined by toroidal ITG modes, with the observed implication for the effect of $\tau$ and $a/L_{T_{i}}$ on the modes. Even though there is usually a shift in the wavenumber range between linear and nonlinear simulation results, PCI generally seems to cover the right wavenumber range to observe the dominant modes. With the given frequencies at the most unstable wavenumbers, those modes have a phase velocity of $v_{\mathrm {ph}} \approx 500 \ \mathrm {m}\,\mathrm {s}^{-1}$ in the ion diamagnetic direction, which has been used in figure 6 as a reference and confirms that the phase velocity of turbulent modes is much smaller than the neoclassical $v_{E\times B}$.

Figure 10.

Maximum growth rates of simulated wavenumbers for a scan of normalised ion-temperature-gradient lengths and temperature ratios, $\tau =T_i/T_e$. The vertical line marks the experimental $a/L_{T_{i}}$ at $r_{\mathrm {eff}}/a=0.75$.

Figure 11.

Simulated growth rates and real frequencies in the relevant wavenumber range for different temperature ratios at $a/L_{T_{i}}=5.4$.

4.2. Nonlinear gyrokinetic flux-tube simulations

Nonlinear gyrokinetic simulations are the best tool available to investigate the turbulent fluctuations that contribute to transport. Fully three-dimensional nonlinear gyrokinetic simulations of the plasma are still at an early stage of their development and prohibitively expensive. In order to study the radial and poloidal distribution of density fluctuations, we select four different flux-tubes with a resolution of $(n_x,n_y,n_z,n_{v_\parallel },n_\mu )=(128, 64, 128, 32, 9)$ and a box of $(L_x,L_y,L_z)=(125,125, 2{\rm \pi} )$ in normalised GENE units at three radial positions outside the half-radius for nonlinear flux-tube simulations with GENE. Three flux-tubes were chosen such that they intersect the PCI line of sight on the outboard side and simulate the turbulence measured by PCI as close as possible. The fluctuations on the inboard side are also captured by the same flux-tubes, since they pass through the stellarator symmetric PCI cross-section in each of the five identical modules of W7-X and very close to the inboard line of sight position in one of them. Figure 12 shows the flux-tubes at the toroidal angle of PCI. The intersection with the line of sight can be observed on the outboard (right-hand) and inboard (left-hand) side. At $r_{\mathrm {eff}}/a=0.75$, an additional flux-tube was selected, which passes through the outboard midplane in the bean shape cross-section of W7-X (and is hence labelled ‘bean shape’). It provides a comparison to assess how representative the PCI density fluctuation measurements are for the strongest fluctuations of this flux surface. It is poloidally shifted to the PCI flux-tube and does not intersect the PCI line of sight. The density and temperature gradients, as well as the temperature ratio for each radial position are listed in table 1 and represent the profiles shown in figure 2. In the edge plasma, $r_{\mathrm {eff}}/a>0.7$, electron and ion temperatures are equal due to good collisional coupling of the two species. The temperature ratio in this region is therefore assumed to be $\tau \approx 1$. Both species were treated kinetically and without collisions, no electromagnetic effects were included. Figure 13 shows the simulation results for the local density fluctuations along the PCI flux-tube as well as the bean shape flux-tube at the same radial location, $r_{\mathrm {eff}}/a=0.75$, where the brackets, $\langle \cdots \rangle$, denote an average in time and perpendicular directions. The fluctuation amplitudes are very similar, despite the poloidal shift between the two flux-tubes. The poloidal shift means that the same values of $\hat {z}$ correspond to different toroidal positions and the fluctuations might not correlate between the two flux-tubes. The outboard side of the PCI line of sight probes the flux-tube close to the maximum of the mode structure, and the inboard side close to its minimum. The simulated density fluctuations in the flux-tubes at $r_{\mathrm {eff}}/a=0.6$ and $r_{\mathrm {eff}}/a=0.9$ have a rather flat mode structure in the parallel direction and the difference between the PCI inboard and outboard position is negligible. A possible reason for this is the fact that the ITG is either not as strongly driven or suppressed by the increasing density gradient and the band of high fluctuation amplitude (Xanthopoulos et al. Reference Xanthopoulos, Mynick, Helander, Turkin, Plunk, Jenko, Görler, Told, Bird and Proll2014), which is a feature of ITG turbulence in W7-X, is not as pronounced as in the strongly driven case at $r_{\mathrm {eff}}/a=0.75$. The band of high fluctuation amplitude at the outboard midplane of the bean shape is wide enough to cover the PCI flux-tube as well and the PCI measurements are a good representation of the general fluctuation level in the stellarator. The asymmetry between the inboard and outboard side fits the experimental observation in the wavenumber–frequency spectrum.

Figure 12.

Intersections of the GENE flux-tubes with the cross-section of the PCI line of sight (thick black line).

Figure 13.

Local density fluctuations along the two selected flux-tubes at $r_{\mathrm {eff}}/a=0.75$ from nonlinear GENE simulations. The vertical lines mark the intersections with the PCI line of sight for the solid line. Here $\hat {z}=0$ is defined at the outboard midplane and $\hat {z}=\pm {\rm \pi}$ at the inboard midplane.

Table 1.

Input parameters for GENE flux-tube simulations. The values are taken from experimental profiles in figure 2. Outside $r_{\mathrm {eff}}/a = 0.7$ the ion temperature and electron temperature are assumed to be equal.

Figure 14 shows the density fluctuations as well as the ion-temperature-gradient for each flux-tube versus the radial position. The density fluctuations are evaluated at the intersection with the PCI line of sight on the inboard and outboard side, but the respective values only show a clear difference at $r_{\mathrm {eff}}/a=0.75$. Both density fluctuations and ion-temperature-gradient are strongest in the $r_{\mathrm {eff}}/a=0.75$ flux-tube and decrease again towards the separatrix, which shows how closely the nonlinear density fluctuations relate to the linear drive of the ITG. This is the numerical confirmation that the fluctuations are indeed strongest at $r_{\mathrm {eff}}/a=0.75$, as it was already derived from the comparison of the dominant phase velocity in the PCI measurement to the neoclassical $v_{E\times B}$.

Figure 14.

Local density fluctuations from nonlinear GENE flux-tube simulations and ion-temperature-gradient at the corresponding radial position. The density fluctuations are evaluated at the intersection with the PCI line of sight on the inboard and outboard side, respectively.