Parameter dependences of small edge localized modes (ELMs)

The plasma density in tokamaks is usually controlled via gas puffing. Since this dominant particle source is outside the confined plasma, the separatrix density and the plasma density are closely linked [24, 25]. This link can be broken if the plasma is fuelled with pellets, which deposits particles further inside the confined plasma region [26]. In figure 2 the time traces of a discharge is shown in which the plasma follows the already mentioned recipe to achieve small ELMs, namely high shaping and high gas puff, with a configuration very close to double null (figure 1). After 3.0 s the stored energy stays constant around 800 kJ (figure 2(a)). At constant heating power the radiated power P_RAD (figure 2(b), purple) stays constant until 5.7 s, after which the reduced ELM frequency leads to tungsten accumulation and thus an increase in P_RAD with a concomitant loss of core density. As ELM monitor we show the divertor current (figure 2(c)). While during the small ELM phase between 3.0 s and 4.0 s the plasma is fuelled only by a high gas puff (figure 2(d), dark blue), the phase between 4.0 s and 5.0 s is fuelled with pellets and a reduced gas puff and finally the phase from 5.0 s to 5.8 s has no gas puff and is fuelled with pellets only. The pellet flux achieved from 4.0 s to 5.8 s was $\Gamma_{\rm pellet}= 1.1\times10^{22}$ mol s⁻¹ (figure 2(d), light blue, multiplied by a factor of 2 for better visibility). The pellet frequency was chosen to achieve the same core density, therefore in all phases the core and pedestal densities do not change significantly. The ELM characteristics, however, are distinctly different. While in the gas fuelled phase only small ELMs are visible, large ELMs appear again as soon as the gas puff is reduced.

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In figure 3, a comparison of edge (a) electron density, (b) electron temperature, (c) electron pressure and (d) ion temperature is shown for these 3 phases, which are called 'gas' (red), 'gas+pellets' (blue) and 'pellets' (black) and the corresponding time windows are indicated in figure 2 by colored bars. The Thomson scattering data points of electron density and temperature are radially binned to reduce statistical noise. Namely, a radial running average of density and temperature was taken for all points in the specified 200 ms time windows. Every point represents the median of 6 neighbouring points. Additionally, profile fits using the modified tanh (mtanh) method described in [27] are shown in figure 3. To increase visibility, experimental error bars have only been plotted for the 'gas' case. For all profiles inside the separatrix, which is indicated by a solid black line, the data are very similar and stay within their scatter width. The only significant difference can be observed in the electron density around the separatrix and in the scrape-off layer. The gas fuelled case shows a distinct shoulder in the scrape-off layer which is decreasing (from red to black) as we reduced the gas puff and switched the fuelling mechanism.

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The electron and ion temperature profiles do not change significantly in the different phases. The ion temperature profile at the separatrix is approximately 2.5 times higher than the electron temperature. This has also been seen in different high density experiments at AUG [28, 29]

More evidence of the difference in the density profiles at the separatrix can be found when computing the fall-off lengths $\lambda_{\rm n_e}$ and $\lambda_{\rm p_e}$, see table 1, using the method described in [30], taking Thomson scattering data in the near SOL. Comparing the 3 phases, the density fall-off length is reduced by more than a factor of 2, while the fall-off length of the pressure profile only changes by around $20\%$.

Table 1. Fall-off lengths of electron density and electron pressure at the separatrix.

Time (s)	Dominant ELM type	$\lambda_{\rm n_e}$ (mm)	$\lambda_{\rm p_e}$ (mm)
3.5	Small	45	12
4.5	Mixed	35	11
5.5	Large	20	9

Figures 4(a)–(c) shows the outer divertor current together with the stored energy (figures 4(d)–(f)) for the three different phases indicating the respective ELM behaviour. Keeping the color code, figure 4(a) shows the purely gas fuelled phase in red. Some larger events with divertor current amplitudes in the range of 10–20 kA are visible with losses in W_MHD of  <6%, but in between the small high frequency ELMs exhibit an amplitude of  ∼5 kA (indicated by two black lines in figure 4(a)) of the divertor current. Type-I ELMs become larger as the gas puff is reduced (figure 4(b)) or completely switched off (figure 4(c)) with losses in W_MHD reaching up to 10% and at the same time the fluctuations between type-I ELMs decrease in amplitude to  ∼2 kA. No strict correlation between the pellet injection time, indicated in light blue, and the appearance of a large type-I ELM can be seen in this discharge. Some ELMs could be directly triggered by a pellet, others are clearly not triggered. This is in agreement with previous findings [31] that at high collisionalities a so-called lag time is observed, i.e. a time after an ELM in which a pellet is not able to trigger an ELM.

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Summarizing, clearly different ELM behaviour is observed for these three cases with different particle fuelling scenarios, in which the plasma shape is kept constant and the pedestal profiles are very similar with the exception of the density around the separatrix and in the scrape-off layer. A high separatrix density leads to strong fluctuations or small ELMs in between some low amplitude large ELMs, whereas the reduction of the separatrix density increases the large ELM amplitude and the small ELM fluctuations decrease in size.

Besides the discharge shown in detail in this section, several experiments comparing pellet and gas fuelling were done at ASDEX Upgrade. The pellet velocity and size was varied while trying to keep a constant pedestal top density, matching the rate of the gas fuelling. The measured data suggests neither a dependence on the pellet size nor the pellet velocity on the ELM behaviour. This non-dependence on pellet parameters was also found for the lag-time mentioned above [31].

The small ELM start-up procedure suggests that closeness to double null is crucial for small ELMs to occur. To further examine this dependence, a second set of experiments was conducted with a constant gas puff, shifting the plasma slowly downwards, away from the double-null configuration.

Figure 5 shows time traces of a discharge exhibiting this downward shift where, following the same procedure as described above, a small ELM phase is achieved with a high gas puff and an upwards shift of the plasma to create a close to double null shape at 3.0 s. W_MHD stays constant at 0.8 MJ (figure 5(a), blue) while the ITER confinement time scaling factor $H_{\rm 98(y, 2)}$ is around 1.1 (figure 5(a), orange). The NBI and ICRF heating power (figure 5(b), red and green) was kept constant (neutral beam sources dropped for short periods with no lasting impact on the plasma parameters 2.2 s and 5.0 s). The outer divertor current shows the ELM behaviour in figure 5(c). The gas puffing rate (figure 5(d) in dark blue) is not changed after 3.0 s and also the plasma density stays constant. The purple time trace in figure 5(d) shows the vertical $(z)$ position of the magnetic axis being shifted downwards from 4.0 s onward. This changes the ELM characteristics with a gradual reappearance of larger ELMs.

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Figure 6 shows the plasma shapes of the two time windows indicated in figure 5. As the magnetic axis is shifted downwards by 1.5 cm the top of the plasma changes significantly while the lower x-point remains in the same position. This shift is also apparent in the plasma elongation, which is reduced by 1.5 cm, from $\kappa= 1.688$ m to $\kappa= 1.673$ m. The aim of the downward shift was to reduce the plasma shape's closeness to a double null configuration. This can best be seen in the difference of the two separatrices $\Delta r_{\rm sep}$ going from 12 mm to 17 mm, representing a shift of the second separatrix from $\rho_{\rm pol}=1.019$ to $\rho_{\rm pol}=1.011$. The downward shift also influences the triangularity. While the upper triangularity is reduced by $\sim$10% from $\delta_{\rm upper} = 0.317$ to 0.285 the lower triangularity stays constant within the margin of error (increases  ∼1% from $\delta_{\rm lower} = 0.423$ to 0.428). In figure 7 the divertor current ((a, b)) and W_MHD signals ((c, d)) are shown for the small ELM case (z  =  7.0 cm in orange) and the mixed case (z  =  5.5 cm in green). In the case with the higher position (orange) the fluctuations or small ELMs dominate the transport in between some irregular large ELMs. Larger type-I ELMs reappear at the lower plasma position (green). The small ELM amplitude also decreases from  ∼5.0 kA to  ∼3.0 kA while the large ELMs grow from  ∼10.0 kA to  ∼20.0 kA.

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A comparison of edge (a) electron density, (b) electron temperature, (c) electron pressure and (d) ion temperature profiles is shown in figure 8 for the two phases representing z positions at 7.0 cm (orange) and 5.5 cm (green) also indicated in figure 5. As the gas fuelling is kept constant, the kinetic profiles do not change at all while the ELM behaviour does change (figure 7). This is in contrast to the profiles shown in figure 3, where a distinct difference around the separatrix density was visible. As above the profiles were fitted using the mtanh function while and the experimental error of the profiles shown in figure 8 is very similar to the one in 3 and has therefore been omitted here for better visibility.

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These two experimental scans show that the separatrix density is a necessary but not sufficient criterion for the existence of small ELMs. While in the first set of experiments (section 2.1), the pedestal top density ($n_{\rm e, ped}=11.5 \times 10^{19}$ m⁻³) and the separatrix density were decoupled by changing from gas fuelling ($n_{\rm e, sep}=4.0 \times 10^{19}$ m⁻³) to pellet fuelling ($n_{\rm e, sep}=2.0 \times 10^{19}$ m⁻³), small ELMs were dominant at high separatrix density, in the second set of experiments (section 2.2) a tiny change in the plasma configuration brought back larger type-I ELMs, without a change in the kinetic edge profiles ($n_{\rm e, sep}$ stayed at $4.0 \times 10^{19}$ m⁻³).

The strong dependence of the small ELMs on the separatrix density suggests, them to be very localized modes near the separatrix. A correlation of the tokamak H-mode density limit with the stability of modes at the separatrix has recently been found at AUG [32]. The ballooning stability of such modes is determined by the pressure gradient as driving term and the magnetic shear as stabilizing term, see $s -\alpha$ diagram in [33, chapter 5.2.2]. As the pressure gradient exhibited no significant change for the second set of experiments (compare profiles in figure 8), the analysis focuses on the evaluation of the magnetic shear.

Figure 9 shows the flux surface averaged magnetic shear $q'/q$ in the relevant pedestal area showing a higher shear for the lower plasma position. The derivative of the safety factor $q'$ was calculated here as $q'=\frac{{\rm d}q}{{\rm d}\rho_{\rm pol}}$. A factor $\frac{{\rm d}\rho_{{\rm pol}}}{{\rm d}R}$ can be be taken into account to see if the effect is due to flux expansion and to compute $\frac{{\rm d}q}{{\rm d}R}$. This factor was found to be lower for the lower shear case and higher for the case with the higher shear and therefore amplified the effect shown in figure 9. The described difference is also visible in the local magnetic shear although more complex and dependent on the poloidal angle. Recently, a local change of ballooning mode stability due to changes (distortion) of the local magnetic shear has also been reported in ASDEX Upgrade experiments with external magnetic perturbations [34].

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In the first set of experiments we show that not the pedestal top but that the separatrix parameters play a role for the appearance of small ELMs. In the second set, however, the kinetic profiles are very similar, so that another parameter must also be important which changes at the slightly lower z-position. To understand the transition from type-I to small ELMs, we first start with the analysis of the pedestal in the frame of ideal peeling–ballooning (PB) modes. The linear ideal stability boundary is calculated using the work-flow described in [35]. Figure 10 shows the operational points of all 5 instances described in the previous section with 15% error bars, again keeping the color code. All operational points lie very close to their respective ideal peeling–ballooning stability boundary. Both regimes with dominantly small ELMs (orange '7.0 cm' and red 'gas fuelling') exhibit not only the highest normalized pressure gradient α, with $\alpha = 5$, but also the highest edge current density $j_{\phi {\rm max}}$. It had already been shown in [36], that the correlation between edge current density and normalized pressure gradient is not easily broken, because the neoclassically driven current density is influenced by two effects in opposite directions: the increased collisionality reduces the drive while the density gradient provides a stronger drive than the temperature gradient. An explanation for smaller ELMs at higher collisionality being due to the effect of a reduced edge current density [16] is therefore not applicable in these presented cases. While it seems counter-intuitive that the small ELM points lie close to the type-I ELM boundary, it is likely that the changes in stability parameters are too small to be identified outside of experimental uncertainties.

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To understand the effect that the separatrix density has on the maximum stable pedestal top pressure $p_{\rm ped}=p_{\rm e, ped}+p_{\rm i, ped}$, a predictive scan using the iPED stability code [35] was performed, in which $n_{\rm e, sep}$, was varied independently with respect to the pedestal top density.

This scan is shown as grey dots with 5% error bars in figure 11 and with three specific expamples for $n_{\rm e, sep}=3.2 \times 10^{19}$ m⁻³ in green, for $n_{\rm e, sep}=3.9 \times 10^{19}$ m⁻³ in magenta and $n_{{\rm e, sep}}=4.8 \times 10^{19}$ m⁻³ in orange. The scan in figure 11(a) shows, that this does not change the maximum pedestal top pressure. However, if $n_{\rm e, sep}$ is slightly changed by varying the out-most ($\rho_{\rm pol} \sim 0.99$) density gradient $\nabla n_{\rm e, sep}$, i.e. shifting the maximum gradient slightly inward, compare magenta and blue profile, p_ped increases by  ∼10%.

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The reason for this behaviour is that the region with steep gradients is narrower and shifted inward, providing better stability against PB modes and consequently allowing for higher pedestal top values.

iPED does not run with separatrix geometry, so the region included is restricted to the closed flux surfaces. The model considers ideal MHD modes, with mode numbers ranging between 1–70. The pedestal gradients are not calculated by transport codes, but the pedestal width is assumed to scale as $\delta_{\rm ped} = c * \sqrt{\beta_{\rm pol, ped}}$, as in the original EPED code [37]. Once a series of pedestal top values have been selected, the width and location of the pedestal are then calculated and the result used as input to the MISHKA ideal-MHD stability code [38]. When an unstable mode is found, a mode structure covering the entire gradient region will be returned, which is the main focus of the iPED code. An actual calculation of the stability of the $n = \infty$ ballooning mode has not been done in this case; it is left for future work to investigate under which conditions this mode is (un)stable.