Experimental evidence for the drift wave nature of the weakly coherent mode in ASDEX Upgrade I-mode plasmas

The improved energy confinement mode (I-mode) is a potential candidate for future fusion power plants, as it combines ELM-free operation with good confinement. The unusual edge transport and turbulence in this regime is still not fully understood. This study analyzes the turbulent structure of the weakly coherent mode (WCM) in ASDEX Upgrade. Measurements from Doppler back-scattering and a thermal helium beam diagnostic are used to determine velocities of the background plasma and the WCM over multiple discharges. A phase velocity of the WCM of the order of 2–5 km s−1 in the electron diamagnetic drift direction is found, quantitatively close to a drift wave assuming negligible temperature fluctuations. A good agreement with a previously proposed mechanism behind the I-mode regime is observed. This marks the first experimental verification of a specific understanding of the WCM and the I-mode regime.


Introduction
In magnetic confinement fusion research, strong focus is set on ELM-free regimes to avoid damaging plasma facing components in future reactors.Of these regimes, the improved energy confinement mode (I-mode) [1,2] is particularly interesting, 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.
as it combines high energy confinement with good impurity transport.These properties are the result of an unexpected separation between heat and particle transport in the edge of the plasma, leading to a pedestal in the electron temperature profile while the density behaves as in the low confinement mode (L-mode).
I-mode is accessed in unfavorable drift configuration and has been achieved in a variety of tokamaks (Alcator C-Mod [2,3], ASDEX Upgrade (AUG) [1,4,5], DIII-D [6], EAST [7,8], HL-2A [9]) and with different heating schemes and magnetic fields.Although much research on the underlying physics of the transport channel separation was done, the results are still inconclusive.Many suggested mechanisms rely on a characteristic change in edge turbulence at the L-I transition to explain the I-mode properties [3,10].
Edge turbulence in I-mode features reduced low frequency density and temperature fluctuations and the appearance of a broad peak of higher frequency turbulence, called the weakly coherent mode (WCM) [2].As with the I-mode, the mechanism of the WCM is not well understood yet.The WCM is often considered a marker of I-mode and closely linked to its special transport properties [10].However, diagnostics using temporal or spatial correlation routinely observe the WCM in L-mode phases preceeding the L-I transition, questioning the WCM as an I-mode-specific feature [11,12].
Different explanations for the WCM have been proposed, mostly based on analyses of simulations.Liu et al [13] used the BOUT++ framework and concluded that the I-mode pedestal is unstable to drift Alfvén wave (DAW) instabilities and resistive ballooning modes.Lang et al [14] also proposed DAW instabilities using the BOUT++ code.Yang et al [15] considered both a trapped electron mode (TEM) and a drift wave instability.Manz et al [16] proposed that the parallel electron heat conductivity is of critical importance, equilibrating temperature fluctuations and thus reducing electron heat transport.In this study, the WCM is found to be the remainder of a scale-dependent suppression mechanism affecting larger scales.Another study by Espinosa and Catto [17] suggests that neoclassical transport alone is sufficient for the I-mode impurity removal, leaving the WCM compatible with, but not responsible for the I-mode properties.
From an experimental perspective, no conclusive results on the WCM are found either.The relative fluctuation amplitude is typically the highest in density (≈10%) [18], with lower values in temperature (2%-4%) [11,12,19].The central frequency varies between experiments (e.g.50-150 kHz at AUG, 100-300 kHz at Alcator C-Mod) and also within a single discharge.This is sometimes attributed to a change in E × B flow velocity [10,12].The poloidal wavenumber was found to be around 1.3(5) cm −1 at C-Mod [3,10,20].Measurements of the n e -T e cross-phase are not conclusive yet and range from close to in-phase [11] to −143 • [12].
This study is dedicated to a more systematic analysis of the WCM in I-mode shots at AUG in order to find a better understanding of the WCM behavior.To that end, a large database of AUG I-mode discharges was set up.Measurements with Doppler back-scattering (DBS) [21] and the thermal helium beam (THB) diagnostic [22] were used to track the WCM frequency and wavenumber and to compare WCM properties with plasma conditions like density, temperature and E × B velocity.

Doppler back-scattering
DBS was used to measure three parameters: The central WCM frequency, f WCM , the WCM amplitude relative to the broadband background, A rel , and the E × B-velocity, v E×B .
The WCM properties were obtained from fits of the power spectrum of the fluctuations in the angular velocity of the complex IQ-signal, as shown in figure 1.Although the DBS is conventionally not used for large scale fluctuations as in the WCM (k ⊥ ≈ 0.5 cm −1 ), the strong density fluctuations of the WCM induce a modulation in the back-scattered beam [23].This does not affect the average Doppler shift, but it modulates the IQ-signal phase and can thus be extracted, as done for instance in [7].The fits of the fluctuation spectrum were done with a 1/f dependence plus a constant as a background and a Gaussian peak for the WCM.From this, the relative amplitude of the WCM can be calculated: where A Gauss is the amplitude of the Gaussian fit and A tot is the total amplitude at the frequency of the WCM, as shown in figure 1.
The E × B-velocity is probed via DBS at small turbulence scales (k ⊥ ≈ 10 cm −1 ).The probed scale is of importance, as the velocity of turbulent structures in the lab frame consists of two contributions, the E × B-velocity and a possible scaledependent phase velocity: ( From comparison with charge-exchange recombination spectroscopy and spectroscopy on He-II in previous studies, the phase velocity at the probed scale of the DBS is found to be negligible in the AUG pedestal [24][25][26].Unfortunately, no reliable data from these diagnostics was available for comparison in the discharges investigated here.However, three identical I-mode shots (#33436-8) have been performed with the DBS set to different k ⊥ between 5-10 cm −1 and no dependence of v lab on k ⊥ was found.Combined with the good agreement with other diagnostics in previous studies, the assumption of a negligible phase velocity for the DBS is considered justified.

Thermal helium beam spectroscopy
The analysis of the THB data [22] is based on the coherence between spectral line intensity ratios (667 nm and 587 nm) at two lines-of-sight (LOS) separated about 2 cm in perpendicular direction.This is very similar to the analysis in [27], although only one pair of spectral lines is used for the intensity ratio and a larger separation of LOS is chosen.The coherence reveals a broad peak at the WCM frequency.Fitting this peak yields f WCM , which is found to be in good agreement with DBS results.The phase difference φ between the signals from the two LOS and the known distance d can be used to calculate the perpendicular wavenumber: Due to diagnostic settings, only one single radial position for k WCM is available.It is reasonable to assume a radially constant k WCM , since f WCM is also found to be radially constant via both DBS and radial coherence between THB LOS.Combining f WCM and k WCM from THB measurements, the perpendicular propagation velocity of the WCM in the lab frame is: Note that, since the THB measures the WCM directly, the phase velocity at the scale of the WCM is relevant.

Radial mode localization
Since the plasma properties vary strongly in the edge gradient region, any analysis of the WCM with respect to the local background plasma requires knowledge of the radial location of the WCM. Figure 2 shows the radial profile for one discharge in the database.The peak of the WCM aligns with the E r well and the steepest gradient of the temperature profile.The radial profiles are obtained using the equlibrium reconstruction [28], the integrated data analysis [29] and the beam tracing [30,31] for the DBS measurement.Since both E r and A rel are obtained from DBS, the analysis yields a selfconsistent mode localization within the electron density gradient and the E r -profile.The localization in the E r well is consistent over different shots in upper single null (USN) and lower single null with reversed I p and B t (LSNrev).
The localization of the WCM in the E r -well is in agreement with previous studies from AUG [12], C-Mod [20] and EAST [7], while one study localized the WCM in the outer E r shear layer [32].The localization in the region of strong gradients and E r minimum is consistent with the common understanding of microinstabilities, such that they drain energy from background gradients, while the generated structures tend to decorrelate in the velocity shear layers around the E r minimum.
For discharges with sufficient data from DBS, the location of the E r minimum was identified.A small region around this location (see figure 2) is then used as a reference for all radially varying properties, such as n e , T e and v E×B , since the THB LOS is not necessarily at the maximum WCM amplitude.

Phase velocity
From a comparison of the WCM lab velocity obtained from the THB and the E × B-velocity at the reference location in the E r well, the phase velocity of the WCM can be determined.Figure 3 shows two exemplary v E×B profiles for the I-mode in LSNrev and USN, as well as the WCM lab velocity measured with the THB.A clear difference is observed, corresponding to a phase velocity in the electron diamagnetic drift direction in both configurations.Additionally, the LSNrev case (#37312) used a heating power ramp, resulting in a significant change in the velocities during the discharge.A notable WCM phase velocity is confirmed at any point during the I-mode.A phase velocity in a similar range was already observed very early in Imode experiments at Alcator C-Mod [3,20], although another study found no phase velocity [32].
Many simulations of I-mode and the WCM suggest DAWturbulence as the dominant driving mechanism behind the WCM [13,16].In the simplest case, i.e. slab geometry and considering gradient and fluctuations in density but not in temperature, a textbook drift wave propagates with a phase velocity of in electron diamagnetic drift direction, with L n = −n e /∇ r n e being the density gradient length.When including electron temperature gradient and fluctuations, the phase velocity is  5)); the gray triangles include the contribution of temperature fluctuations in the theory (equation ( 6)).A good match is obtained in some cases when neglecting Te, with a slightly poorer agreement in others.A sizable mismatch is found when including Te in v ph,DW .As described in section 4.2, the colored points are considered to be in line with a proposed mechanism behind the I-mode regime.The gray triangles are shown to highlight the discrepancy with the drift wave model including temperature fluctuations (equation ( 6)).
increased by a temperature dependent term, yielding the electron diamagnetic drift velocity v ph,DW,nT = T e eB with the electron temperature gradient length L T .Figure 4 shows the comparison of experimental values with both the 'simple' drift wave phase velocity (5) and the electron diamagnetic velocity (6).A reasonable match is obtained for most intervals under the assumption of negligible temperature fluctuations (colored), with slightly smaller phase velocities in the experiment than expected.Interestingly, there is a significant disparity to the electron diamagnetic velocity ((6), grayed out).These findings are further discussed in section 4.2.
The measurement of the phase velocity relies on data from both DBS and THB measurements in the same discharge, which limits the number of usable discharges.However, the perpendicular wavenumber k WCM of the WCM is found to be almost constant in the database, with small differences between LSNrev at k WCM = 0.56(5) cm −1 and USN at 0.69(5) cm −1 .Assuming k = const, a combination of the E × Bvelocity and the 'simple' drift wave velocity (5) can be used to predict the WCM frequency in the lab frame: This frequency can be compared to the measured frequency from DBS, which expands the analysis to shots without sufficient THB data.The result shown in figure 5 yields good agreement.

Phase velocity
The measured phase velocities (or the WCM frequency) tend to be smaller than predicted by theory, especially at higher velocities.This could be interpreted as a non-ideal drift wave caused by a non-adiabatic electron response.A non-adiabatic  7).The continuous line shows f WCM for the measured values of k in USN and LSNrev.A good agreement is found for low f WCM , whereas the velocities for higher f WCM in USN are overestimated.response would lower the resulting phase velocity, qualitatively matching the experiment.More generally, the lower phase velocities/frequencies could be an indication of different dominant micro-instabilities with slightly different phase velocities, such as TEMs or micro-tearing modes (MTM).For drift waves, TEM and MTM, the parallel electron dynamics are essential, and the mechanism of separating density and temperature dynamics through heat conduction in [16] would affect these instabilities similarly.Gradients, collisionality and plasma beta influence which microinstability is dominant, hinting towards further studies of these parameters and their influence on the WCM.

Temperature fluctuations and agreement with theory
The phase velocity agrees well with the 'simple' drift wave, i.e. neglecting temperature fluctuations.When including temperature fluctuations in the model, the comparison yields a large disagreement.This is somewhat surprising, since the temperature gradient is strong in the I-mode pedestal.However, the observations of the WCM presented here are consistent with the mechanism behind the I-mode confinement regime proposed in Manz et al [16].
Accordingly, I-mode requires two ingredients: (a) unfavorable magnetic drift configuration and (b) sufficient heating power, as shown in figure 6.The unfavorable configuration is a necessary condition to allow for strong enough heating without entering the high confinement mode (H-mode).As Hmode is avoided, the turbulence suppression is less severe and some broad-band turbulence can persist.
Sufficient heating leads to an increase in edge electron temperature compared to L-mode, resulting in reduced collisionality.This has two important effects: first, electron-ion coupling is reduced, leading to steeper gradients in T e than in T i and higher T i /T e around the separatrix [33,34].This results in a reduction of ion temperature gradient (ITG) driven instabilities Figure 6.Proposed mechanism behind the I-mode confinement regime, following [16].Marked are the properties found to agree with the experimental WCM studies presented here.For details refer to the main text.
that are a primary cause for turbulent transport in L-mode.As ITG turbulence is reduced, the main interchange-type instability is weakened, which allows for drift-wave turbulence to develop.The second consequence of a high electron temperature is an increased parallel heat conductivity.This can equilibrate temperature fluctuations, which-combined with the low ITG drive-leads to a reduced energy transport.This improves the energy confinement, which leads to a positive feedback loop that often results in an I-H-transition.In present day Imode experiments, external control of β pol is frequently used to counter the effect and to obtain stationary I-mode plasmas [35].
The WCM study presented here fits well into the described I-mode mechanism [16], as its results are explained by driftwave turbulence at low temperature fluctuations.One should stress that temperature fluctuations-albeit smaller than density fluctuations-are consistently measured in I-mode [11,12,19].However, as long as their relative amplitude Te /T e is small compared to ñe /n e (10% versus 2%-4% in the experiment [12,18,19]), the drift wave dynamics are dominated by the density fluctuations and the phase velocity is best described by the 'simple' drift wave model (5).

Appearance of the WCM
As a last part of this study, the appearance of the WCM at the L-I-transition is discussed.In Manz et al [16], the prominence of the WCM in the frequency spectrum was explained by a dominant suppression mechanism due to phase randomization likely caused by magnetic flutter.This would primarily affect large scales, dependent on the square root of the plasma beta: k WCM ∼ √ β e .However, this dependence on β e could not be verified with the database in this study.Therefore, the prominence of the WCM (or the reduction of turbulence below the WCM frequency) has to be explained by another mechanism.Turbulence suppression due to phase randomization also occurs due to shear flows [36].In the case of I-mode, the geodesic acoustic modes (GAM) is a natural candidate [10,20,37] for most of the discharges.This type of turbulence suppression mechanism regarding GAMs and similar low frequency modes like the edge temperature ring oscillation [38] with shear flow should be studied in more detail.
It has been known for a while that WCM features can be detected in L-mode plasmas prior to an L-I-transition [11,12].The WCM transitions smoothly from L-to I-mode, with a constant or even slightly decreasing poloidal wavenumber and a significant increase in frequency in the laboratory frame.During this, the normalized wavenumber k WCM ρ s stays fairly constant, as the hybrid Larmor radius ρ s increases with √ T e during the temperature rise when entering I-mode.The increase in frequency combined with the constant or decreasing wavenumber requires the lab velocity of the WCM to increase substantially at the L-I transition.With the match to a drift wave presented here, this increase in v lab,WCM is attributed to both the known increase in v E×B and the impact of T e (cf (5)) on the phase velocity.Hence, the prominence of the WCM after the L-I transition is most likely not only caused by a suppression of structures larger than the WCM due to phase randomization, but an acceleration of the underlying drift-wave microinstability.Further quantitative studies on the WCM during the L-I-transition are planned.

Conclusion
The study presented here sheds new light on the understanding of both the I-mode confinement regime and the WCM.The experimental results strongly point towards a drift-wave turbulence regime with suppressed temperature fluctuations as the drive behind the WCM.This is also in good agreement with a suggested mechanism behind the peculiar edge transport of the I-mode regime.The appearance WCM is thus likely caused by an acceleration of the underlying structure at the L-I-transition.

Figure 1 .
Figure 1.Fit of the WCM on the fluctuation power spectrum obtained with DBS.The shaded area illustrates the 25th and 75th percentile.

Figure 2 .
Figure 2. Exemplary edge profile of various plasma parameters and the relative WCM amplitude.The I-mode-specific gradients can be seen in (a).(b) Profile of −Er and A rel from DBS.A good match of steepest gradients, Er minimum and maximimum WCM amplitude is found.The gray shaded radial region is used for all further analysis of the WCM.

Figure 3 . 4 .
Figure 3. Velocity profiles for two shots in LSNrev (a) and USN (b).A clear difference in the electron diamagnetic direction between the E × B background velocity and the lab velocity of the WCM is found.Different colors in (a) correspond to a heating power ramp over the course of the I-mode interval.For this study, the Er minimum is taken as a reference position.

Figure 5 .
Figure 5.Prediction of f WCM according to equation(7).The continuous line shows f WCM for the measured values of k in USN and LSNrev.A good agreement is found for low f WCM , whereas the velocities for higher f WCM in USN are overestimated.