The I-mode is an edge-localized mode (ELM)-free confinement regime showing enhanced heat confinement. Nevertheless, also in I-mode, events similar to ELMs—albeit less harmful to the divertor—can occur, relaxing the pedestal gradients. A global electromagnetic gyrofluid simulation of such a pedestal relaxation event (PRE) in I-mode conditions is presented and qualitatively compared with previous measurements of PREs in ASDEX Upgrade [Silvagni et al., Nucl. Fusion 60, 126028 (2020)]. The gyrofluid simulation shows a precursor-oscillation of the characteristic instability of the I-mode, the weakly coherent mode (WCM). Prior to the PRE, the drive of the WCM-precursor changes from drift-wave to interchange. A sudden jump in cross-phase between electron temperature and plasma potential fluctuations at the WCM scale leads to the transport enhancement of the PRE.
Future fusion devices have to operate at confinement levels similar to that in the high confinement mode (H-mode) regime. The strong edge pressure gradients due to the high confinement induce periodically appearing magnetohydrodynamic (MHD) instabilities, called edge localized modes (ELMs), causing fast relaxations of the steep edge pressure gradients. ELMs can be of different types. Type-I ELMs are the strongest ones, and it is generally recognized that the heat and particle fluxes induced by type-I ELMs are not acceptable for future tokamak based fusion reactors.1 Therefore, the mitigation or even full suppression of type-I ELMs is mandatory. Type-I ELM-free high-confinement regimes are of major interest. Possible examples are the enhanced Dα (EDA) H-mode regime2 or the small-ELM regime.3,4 A recent review article about ELM-free regimes can be found in Refs. 5 and 6. Another such ELM-free regime is the improved energy confinement mode (I-mode)7,8 which is an attractive confinement regime for future tokamak-based fusion reactors. Compared to the low confinement mode (L-mode), the I-mode9,10 is characterized by similar particle transport at reduced heat transport. Consequently, a pedestal is formed in the temperature, while no pedestal is observed in the density. Whereas turbulence in L-mode is broadband, turbulence in I-mode is characterized by the appearance of the weakly coherent mode (WCM).11–13 The turbulence is more intermittent in I-mode compared to L-mode.14
However, the I-mode can exhibit events showing some similarities with ELMs.7,15,16 Those events occur close to the transition to H-mode. Whether the bursts in the different devices are of the same nature is still being investigated. In ASDEX Upgrade, they appear at the upper limit of electron pressure within the operational range of the I-mode (around pe = 4 kPa measured at ).16 These events have been called16 I-mode pedestal relaxation events (PREs) as they lead to a relaxation of edge profiles, observed in both the electron temperature and electron density. This leads to an increase in the energy deposited onto the divertor targets. These events appear at a similar timescale between 100 and 500 Hz, which is similar to that of type-I ELMs. Electromagnetic activity is observed; however, it occurs after the onset of the PRE.16 This is different to ELMs, which are characterized by clearly detectable magnetic precursors. Also the frequency of occurrence of PREs increases with increasing heating power,17 which is similar to type-I ELMs and inverse to type-III ELMs.
A typical experimentally observed time trace of such a PRE in I-mode is shown in Fig. 1. The PRE shows up as a burst (at about t = 285 μs), for example, in the He 587.6 nm line intensity by the thermal helium beam diagnostic18 [Figs. 1(a) and 1(b)]. The PRE is the same as shown in Ref. 16. Prior  to the PRE, fluctuations at the frequency of the WCM with about f = 75 kHz can be observed [Figs. 1(a) and 1(b)]. As the WCM precursor activity stops (at t = 225 μs), the signature of the radial magnetic field fluctuation measured at outboard midplane changes from rather coherent to more incoherent behavior [Fig. 1(c)]. The incoherent magnetic field fluctuations can be seen as an intermediate precursor. After this change in coherency, the PRE occurs.
Also recently,19 the I-mode regime has been explained as a regime of drift-Alfvén19,20 turbulence where the level of electron temperature fluctuations is suppressed by electron heat conduction along the magnetic field lines. The electron heat conduction is responsible for the decoupling of heat and particle transport. This decoupling gets particularly strong when interchange driven modes, such as, in particular, the ion temperature gradient (ITG) mode, are weak. In Ref. 19, a weak ITG has been justified by a reduced ion temperature gradient close to the separatrix induced by higher electron than ion conduction toward the divertor targets. Large and small-scale fluctuations are suppressed by phase randomization and finite-Larmor-radius effects, respectively. The remaining intermediate scales form a broad peak in the frequency spectrum, which features the same properties as the characteristic WCM. By means of gyro-fluid simulations, not only the WCM, but also several other experimental I-mode observations have been reproduced,19 such as decoupled energy and particle transport, and intermittent turbulent bursts with precursors.14 An explanation of the operational window widening with magnetic field strength8,15 is also given in Ref. 19.
Drift-Alfvén turbulence is very calm and characterized by rather low transport. How this kind of turbulence can lead to pedestal relaxations at all is not obvious. In the present contribution, I-mode PREs are simulated by means of global electromagnetic gyrofluid simulations and are placed in the framework of Ref. 19. The major difference compared to the previous simulation results19 is the much higher plasma beta at which PREs occur. We will see that the turbulence transits to a more violent regime.
II. SIMULATION SETUP
Global nonlinear simulations have been carried out using the δ–f, electromagnetic, gyrofluid code GEMR.21,22 GEMR simulates the three-dimensional evolution of densities, parallel velocities, parallel and perpendicular temperatures, and parallel–parallel and perpendicular–parallel heat fluxes for ions and electrons in a time-dependent self-consistent equilibrium in the very edge of a tokamak with a circular poloidal cross section. The kinetic gradients evolve freely, as required by the strength of fluctuating dynamics in the plasma edge region. Transport by gradient-driven turbulence leads to a degradation of the gradients. Profiles are maintained by source/sink zones at the radial boundaries, which are feedback controlled toward the initially specified values.23 The simulations reported here are similar to the ones previously reported in Ref. 19. Due to the implemented circular plasma cross section, the missing terms related to neutral particles, and the limitation due to the δ–f approach, the presented simulations allow for qualitative investigations only. Quantitative deviations from experiments are expected.
The input parameters are , plasma beta , and normalized collisionality . Here, is the hybrid Larmor radius, the ion sound speed, a the minor radius, and are the reference electron density and temperature, B the magnetic field strength, and νe the inverse Braginskii collision time. Simulations have been carried out at ASDEX Upgrade parameters (major radius R = 1.65 m, a = 0.5 m). One case showing similarities to a PRE is shown here. The nominal input parameters at the reference position in the present study are with the reference density m−3, electron temperature eV, and a magnetic field of B = 2.5 T. The profile is initialized with an electron temperature gradient decay length of LTe = 3 cm, , and . The coordinate system (x, y, s) is in the radial, binormal, and parallel direction of the magnetic field. Simulations have been carried out on a grid, the spatial resolution of the drift-plane is , and the time step is .
In general, the gradients do not represent the experiments well as they are too shallow as discussed in Ref. 19. An elevated ion to electron temperature ratio and reduced ion temperature gradient length have been chosen to reduce ITG turbulence similar to the previous attempt.19 Compared to the simulations presented in Ref. 19, both τi and are a bit smaller. The present simulation investigates the dynamics in the pedestal; therefore, the choice of the reference values has been oriented by the values at mid-pedestal. As PREs appear at the upper limit of pressure within the operational range of the I-mode, the plasma beta has been chosen relatively high, significantly higher than those studied in Ref. 19 but just below the expected threshold of the finite beta regime of turbulence at .
The simulation is obtained in the following way.21 After a short phase of pre-equilibration without the E × B and magnetic flutter nonlinearities, the nonlinearities are activated. The activation of the nonlinearities directly leads to a transient event. After this first event has decayed, a close to stationary state is reached. At some point, a second transient is observed. This second event is presented here. The profiles do not recover completely after these transient events. Also a third transient event appears, which is very similar to the one discussed here in detail, but at lower amplitude.
III. SIMULATION RESULTS
A pedestal relaxation event (PRE) is qualitatively characterized by two main features: it is (i) an incident of a short duration during which (ii) the entire pedestal area relaxes. These features can be seen in Fig. 2 showing the density [Fig. 2(a)] as a function of radius and time at the outboard midplane at one arbitrary toroidal position. This representation corresponds roughly to the typical representation of the measurements of the He beam diagnostic of AUG. An example for a measurement of PRE can be found in Fig. 9 of Ref. 16 reproduced in Fig. 1(a) of this contribution. The 587 nm He-line is often used as an approximation for changes in density. Similar to the experiments, the PRE in the simulation takes place on a timescale of milliseconds. In the simulation, a transport event at t = 200 μs affects the entire simulation domain covering roughly the complete pedestal area. Thus, this can be called a PRE. Comparing qualitatively to the experiment,16 the profiles relax by about 10%, which can be seen by comparing the relative values before and after the PRE in Fig. 2. Similar values can be seen for the electron temperature in Fig. 2(b).
The PRE consists of several filamentary bursts [Fig. 1(a)]. In the simulation, the strongest one appears around t = 0.22 ms (Fig. 2). The filamentary bursts cause transport across the last closed flux surface (LCFS) into the scrape-off layer. The dynamics in the confined region is much faster than in the scape-off layer. The filaments move much slower in the far scrape-off layer. Also due to the decrease in temperature, the first filaments are faster than the later ones. The timescale changes across the radius due to the difference in electron temperature.
Before the PRE onset (), a precursor oscillation localized in the very edge of the confined region of the plasma is observed (Fig. 2). Compared with a Fourier-based analysis, a wavelet analysis avoids averaging out temporally localized events important for short-lived events such as the PRE. The wavelet power spectrum shown in Fig. 3 has been calculated with Morlet wavelets. Even prior to the precursor phase (), we observe the WCM, which for ASDEX Upgrade is typically in the range of 100 kHz. Also a mode at around 25 kHz can be seen as a radial movement in the electron density and temperature iso-layers around the last closed flux surfaces (LCFS) at ρ = 1 in Fig. 2. In the precursor phase (), the low frequency mode at around 25 kHz fades away. The WCM precursor mode chirps down in frequency. Also the WCM gets broader. In this phase, the perturbation of the WCM precursor propagates inwards (Fig. 2). The WCM precursor oscillation excites small filamentary bursts.
A. Appearance in the drift-plane
Figure 4 shows the drift plane (the plane perpendicular to the magnetic field) at the outboard midplane, which is in structure similar to a poloidal cross section of the plasma edge. It shows the temporal evolution leading up to the PRE through four consecutive images. The PRE onset is at about 200 μs. At 50 μs, a small wave localized close to the separatrix is observable. This is at a wavenumber () and frequency typical for the WCM. Due to its drift-wave nature, it does not lead to much transport by itself. Its appearance is hardly detectable in any transport channel. Likewise, no effects on the gradients can be detected. The appearance of the WCM as a drift-Alfvén wave is consistent with the previous study.19 The initial perturbation grows. At 100 μs, the perturbation is larger in amplitude as well as in its radial extent. It is still wave-like although some wave-steepening can be observed. To estimate the propagation direction wavenumber-frequency spectra have been estimated. The precursor oscillation is propagating roughly with the E × B background flow in the electron diamagnetic direction in the laboratory frame of reference. The disappearance of the phase velocity is discussed in Ref. 24. A bit later, at 150 μs, the shape of the perturbation changes slightly. These are the beginnings of the typical mushroom shape of an interchange instability. The precursor affects the region . At 180 μs, the perturbation still grows in amplitude and radial extent. The perturbation affects the region . The typical mushroom-like shape of the Rayleigh–Taylor instability can be observed more clearly. Also, smaller elongated jet-like structures are observed in the scrape-off layer. These are called streamers and are characteristic for interchange, too. They are tilted due to propagation with the E × B background flow. The E × B background flow is in the ion diamagnetic direction in the scrape-off layer.
B. Two growth phases of the PRE
It is important to make clear that the PRE leads to a reduction in density and temperature in the pedestal. During the PRE, the main property of the I-mode (H-mode-like heat transport and L-mode-like particle transport) does not apply. Therefore, one must distinguish the I-mode in inter-PRE and during PRE. The model in Ref. 19 is valid for the inter-PRE case.
Since the PREs appear at elevated plasma beta, it can be suspected that the dominant drift wave undergoes a transition to the micro-tearing mode (MTM) due to an increase in plasma beta. MTMs mainly lead to electromagnetic electron heat transport; particle transport is rather little affected.25 Figure 5 shows the particle and heat fluxes crossing the last closed flux surface. The PRE induces significant heat and particle transport. The transport induced by PRE is mainly electrostatic. The electromagnetic contribution to the particle and heat flux is about % and %. Since here both particle and heat transport are affected, both gradients degrade and the heat transport is rather electrostatic in nature; the PREs are presumably not due to MTMs in this simulation. In the WCM precursor phase, the total transport is much weaker, but the ratios and are significantly higher. One may describe the WCM precursor phase by a mixture of an ideal drift-wave and an ideal MTM, a so-called drift-tearing mode.26,27
C. Breaking of the WCM precursor
What happens at the first growth phase (Fig. 5)? Figure 7 shows the time evolution of the wavenumber spectrum of electron temperature fluctuations. At the beginning (t < 130 μs), the spectrum is dominated by the WCM. Then (around t = 130 μs) a train of gradually emerging higher harmonics of the WCM are observed. This is due to wave steepening of the WCM, which can be seen in Fig. 4. The excitation at larger wavenumbers happens when the I-mode bursts get more pronounced.14,28 At lower plasma beta, these higher harmonics do not induce magnetic fluctuations, and at higher beta, the excited modes at higher wavenumbers also induce magnetic field perturbations. The superposition of magnetic field fluctuations at different wavenumbers let the magnetic field fluctuations appear incoherent.
D. Transition of the underlying instability
Next we want to address the reason for the second enhanced growth phase (Fig. 5). Instabilities can be classified according to their cross-phases.29 Drift-waves are characterized by close to zero cross-phases between plasma potential and electron pressure fluctuations. They lead to low transport. Interchange instabilities exhibit a cross-phase close to between plasma potential and electron pressure fluctuations and lead to strong transport. The cross-phases are estimated by the cross-spectra. The cross-spectrum of two quantities X and Y can be decomposed, , in amplitude pXY and cross-phase α. The cross-coherency is given by . The observation of typical mushroom shapes (Fig. 4) suggests that the underlying instability of the WCM changes as the precursor oscillation grows. This much likely triggers the PRE as the interchange instability is much more violent than the drift-wave instability. From Ref. 19, we know that due to the low collisionality in I-mode, potential fluctuations are forced toward the density fluctuations . This is one main characteristic of a drift-wave instability. Due to the non-adiabaticity induced by electron thermal conductivity, remains finite, driving the WCM.19 As the difference in cross-phase between electron temperature and potential fluctuations is larger compared to that between density and potential fluctuations, we have a detailed look at the electron temperature fluctuations. The temporal evolution of the cross-phase and cross-coherency between electron temperature and potential fluctuations at the maximum spectral power of the electron temperature fluctuations in wavenumber ky space is shown in Fig. 6. Prior to the PRE, the coherency increases from 0.4 at the beginning (t = 0 μs) to about one at the onset of the PRE at t = 200 μs. The precursor oscillation is the WCM which gets more coherent. Initially (t < 160 μs), the cross-phase fluctuates within and , which corresponds to the drift-wave range. Drift-waves are associated with low transport levels. A cross-phase of is the characteristic value for the WCM in I-mode as reported in Ref. 19). After t > 150 μs, the cross-phase makes a jump and fluctuates afterward (t > 160 μs) around the ideal interchange value . As soon as there is a transition to the interchange regime, the transport is determined by the local dynamics at the outboard midplane, and a significant reduction of the heat transport by conduction as described in Ref. 19 cannot be expected. Interchange instabilities are associated with much stronger transport; this is the reason why the transports explode and the pedestal gets more or less suddenly relaxed. This jump in cross-phase is responsible for the PRE in the simulation. Due to the enhanced transport in the outer edge, close to the separatrix density and electron temperature are reduced. This leads temporarily to an enhanced pressure gradient across the pedestal region, which leads to additional enhanced transport. This is also the reason why the radial extent affected by the WCM precursor increases over time. The process is highly self-amplifying.
It should be discussed how it could come to the jump in the cross-phase. The cross-phase is determined by the competition between the local perpendicular dynamics possibly leading to the interchange growth and the parallel dynamics leading to the drift-Alfvén wave. In the electrostatic case, this competition is controlled mainly by the curvature radius and the collisionality30,31 (see also Appendix E in Ref. 32). Similar to collisionality, induction can also increase non-adiabaticity. The plasma beta in the simulations is relatively high. Before the PRE, radial magnetic field fluctuations can be observed. These appear at roughly twice the frequency of the WCM. Any deviation from an ideal parallel electron response leads to stronger interchange drive. Induction, necessary to excite magnetic field fluctuations, enhances non-adiabaticity. At first (t < 130 μs), the amplitude of magnetic fluctuations decreases over time; therefore, the induced non-adiabaticity by these rather coherent fluctuations much likely does not lead to the jump. At around t > 130 μs, the radial magnetic field fluctuations change their nature from coherent to incoherent. Such a change in coherency can be also observed in the experiment [Fig. 1(c)]. This coherency change is accompanied by a reduction in [Fig. 6(b)]. The dynamics gets even more adiabatic during the first growth rate . However, just before the jump in cross-phase, an event in the radial magnetic field fluctuation is observed (t = 150 μs). Magnetic field fluctuations lead to deviations from the dynamics along the magnetic field lines. Mathematically, this means that the parallel derivative operator is extended, . Magnetic field fluctuations act even on ideal interchange flute modes with . Thus, radial magnetic field fluctuations disturb the parallel electron response in different ways. In particular, the parallel electron response is central to the WCM in the case of Ref. 19. As a consequence a dropout in the coherency is observed. In turn, the stabilizing parallel electron dynamics cannot compete with the destabilizing local drive of the interchange mechanism.30 The WCM transits from the drift-wave to the interchange regime. The cross-phase of the WCM makes a jump. As long as these radial magnetic field fluctuations are present , the cross phase fluctuates around the ideal interchange value . After the phase of enhanced radial magnetic field fluctuations (t > 250 μs), the cross-phase is around between the drift-wave and interchange regime. A cross-phase of is the typical value observed in the stationary I-mode-like simulations previously presented in Ref. 19.
E. Increased coherency prior to the WCM precursor
Finally, the increase in coherency in the phase prior to the WCM precursor phase is discussed. The generation of gradually emerging higher harmonics (Fig. 7) is reminiscent of the transition from the phase-locked regime to the weakly turbulent regime of the Ruelle–Takens scenario, describing the transition to drift-wave turbulence.33–35 The nonlinear interaction in the phase-locked regime is dominated by resonant three-wave interactions. Nonlinear interactions in frequency space have to fulfill the three-wave-coupling condition . Due to phase locking, a quasi-coherent mode is observed in the phase-locked regime in the Ruelle–Takens scenario. A quasi-coherent mode is also observed in I-mode; this is the WCM. The bicoherence measures how phase-locked modes are with values in [0,1]. Therefore, the bicoherence is a direct measurement of the strength of phase synchronization. As density fluctuations are advected by the E × B flow, the most important nonlinearity is based on the interaction between density and potential fluctuation. This can be measured by the cross-bicoherence between the plasma potential and density fluctuations. A plasma potential fluctuation at low frequency is necessary to induce the phase synchronization with the density fluctuations at the WCM frequency. Such a low-frequency perturbation is present prior to the precursor phase () at around 25 kHz (shown for the density in Fig. 3, but also observed in the plasma potential not shown here). The mode at 25 kHz has not been identified yet. It appears in a similar frequency range as the geodesic acoustic mode (GAM), but it does not occur in the zonal average as a GAM would. The binormal wavenumber ky is similar to that of the WCM. It does not lead to significant transport. To investigate the time evolution, the wavelet cross-bicoherence between plasma potential and density fluctuations has been estimated according to
For the ensemble average, different points on the drift-plane are used. These are of course not independent. An interpretation of the absolute values should be refrained from. Nevertheless, the development over time should allow conclusions to be drawn. The time evolution of the total cross-bicoherence,
is shown in Fig. 8 on the top. Prior to the precursor phase (), the WCM around 100 kHz and a mode at around 25 kHz are observed (Fig. 3). In the total cross-bicoherence , the WCM shows enhanced coupling. The coupling at lower frequency seems to be at a slightly higher frequency (at 30 kHz) than the amplitude (25 kHz) for t < 50 μs. For , the enhanced coupling is at 25 kHz. A closer look at the cross-bicoherence itself shows that both modes also couple with each other. Besides the coupling between both, we find increased coupling of both modes to low frequencies. Comparing the time points t = 50 and t = 100 μs, we see an increase in the bicoherence between the WCM and the mode, which is now closer to 25 kHz. This indicates that both modes are phase synchronized. The increase in phase synchronization leads to the coherency increase observed in Fig. 6. After t > 100 μs, the mode at 25 kHz fades away (Fig. 3). Since the WCM is no longer phase-synchronized to the 25 kHz mode, it can now be coupled with other frequencies. The coupling area of the WCM in (f1, f2) space (this is called the resonance manifold) widens considerably. This is a feature of a transition to the weakly turbulent regime. This also ultimately allows the WCM to excite higher harmonics. After t > 150 μs, the total cross-bicoherence is low. The actual PRE does not have a strong fingerprint in the nonlinear coupling. The actual PRE is due to the linear growth of the interchange instability.
IV. SUMMARY AND CONCLUSIONS
Pedestal relaxation events (PREs) in I-mode16 have been investigated using global electromagnetic gyrofluid simulations. The simulation presented here is similar to the ones previously reported in Ref. 19 and have the same constraints (i.e., small gradients) as discussed therein. The simulation compares well to experimental observations. The pedestal relaxed by about 10%. The PRE is preceded by a precursor-oscillation at the frequency of the weakly coherent mode (WCM) localized close to the separatrix. While the WCM grows, the amplitude of the train of higher harmonics increases. The excitation at larger wavenumbers is observed for I-mode bursts at moderate plasma beta.14,28 As PREs occur close to the transition to the high confinement regime of operation, the plasma beta in the present simulations is significantly higher than the simulations presented in Ref. 19. At higher plasma beta, the excited higher harmonics induce magnetic fluctuations, and the radial magnetic fluctuations appear more incoherent. These radial magnetic field fluctuations can disturb the parallel electron dynamics which is stabilizing the interchange effect. Then the cross-phase between electron temperature and plasma potential fluctuations of the WCM-precursor changes from drift-wave to interchange. The cross-phase between electron density and plasma potential fluctuations undergoes the same transition (not shown here). The jump in cross-phase to the ideal interchange value leads to the enhanced transport in the very edge. This leads to a steeper gradient across the pedestal region. Together with the enhanced cross-phase, this leads to the PRE.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
The authors would like to thank B. D. Scott for providing the code GEMR. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 and 2019–2020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.