Structure and dynamical steady state of the peeling mode in edge-localized mode-free, high-confinement tokamak plasmas

Explosive dynamical events in controlled-nuclear-fusion devices (known as edge-localized modes) display many similarities to solar-flare events on the sun, revealing a new connection between laboratory plasma physics and astronomy. However, to date there has been no direct evidence for the peeling mode structure, due to the lack of decisive diagnostics. Here we report the first evidence for the structure and dynamical steady state of a peeling mode for low- n edge-harmonic oscillations (EHOs) in the quiescent H-mode. EHOs are dominated by the fundamental mode (1f EHO ) at both the low- and high-field sides. 1f EHO edge perturbations are confirmed to have kink parity and exhibit the frozen-in-condition predicted by a linear stability analysis. The envelope signal of the 1f EHO mode exhibits repeated cycles of growth and damping to the order of a few hundred Hz associated with small changes in an edge gradient, and results are quantitatively consistent with a limit-cycle-oscillation model.

Studies of explosive dynamical events in controlled-nuclear-fusion devices (known as edge-localized modes (ELMs) 1-3 ) display many similarities to solar-flare events on the sun, revealing a new connection between laboratory plasma physics and astronomy [4][5] . Improved understanding of the common physics of ELMs and solar/stellar flares not only can contribute to our intellectual understanding of plasma physics, but also can improve the controllability of burning plasmas in selfregulating, combined/complex next-step devices such as ITER [6][7] and DEMO [8][9] . In particular, highconfinement-mode (H-mode) operation, which has a high pedestal pressure, may be impossible without a proper physics understanding of the "peeling" mode (and/or coupled peeling-ballooning modes). This seems essential for predicting the ELM stability limit, because the instability is driven by the pressure gradient and the parallel current (the so-called bootstrap current) [10][11][12][13] .
The quiescent H-mode (QH-mode) regime was discovered first in DIII-D 14 , and it was subsequently reproduced in JT-60U 15 and ASDEX-Upgrade/JET 16 , in which ELMs are replaced by continuous MHD oscillations (the so-called edge-harmonic oscillations, EHOs). Theoretical models, together with experimental support, predict that the QH-mode edge condition is always near the low-n (≤ 5) boundary of the peeling mode instability [17][18][19][20] . However, to date there has been no direct evidence for the structure of the peeling mode, due to the lack of decisive diagnostics. This is especially true for fluctuation measurements at the high-field-side (HFS) of the tokamak edge region, and measurements have been limited solely to the low-field-side (LFS).
To identify the peeling mode, we present here the following five criteria for the associated EHOs: They must (1) be lower-n modes (≤ 1, 2, 3…), (2) have a mode structure that extends to all of the plasma surface (i.e., not just be localized at the LFS), (3) result in perturbation structures that are close to the edge current and/or corresponding eigenmode, (4) have "kink" parity (rather than "tearing" parity), and (5) be EHOs that are sustained for long times; i.e., times comparable to the currentdiffusion time in the edge region. New information about the dynamical steady state that occurs during the QH-mode is also given. The envelope signal of the dominant fundamental mode of the EHOs exhibits repeated cycles of growth and damping on the order of a few milliseconds that are associated with small changes in the pedestal gradient, and the results are quantitatively consistent with a limit-cycle-oscillation model.

Results
Observation of low-n EHOs at both the LFS and HFS. Figure 1 (a)-(d) shows that the dominant fundamental mode of the EHOs (1fEHO)-obtained from various diagnostics measured at different poloidal (toroidal) angles-occurs at a frequency ~ 10 kHz during a stationary QH-mode phase without large ELMs, keeping the plasma density constant. The magnetic diagnostics identify this 1fEHO mode as n = 2. This QH-mode phase-with continuous EHOs (tEHOs ≥ 3 s)-is sustained with a neutral-beam-injection power PNBI = 14 MW (4 MW counter-tangential, 5 MW co-perpendicular, and 5 MW counter-perpendicular injections) for t = 4.5-7.5 s (Fig. 1d). This is significantly longer  As illustrated in Figs. 1 (a)-(c) and (e)-(f), the spectra for the ECE, FIR-U1, and -U2 show that they manifest high coherence among them at the frequency 1fEHO ~ 10 kHz (n = 2). This observation suggests that the 1fEHO mode has a uniform mode structure over the entire circumference of the peripheral region of the plasma (Fig. 1g), and hence it is more likely to be a peeling mode (i.e., it is non-ballooning at the LFS). On the other hand, the power (and coherence) for the 2 nd -harmonic component that exists at the frequency 2fEHO ~ 20 kHz (n ~ 4) becomes weaker than that of the 1fEHO mode, and there are no higher-order modes. It is worth noting that the power spectrum of the EHOs observed in the QH-mode phase at JT-60U is characterized by a dominant fundamental MHD mode rather than containing strong higher-order harmonics (as seen in other devices 14 ), which suggests weak nonlinearity (i.e., it is a quasi-linear mode).

Effects of GAPOUT and GAPIN values on ELMy/QH operational regimes. The GAPOUT value
needed to access the stationary QH-mode regime in E042870 was determined by sweeping the edge plasma slowly in another discharge (i.e., E042868), with the cross-sectional shape of the plasma remaining fixed; hence, the GAPIN value is inversely correlated with the GAPOUT value. Here, the GAPOUT/GAPIN values are defined by the distance between the last closed flux surface and the wall at the outboard/inboard mid-plane.  This observation suggests that the condition of a locally wide GAP only at the LFS seems to be insufficient for accessing the stationary QH-mode regime. In any case, this is a necessary condition (known to be one of the "control knobs" for accessing the QH-regime 14 ). By comparison with other devices, the GAPOUT value seems to be wide enough for this ELMy phase in the pre-QH phase (t ≤ 6.4 s) to occur, while the ELMs reappear in the post-QH phase (t ≥ 6.7 s) as the GAPOUT value becomes narrower (but, still wider than that of other devices) as is well known. This was pointed out in Ref. 15, which discussed the role of an optimized GAPOUT condition. On the other hand, the conditions of non-local GAPs at both the LFS and the HFS seem to be more important for the excitation of EHOs that have uniform mode structures over the entire circumference, absent the inhibition expected from a peeling mode structure. This observation is still challenging for the theoretical understanding of the conditions necessary and sufficient for accessing the QH-mode regime.
The theory of the stability of peeling-ballooning (kink) modes at the plasma edge allows us to confirm one of the peeling mode features for the EHOs in the QH-mode. As illustrated in Fig. 2 (c), the QH-mode data from JT-60U are consistent with the stability predictions of the theory, which have been discussed in previous publications from DIII-D 21-23 , because the operating points for the QH (and/or ELMy) phase are on or below (up to) the low-n peeling boundary to within the experimental error bars. On the other hand, the dependence of the marginal-stability boundary on the pressure gradient (aped.) for JT-60U is stronger than that for DIII-D, suggesting that these two devices have different mechanisms for accessing the QH-mode regime. Furthermore, the radial structures of the toroidal flow ( ! "#$ ) and of the radial electric field ( % ) in the QH-mode phase at the pedestal region are observed to become slightly weaker than those in the ELMy phase (Figs. 2d and e), suggesting that a strong E×B shear flow is unnecessary. Instead, the appearance of EHOs seems to be strongly influenced by the optimization of the GAPs, as described above. Indeed, the phase transition from ELMy to the QH-mode occurs solely due to changes in the GAPs, while other external parameters are kept fixed (e.g., the cross-sectional shape of the plasma, its density, and the external input torque).
Radial structure of the peeling mode. )-seems to be reasonably consistent with the radial structure of the 1fEHO mode, exhibiting the so-called "frozen-in" condition of the MHD mode (i.e., the field lines are frozen into the plasma and have to move along with it) [25][26] . This is one of the criteria for identifying the peeling mode. Furthermore, the radial structure of the relative phase difference between the ECE and the magnetic diagnostics (i.e., the saddle coil located on the vacuum vessel) is observed to be almost constant radially, exhibiting a non-tearing type of kink parity of the mode, as illustrated in Fig.   3 (d).  Note that the experimental identification described above was made possible because the EHOs were in a state close to that of a linear mode, with almost no higher harmonics, although the observation of a QH-mode phase that remained stationary longer than its MHD stability growth rate seems strange from a common-sense point of view. Thus, there must be a mechanism to keep the peeling mode from growing explosively. Here, this is provided by a "dynamical" stationary phase that has a timescale of variation of a few ms (~1/ fenvelope), which is a much longer span of time than that of the peeling mode: 1/ fEHOs ~ O(0.1 ms). We conclude that the peeling mode (which has almost no higher-harmonic components) has been identified as a new "dynamical" steady state of a QHmode phase that does not have any large ELMs.
This dynamical steady state is realized as a limit-cycle oscillation, as shown by the relationship In conclusion, we have presented the first evidence of the peeling mode for EHOs seen in the peripheral region of the plasma for the QH-mode in JT-60U, and it meets the five criteria described above. New information about the saturation mechanism that enables the EHOs to peel off the pedestal (without the effect growing explosively) is also shown. In particular, the dominant fundamental mode of the EHOs is observed to repeat cycles of growth and damping, in association with changes in the mean temperature gradient, on the order of a few 100 Hz. This observation is quantitatively consistent with the new dynamical quasi-linear steady state model of limit-cycle oscillations whether or not a strong E×B shear flow exists. This sheds further light on the "peeling nature" of both laboratory plasmas and astronomical plasmas (such as ELM events in magneticconfinement-fusion plasmas and solar/stellar flares).

QH-mode operation on JT-60U.
The JT-60U tokamak is a single null divertor tokamak device having the plasma major radius R P =

Fluctuation measurements for edge-harmonic oscillations (EHOs).
The temporal evolution of the multi-mode structure FFT spectra during a stationary, long-sustained QH-mode is obtained from the electron-cyclotron-emission radiometer (ECE) used for local electrontemperature measurements at the LFS and from the FIR laser interferometers used to obtain lineintegrated electron density measurements along two vertical chords, which pass through the peripheral region of the HFS plasma (FIR-U1) and from top to bottom of the plasma center (FIR-U2), respectively.

Radial profile measurements for toroidal plasma flow and radial electric field.
In the study of the JT-60U tokamak for the fiscal year FY2003 experimental campaign, the radial profiles for the density, temperature, and poloidal/toroidal plasma flows of fully stripped carbon impurity ions are measured by means of the Charge eXchange Recombination Spectroscopy (CXRS) diagnostic method with time resolution up to 60 Hz at 59 spatial points (23 toroidal and 36 poloidal viewing chords). With regard for determining the E r structure at the pedestal region, we measured the pressure gradient, and plasma velocity perpendicular to the magnetic field, and the E r was evaluated by the radial force balance equation.