Introduction

In the turbulence of magnetized high-temperature plasmas, fluctuation components at several scales coexist. Multi-scale turbulence studies have been actively conducted on laboratory1 and space2 plasmas. Studies on multi-scale turbulence in magnetically confined high-temperature plasmas3, for which detailed data are available, are particularly valuable because they provide fundamental knowledge essential for fusion reactor development.

Micro-scale (i.e., scale of ion gyroradius, ρi) instabilities in magnetically confined high-temperature plasmas and the turbulent transport driven by these instabilities have attracted much attention4. The coexistence of nonlinearly excited meso-scale fluctuations (e.g., zonal flows)3 has also shed light on multi-scale turbulence. Hyper-fine (HF) scale turbulent fluctuations, such as those on scales near the collisionless skin depth5,6 and electron temperature gradient (ETG) mode7 turbulence, which is characterized by the electron gyroradius ρe, have been theoretically predicted. The existence of ETG instabilities has been confirmed in basic experimental plasmas at low temperatures8,9,10. Although observations of HF-scale turbulence in magnetically confined high-temperature plasmas have been reported11,12,13,14, there are few experimental studies compared to those on micro-scale turbulence.

Transport due to HF-scale turbulence has been considered insignificant based on the assumption that the diffusion coefficient D ~ γ/kr2 (where γ is the growth rate and kr is the radial wavenumber for the turbulence)4, with the order of magnitude of krρe estimated to be ~O(1). However, nonlinear simulations15,16 and a theoretical analysis17 pointed out the possibility that the ETG instabilities may have a short wavelength in the poloidal direction but a long and elongated structure (a kind of streamer) in the radial direction. That is, the quasi-linear model γ/kr2 predicts the possibility of large transport. Another issue is that HF-scale turbulence may have completely different behavior from that of the previously-studied multi-scale turbulence18. That is, the authors of reference18 pointed out that HF-scale turbulence is stretched by micro-scale turbulence due to the nonlinear effects of cross-scale interactions. From this nonlinear coupling, two states can exist: one in which HF-scale turbulence is dominant and one in which micro-scale turbulence is dominant. A macroscopic non-uniform radial electric field can control which of the two states is realized, because it suppresses the low-wavenumber turbulence more effectively; as a result, it enhances the high-wavenumber turbulence by suppressing the low-wavenumber turbulence. These results are supported by a study on nonlinear effects that may induce streamers in ETGs19. In addition, nonlinear simulations that cover both the micro- and HF-scales have recently become available20,21. The results confirm the importance of coupling between turbulence components at different scales. It is thus extremely important to experimentally observe the HF-scale turbulence component and to understand the mechanism of the cross-scale nonlinear interactions in which they are involved and their impact on turbulent transport. For example, in the transport barrier of H-mode plasma22, the strong inhomogeneity of the radial electric field suppresses ion-scale micro-instabilities23,24,25. The HF-scale turbulence component might play a key role in controlling the structure of the transport barrier in plasma density.

We previously observed HF-scale turbulence fluctuations in the high-temperature plasma of the Large Helical Device (LHD)26 using a recently developed measurement method27 and confirmed that these fluctuations substantially contribute to anomalous electron transport28. In the present study based on our previous experimental progress, we report the results of experiments on cross-scale nonlinear interactions obtained by simultaneously measuring the cross-scale turbulence response at a given spatial location and also measuring the anisotropy of the HF-scale turbulence. A cross-scale bifurcation occurs in which the micro-scale turbulence is suppressed and simultaneously the amplitude of the HF-scale turbulence increases, and an abrupt change in the isotropy of the HF-scale component is also observed. Considering its fast time scale, this bifurcation is considered to be caused by the cross-scale interaction of the plasma turbulence.

Results

Experimental set-up

The experiments were conducted in the LHD. In the LHD, the magnetic field required for plasma confinement is entirely covered by the current supplied to the coils from the outside (i.e., no magnetic field from the current flowing through the plasma is required). The characteristics of plasma instability can thus be measured stably and reproducibly. Discharges with slowly increasing plasma density were used to investigate the onset of bifurcation phenomena and the response of turbulence under the following conditions: the magnetic axis position in the vacuum field was (Rax) was 3.55 m, the magnetic field strength (Bt) was 1.0 T, the helical coil pitch parameter (γc) was 1.2538, and the ratio of the quadrature field (Bq) was 100%. For plasma heating, 0.35-MW electron cyclotron heating (ECH) and 15-MW neutral beam injection (NBI) were used. To study the cross-scale interactions of micro-scale turbulence and HF-scale turbulence, simultaneous measurements at a given spatial location are required. Such measurements can be conducted by operating a millimeter-wave backscattering (BS) system27 in combination with a microwave Doppler reflectometer (DR) system29,30. For example, under the experimental conditions shown in Fig. 1a, the 57-GHz frequency channel of the DR system was used to observe micro-scale turbulence (k⊥ ρs ~ 1.5) near R = 4.4 m from the plasma cutoff layer, and the BS system was used to observe HF-scale turbulence (k⊥ρs ~ 7.0) at the same location. Furthermore, the anisotropy of turbulence in the HF-scale component was investigated via the simultaneous observation of various wavenumber components using the receiving antennas of the BS system at the positions and in the directions indicated by ks in Fig. 1c, d. The wavenumber vector for the turbulence observed at Receiver 1 is k1 and that observed at Receiver 2 is k2; these vectors have approximately the same magnitude and different directions. Anisotropy here refers to the anisotropy within the cross-section perpendicular to the magnetic field lines (i.e., the anisotropy with respect to the radial and poloidal directions).

Fig. 1: Coordination of multi-scale turbulence observations.
figure 1

a Radial profiles of electron cyclotron frequency fce and electron plasma frequency fpe. Probing wave frequencies of Doppler reflectometer (DR) and backscattering system (BS) are 57, 90, and 140 GHz, respectively. All waves use the ordinary polarized mode. b Magnetic flux surfaces in horizontal plasma cross-section with eye guide for each scattered wave ray. Relation between incident wave ki and scattered wave ks vectors at each BS receiving condition for (c) Receiver 2 and d Receiver 1. Here, kr is the wavenumber in the radial direction and kθ is the wavenumber in the poloidal direction. (e) Schematic diagram of backscattering antenna configuration. Probing millimeter waves with frequencies of 90 and 140 GHz are injected from launcher antenna. The scattered waves are received by two antennas namely Receiver 1 and Receiver 2. Receiver 1 receives 90-GHz backscattered waves at about 160 degrees, consists of two metal lenses. It is movable, allowing the observation position to be changed. Receiver 2 is located above this cross-section and receives the scattered signal at about 70 degrees. It has a parabolic antenna at the tip and focuses the beam to R = 4.4 m. In this experiment, Receiver 2 received millimeter waves at 140 GHz, making the wavenumbers received by the two antennas nearly the same. Here, the plasma shape is shown by the magnetic surface layers (pink region).

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Observation of bifurcation in turbulence state

Figure 2a–g respectively show the time evolutions of the heating power of ECH and NBI, plasma current, plasma stored energy, line-averaged electron density, electron temperature, and line emission of Hα. The plasma gradually increased in density after being started by ECH. The fluctuation of the Hα signal increased at around t = 3.82 s. At this time, the micro-scale turbulence intensity decreased rapidly and the HF-scale turbulence intensity increased rapidly, as shown in Fig. 2j, k, respectively. Here, we refer to the state before this phenomenon occurs as state A and the state after it occurs as state B. The temperature and density profiles just before and after the bifurcation into these two states show no significant change, especially in the edge region (reff/a99 > 0.8, where, reff is the effective minor radius and a99 is the average minor radius in which 99% of the electron kinetic energy is confined), as shown in Fig. 3. The relative variation between the two turbulence components is shown in Fig. 4. The micro-scale turbulence decreases and the HF-scale turbulence increases at the instant of bifurcation (<5 ms). This phenomenon is described in this paper as a bifurcation in turbulence states between those with lower and higher amplitudes of HF-scale turbulence.

Fig. 2: Temporal evolution of plasma parameters.
figure 2

The evolution of (a) ECH input power PECH and b tangential (red) and perpendicular (blue) NBI input power PNBI, c plasma current Ip, d stored energy Wp, e, h line-averaged electron density ne, f center electron temperature Te, g, i light emission of Hα line, j micro-scale turbulence intensity Iamp(micro-scale), and k HF-scale turbulence intensity Iamp(HF-scale). Graphs on right show magnified views of regions enclosed by green dotted line in graphs on left. State A is before the bifurcation, which occurred at around 3.82 s, and state B is after the bifurcation. The radial profiles at pre- and post-bifurcation times, indicated by blue and red arrows, are shown in Fig. 3.

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Fig. 3: Radial profiles of temperature and density before and after bifurcation.
figure 3

Radial profiles of (a) ion temperature Ti, b electron temperature Te, and c electron density ne before (state A) and after (state B) bifurcation. Error bars show the standard deviation of each parameter.

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Fig. 4: Response relationship of turbulence intensities before and after the bifurcation.
figure 4

Relationship between micro-scale turbulence intensity and HF-scale turbulence intensity. The numbers in the figure show time. The time period in which the bifurcation occurred (from 3.815 to 3.820 s) is indicated by the green dotted line.

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It should be noted that this bifurcation of turbulence states is associated with a change in the global properties of plasmas. As shown in Fig. 2g, the Hα signal, which indicates the loss of energy from the main plasma, shows a transition to grassy-edge-localized-mode (ELM)-like behavior31,32,33 at the onset of the bifurcation of the turbulence states. To categorize the time scales involved, the bifurcation phenomenon occurs in a few milliseconds or less, whereas macroscopic parameters such as the core density and temperature change more slowly (a few ten to a hundred milliseconds) and the burst-like response of the Hα signal is faster than the energy confinement time but longer than the bifurcation time scale. The intermittent fluctuation of Hα signal may be evidence of enhanced radial transport caused by HF-scale turbulence at the plasma periphery after the bifurcation.

Although the causal relationship is not clear in this figure, one might conjecture that the decrease in micro-scale turbulence may have caused the increase in HF-scale turbulence. In other words, the cross-scale interactions induced the changes in the intensity of micro-scale turbulence and that of HF-scale turbulence, which was deformed and suppressed by the micro-scale turbulent eddies. This hypothesis is supported by the observation of turbulence anisotropy, as described below.

Anisotropy in HF-scale turbulences

Theories have predicted that HF-scale turbulence might produce directional structures, such as long radially elongated formations (e.g., streamers) or eddies stretched by velocity shear. In this study, the issue of whether the structure of turbulence changes before and after the bifurcation in the turbulence state was investigated by simultaneously observing turbulent wavenumber components in two directions (k1 and k2), as shown in Fig. 1c, d. In other words, the anisotropy was investigated by simultaneously observing the turbulence intensity of wavenumber k components with the same magnitude but different directions. As shown in Fig. 5a, b, the intensity of the HF-scale turbulence increased as the micro-scale turbulence decreased at around t = 4.27 s for both directional components, as was the case in the previous section. Furthermore, the ratio of the intensities due to the observed differences in wavenumber direction changed, with the HF-scale turbulence being more isotropic after the bifurcation as shown in Fig. 5c. That is, the intensity ratio (k2/k1) exceeds unity when the turbulent eddies are radially elongated and becomes unity when the turbulent eddies are isotopic.

Fig. 5: Anisotropy of turbulence.
figure 5

Temporal behavior of HF-scale turbulence intensities in observed signals of (a) Receiver 2 and b Receiver 1. Here, the wavenumber k2 and k1 observed by Receiver 2 and Receiver 1 are wavenumber components in the directions shown in Fig. 1c, d, respectively. c Time variation of ratio of signal intensities in panels (a) and (b). The dotted ellipses are eye guides indicating the deformation of the HF-scale turbulent eddies. and the inclination of the propagation direction with respect to the magnetic surface before and after the bifurcation. After the bifurcation, the isotropy is recovered.

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When the micro-scale turbulence is strong before the bifurcation, the HF-scale turbulence is more anisotropic, and when the micro-scale turbulence is weak after the bifurcation, the HF-scale turbulence is more isotropic. This result qualitatively supports the theoretical model’s depiction of HF-scale turbulence being stretched by the background turbulent eddies (i.e., the dynamic shearing by micro-scale turbulence model)18.

Discussion

We now discuss the possibility that dynamic shearing by micro-scale turbulence as described above could actually occur. In addition, we make comparisons with the shear effect of background flow.

First, we evaluate the rate of dynamic shearing by micro-scale turbulence. From the mixing length estimation based on the gyro-Bohm diffusion model4, the amplitude of micro-scale turbulence can be expressed as

$$\frac{{e\phi }_{m}}{T} \sim \frac{{\rho }_{i}}{L}$$

and the decorrelation rate γm of micro-scale turbulence can be assumed to take the following value:

$${\gamma }_{m} \sim {C}_{m}{V}_{{drift}}/{\rho }_{i}$$

Here, e, ϕm, T, L, Cm, and Vdrift are the electric charge, electrostatic potential, temperature, scale length of the global gradient, coefficients smaller than 1, and E × B drift velocity, respectively. The electric field Em produced by the micro-scale turbulence eddies is estimated to be Em ~ ϕm/ρi. The shearing rate of HF-scale turbulence, V’m, induced by this micro-scale turbulence, is evaluated as follows:

$${V}^{\prime} _{m}\, \sim \,\frac{{V}_{{drift}}}{{\rho }_{i}}\,$$

Based on the plasma condition (Ti = Te = 200 eV, B = 0.65 T) at the time of turbulence bifurcation in this experiment, we obtain, L = 7.6 × 10–2 m, and ρi = 2.2 × 10–3 m.

Next, we investigate the decorrelation rate of the HF-scale turbulence obtained in the experiment, which is compared to the shearing rate caused by the micro component. Fig. 6 shows the auto correlation function of HF-scale turbulence during state A. The decorrelation rate can be estimated as the inverse of the decorrelation time, which is about 4 × 104 s–1. The estimated electric field produced by the micro-scale turbulence eddies Em is about 0.1–1 kV/m and its shearing rate is on the order of 104–105 s–1. Thus, the estimated decorrelation rate is well within the range where this dynamic shearing caused by micro-scale turbulence can be effective.

Fig. 6: Estimation of decorrelation rate.
figure 6

Auto correlation function of HF-scale turbulence in state A. The mean value calculated every 1 ms during 20 ms is shown by the red line and the standard deviation is shown by the error bar. The decorrelation time was defined as the time to the first peak (2.5 × 10–5 s) and its reciprocal was estimated as the decorrelation rate.

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This argument is compared to the suppression by a background DC electric field. When the radial electric field Er changes and Er × B flow shear V’E×B becomes larger than γm, the micro-scale turbulence, which acts as a trigger for bifurcation is considered to be suppressed23. However, the effect of this radial electric field on the HF-scale turbulence is relatively small because it needs to be (ρi/δ) times stronger than that for the micro-scale turbulence to suppress the HF-scale turbulence. Here, δ is the scale length of the HF-scale turbulence. Note that the Er distribution could not be measured accurately in this experiment; this is a topic for future work.

The results show that, the suppression mechanism by the micro-scale component is highly plausible and important for the dynamics of the HF-scale component.

Conclusions

In this experimental study, we succeeded in simultaneously measured the micro-scale turbulence and the HF-scale turbulence, as well as the anisotropy of the HF-scale turbulence, at a given spatial location. The main results can be summarized as follows.

(i) A bifurcation in the turbulence state occurs between the micro-scale turbulence and the HP-scale turbulence. When this bifurcation occurs, the amplitude of the micro-scale component decreases rapidly and that of the HF-scale component increases rapidly. Before the bifurcation, the anisotropy of the HF-scale component is large (deformed by the micro-scale component). After the bifurcation, however, the anisotropy becomes smaller. This turbulence bifurcation qualitatively supports the theoretical depiction18. Associated with this turbulence bifurcation, the plasma shows the change to grassy-ELM-like feature.

(ii) The decorrelation rate of the HF-scale component was estimated. The degree of deformation due to a vortex motion of the micro-scale component, as evaluated by the commonly used mixing length model, was found to be as large as or larger than the experimentally measured decorrelation rate of the HF-scale component. The suppression mechanism by the micro-scale component is highly plausible and important for the dynamics of the HF-scale component.

(iii) The experimental results show that it is essential to study cross-scale nonlinear interactions, including those of micro- and HF-scale components.

In addition, we found anotherphenomenon involving seesaw oscillation of the turbulence intensity at both scales in experiments with lower density and stronger electron heating than in this study, which will be reported in a future paper. In the near future, we plan to increase the number of spatial observation points and the number of observed wavenumber components to obtain wavenumber spectra and measure the radial electric field with high precision to develop a more detailed understanding of physical phenomena.

Methods

Large helical device

The Large Helical Device is a heliotron type device for the magnetic confinement of high-temperature plasmas. The representative major and minor radii of the torus plasma are 3.75  and ~0.6 m, respectively. The confinement magnetic field is mainly produced by the external helical coils; therefore, the plasma current need not be maintained. The plasma shape can be controlled by the quadrupole magnetic coils. The percentage cancellation of the quadrupole component, Bq, is less than 100% when the plasma cross-section becomes elongated vertically. The plasma shape can also be controlled by the helical pitch parameter defined as γc = (m/l)(ac/Rc), where m, l, ac, and Rc are the toroidal pitch number, poloidal pitch number, and average minor and major radii of the helical coils, respectively. For an LHD-type (l = 2, m = 10) heliotron system, γc corresponds to the inverse aspect ratio of the helical coils (ac/Rc) and γc = 1.2538 is the standard configuration. The magnetic field strength is up to 2.8 T. The device can be used to perform experiments in a variety of magnetic field configurations. The LHD features a total of five gyrotrons with frequencies of 56, 77, and 154 GHz for electron cyclotron heating. In this experiment, only the 56-GHz gyrotron was used for the plasma startup. The plasma was also heated by five neutral beams: three were tangentially injected and two were perpendicularly injected. Vertical neutral beam injection was also used to diagnose the ion temperature Ti, perpendicular velocity Vp, and radial electric field Er measurements. The total port-through power was ~20 MW. In this experiment, the plasma density was 1–4 × 1019 m−3 and the central temperature was about 1 keV.

Millimeter-wave back-scattering measurement

A 90-GHz (W-band) and 140-GHz (D-band) millimeter-wave backscattering system was used to measure HF-scale turbulence characterized by collisionless skin depth and the electron gyroradius ρe. This system is crucial for accurately detecting the intensity of the scattered signal, which correlates with the square of the fluctuations in electron density within the plasma34,35. The system consists of one transmitting antenna and two receiving antennas placed inside the LHD vacuum vessel, as shown in Fig. 1(e). To achieve high spatial resolution in these measurements, one of the receiving antennas had collinear focusing optics with metal lenses and the other had focusing optics with a rotating parabolic mirror. This setup kept the beam diameter below 30 and 50 mm, respectively. The collinear antenna (Receiver 1) design facilitates the adjustment of the observation position from the plasma core to its edge by means of a remote steering mechanism. The scattering volume’s estimated size is approximately 100 mm at the plasma edge. This dimension corresponds to a length of approximately 0.1 in terms of reff/a99. The receiving antenna with a parabolic mirror (Receiver 2) is fixed to the vacuum vessel to observe at R = 4.4 m. In addition, the millimeter-wave heterodyne detection circuit within this system incorporates a modulation function. This function is pivotal for distinguishing and identifying noise components in the scattered signal. These noise components are primarily attributed to the cyclotron radiation emitted by electrons from the plasma, making this function essential for precise signal analysis.

Microwave Doppler reflectometer

Doppler reflectometry (DR)35,36,37 can measure density fluctuations with sensitivity to ion-gyroradius ρi scale turbulence. The perpendicular flow velocity, radial electric field, and perpendicular wavenumber spectrum within the range k⊥ = 2–20 cm-1 as determined by the Bragg condition can be simultaneously observed with high wavenumber and spatial resolution. Since the observation position is determined by the position of the cutoff layer corresponding to the microwaves emitted into the plasma, to measure turbulence at various positions, the LHD applies a multi-frequency channel DR system that uses the frequency comb method. This system has sufficient time resolution and allows simultaneous multi-point spatial measurements. At the same toroidal position as that of the BS, both Ka-band and U-band DR systems are operated, allowing observation of the same plasma cross-section.