1 INTRODUCTION

To achieve the conditions of the Lawson criterion in thermonuclear facilities for controlled thermonuclear fusion (CTF), it is necessary not only to extensively increase the dimensions of the facilities, but also to improve the physical characteristics of confinement: an increase in the electron density of the plasma ne, the relative plasma pressure β, and the energy lifetime τE. These parameters may change abruptly when switching to the improved confinement (L–H) mode.

Since the discovery of the H-mode in the ASDEX tokamak [1], a wide variety of modes with improved confinement have been found in various experiments. In stellarators, the H-mode was first demonstrated in [2], and a significant increase in the plasma energy W was demonstrated, e.g., in [3]. The generally recognized features of the L–H transition are the decrease in the level of fluctuations in the edge parameters of the plasma (density, potential, electric field, particle flux) and radiation intensity Hα, as well as an abrupt increase in the plasma energy W and the parameter β.

It is well known that low- and moderate-order rational magnetic surfaces can play an important role in confining the plasma in toroidal CTF facilities. For example, in stellarators with a small shear (angle of rotational transformation profile), better confinement conditions can be achieved if the rotational transformation is close to but below the rational value [4]. In systems with a large shear, the existence of low-order rational magnetic surfaces can provide a reduction in transport [3].

As regards low- and moderate-order rational magnetic surfaces at the plasma boundary, their most typical consequence is the development of edge localized modes (ELMs). ELMs are short periodic perturbations of the plasma front, which are observed in many experiments in tokamaks and stellarators. Various magnetohydrodynamic (MHD) instabilities are considered as the driving mechanism of the ELM activity (e.g., ballooning, interchange and peeling modes [5]).

The appearance of ELMs was also noted in the MHD-resistant TJ-II heliac [6]. To explain this effect, it was necessary to consider drift oscillations within the two-fluid hydrodynamics, and the explanation was found taking into account the interaction of drift modes with Alfvén and acoustic waves [7]. That is, it should be noted that the physical phenomena that affect the transport transitions can be very complex, and, first of all, an analysis of the characteristic frequencies and increments of perturbations is required.

The transition to a mode with improved confinement is necessarily accompanied by a decrease in turbulent transport, i.e., stabilization of plasma instabilities. However, the roles played by instabilities can be quite varied. It is well known that even a transition between closely related states cannot occur without some kind of trigger. In particular, this can be a local perturbation of the profile of plasma parameters as a result of the development of instability, as well as changes in the particle flux from the chamber wall [8]. The L–H transition in the stellarator accompanied by a noticeable increase in the plasma energy and the evolution of edge perturbations of the parameters, was observed, e.g., in TJ-II [9]. In most toroidal facilities, the transport transition is not accompanied by pronounced bursts of the MHD activity, but, e.g., at the HL-2A tokamak, an internal MHD mode was found as a precursor of the L–H transition [10]. Transport transitions in toroidal facilities usually exhibit features typical of bifurcations [11]. This implies the existence of two stable stationary states.

The work is organized as follows: Section 2 briefly describes the experimental facility and the characteristic features of plasma confinement, Section 3 shows the experimental data and the results of the wavelet analysis. Possible theoretical explanations for the observed effects are outlined in Section 4. Conclusions are presented in Section 5.

2 EXPERIMENTAL

The L-2M facility (Prokhorov General Physics Institute, Russian Academy of Sciences) is a classic stellarator with a large magnetic field shear and a flat geometric axis. The total number of periods of the helical field N = 14, number of starts l = 2, major radius of the geometric axis of the torus R0 = 1 m, average separatrix radius ap = 0.115 m, maximum (resonant) magnetic field on the axis B0 = 1.34 Т. We take x = a/ap as the relative coordinate of the minor radius, where a is the average radius of an elliptical magnetic surface. See [12] for details on the facility. The plasma is created and heated out using microwave radiation in the power range of 0.1–1 MW by the method of electron cyclotron resonance at the second harmonic of the gyrofrequency (75 GHz) [13].

The experiments described here were carried out at 〈β〉 ≤ 0.25%, where 〈β〉 is the ratio of the gas-kinetic pressure of the plasma to the magnetic pressure, averaged over the volume of the plasma column. In L-2M, the plasma is almost currentless, the low current Ip ~ 1 kA (bootstrap) cannot noticeably change the geometry of magnetic surfaces or affect the stability conditions.

To analyze possible peripheral perturbations, one needs to understand the position of resonant surfaces near the separatrix, internal and external ones. The dependence of the angle of rotational transformation μ on the average radius of magnetic surfaces and the position of low-order rational magnetic surfaces are shown in Fig.1. The diagnostic complex of the facility makes it possible to determine the global and radial plasma parameters (W, Te(a), ne(a)), as well as the fluctuating electric field and density values using probe diagnostics, reflectometry, and scattering of gyrotron microwave radiation by density fluctuations [14]. Figure 2 shows a scheme of high-frequency diagnostics in the facility.

Fig. 1.
figure 1

Radial distribution of rotational transformation angle μ at β = 0.2% and position of rational magnetic surfaces.

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Fig. 2.
figure 2

3D model of the facility and position of high-frequency diagnostic systems: magnetic probes (st), Langmuir probes (Vf), and reflectometry (Refl).

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Numerical methods of the Fourier analysis and wavelet analysis (based on Morlet and Haar basic wavelets) were used for the spectral and correlation analysis of high-frequency signals [15].

3 RESULTS

The work presents studies of the characteristics of plasma pulses with transport transitions, which were observed in two experimental sessions. We analyzed pulses no. 16081 (with transition) and no. 16082 (without transition) at the microwave heating power P = 450 kW and medium linear density ne = 2 × 1019 m–3. Pulse no. 57442 with a fast transport transition (FTT), for which the structure and evolution of the peripheral electric field were previously analyzed in detail: P = 180 kW and ne = 1.6 × 1019 m–3, was chosen for the comparative analysis [16].

Figure 3 shows oscillograms of ne, Hα and floating potential of the Langmuir probe Vf at a position 1 cm deep from the separatrix for pulse no. 16081. It can be seen that the discharge has features of a spontaneous L–H transition; at time of about 57 ms, there is a sharp increase in ne, a more gradual decrease in Hα, and a decrease in the amplitude of the electric field oscillations. It should be noted that there are no changes in the main set parameters: microwave heating power P and gas purge mode, i.e., the transitions are not forced. Figure 3d shows the power of the Fourier spectrum of the signal Vf, where there is a frequency band of 10–30 kHz, which almost disappears after the transition. For comparison, the parameters of a similar pulse no. 16082, in which no transition is observed, are presented (Fig. 4).

Fig. 3.
figure 3

Oscillograms of plasma parameters in the pulse no. 16081 with transport transition: (a) average plasma density ne, (b) radiation intensity signal Hα, (c) floating potential Vf of the Langmuir probe, (d) Fourier spectrum power Vf. The vertical line marks the time of the beginning of the transport transition.

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Fig. 4.
figure 4

Oscillograms of plasma parameters in the pulse no. 16082 without transport transition. Parameters are similar to those shown in Fig. 3.

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Further, the experimental results of other diagnostics, which record fluctuations in plasma parameters, were analyzed: magnetic (Mirnov) coils and Doppler reflectometry. Figure 5 shows the 3D Fourier (3d FFT) spectrum and coherence between magnetic probe signals. The magnetic probes are separated by 13/14π along the torus (along the toroidal coordinate φ) and are in the same position along the poloidal coordinate θ. The color scale corresponds to different Fourier coefficient and coherence coefficient values. One can see the appearance of a disturbance with f ~ 80 kHz at the 56th ms, its disappearance and the appearance of a disturbance with f ~ 30 kHz during the transport transition. In a pulse without a transport transition (pulse no. 16082), the mode with  f ~ 80 kHz also appears at 56th ms and decays as the plasma cools down at 61st ms. The question about the origin of the appearance of the mode at 56th ms is important, since there are no corresponding features on the density and energy content oscillograms at this time.

Fig. 5.
figure 5

Fourier spectrum of magnetic probe signals in pulses no. 16081 with transport transition (a) and no. 16082 without transport transition (b) and coherence between the signals of magnetic probes for them (c, d).

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Correlations between different magnetic probes show an increase in cross-coherence at f ~ 30 kHz. The 80 kHz band on some probes disappears during the transition, and it is transformed on others. Previously in [17], the attenuation of modes with a frequency f ~ 35 kHz and a structure m = 5, n = 4 was found in the case of FTT; m and n are the toroidal and poloidal wave numbers, respectively. There is no corresponding rational magnetic surface inside the separatrix, perhaps the perturbation is due to external resonance.

We consider pulse with a fast transport transition (FTT) no. 57442 (Fig. 6). A stable disturbance with a frequency of about 30 kHz was recorded on magnetic probes, which is attenuated during the transport transition. Previously, a mode with f ~ 30 kHz was found in pulses with FTT; upon transition, it was transformed into an electromagnetic mode with a frequency of f ~ 10 kHz, which is characteristic of a ge-odesic acoustic mode (GAM) but has a three-dimensional geometry. Before the transition, this frequency is on the coherence between two signals Vf from different Langmuir probes or Vf and a fluctuating magnetic field. After the transition, it disappears there and appears on the coherence between two magnetic probes [17].

Fig. 6.
figure 6

Signals of magnetic probes in the pulse no. 57442 with FTT (a, b), their Fourier spectra (c, d), and coherence between signals (e).

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In addition, the coherence pattern between the two magnetic probes (Fig. 6) shows the disappearance of the 130 kHz perturbation and the appearance of the 110 kHz perturbation. Figure 7 shows the results of measurements on a Doppler reflectometer [18]. For the mode with a transport transition, one of the diagnostic signals during the plasma pulse and the change in the wavelet spectrum plotted from the basic Morlet wavelets are given.

Fig. 7.
figure 7

Reflectometer signal (a) and its Fourier spectrum (b).

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4 DISCUSSION

We consider the characteristic times of transient processes at different plasma parameters during a pulse with a transport transition. The drop in the Hα intensity begins at the 56th ms, the flash on magnetic probes occurs at the 56.5th ms, the decrease in fluctuations and the beginning of the growth of ne and the fluctuations of Vf occur at the 57th ms, the decrease in the envelope of Vf and the termination of the growth of ne occur at the 58th ms. The delay between the current and potential during the transient process is 2 μs [19], which is logical, since the rearrangement of the charges leads to the transformation of the electric field.

The coherence between the signals of high-frequency diagnostics has a flash character. The characteristic frequencies of perturbations transforming during transient processes are in the bands  f ~ 30, 70, 120 kHz.

Local MHD modes, drift instabilities and tearing modes associated with non-inductive current, as well as their possible combinations, can be considered as mechanisms that stimulate transient processes. In principle, near-boundary internal and external modes (with respect to the separatrix) can be responsible for the release of energy near the plasma boundary. The study of the near-boundary plasma using Langmuir probes made it possible to eliminate the influence of the internal mode m = 4, n = 3. The perturbation of the external mode can be associated with the behavior of particles in the stochastic layer, in particular, with the input of particles of the boron-carbon mixture, which covers the inner surface of the chamber, into the edge plasma. Therefore, the problem of the cause-and-effect relationship of the phenomena accompanying the spontaneous transition process is important.

Previously, local MHD perturbations were considered as a mechanism stimulating transient processes. For the case considered in this paper (ECR discharges without induced current), the plasma is stable with respect to large-scale ideal MHD modes. The vacuum magnetic configuration has a maximum of the averaged magnetic field B, which has a strong destabilizing effect on the plasma configuration even at small β (the so-called magnetic hump), but there is a magnetic well in the central part of the plasma column that stabilizes large-scale internal MHD modes (interchange and ballooning ones). In the outer region, their stability is ensured by the high shear. However, there is a zone of a magnetic hillock at the edge of the plasma column (x > 0.6), and in this region, resistive permutation MHD modes are unstable, which are not stabilized due to the shear [20]. A model of external local MHD instabilities (peeling modes) was proposed as a FTT trigger, at which there occurs a fast short-term reset on a dW/dt diamagnetic signal similar to ELM [21].

The ideal peeling mode is rigidly tied to the plasma boundary but the main perturbation has toroidal satellites. For the mode m = 5 and n = 4, the satellite is the mode m = 6 and n = 4 localized on the surface with μ = 2/3. Unstable perturbations are almost constant along the magnetic field line. In this case, numerous regions with favorable and unfavorable curvature on the field line are averaged. Thus, due to the satellites, the characteristic size of the perturbation increases to a thickness of about 1 cm, which is taken as the maximum penetration depth of the Langmuir probe into the plasma from the separatrix.

It is also necessary to analyze the possible influence of drift instabilities. We restrict ourselves to consideration of ion temperature gradient (ITG) and electron temperature gradient (ETG) modes. In classical works, the characteristic parameters of ITG and ETG modes are defined as follows: for ITG, the frequency ω ~ \(\omega {\kern 1pt} {{{\kern 1pt} }_{{*{\text{i}}}}}\), the transverse (with respect to the magnetic field) wave number k⊥ ~ 1/ρTi; for ETG, ω ~ \(\omega {\kern 1pt} {{{\kern 1pt} }_{{*{\text{e}}}}}\), k⊥ ~ 1/ρTe, where \(\omega {\kern 1pt} {{{\kern 1pt} }_{{*{\text{i}}}}}\) and \({{\omega }_{{{\kern 1pt} {\kern 1pt} *{\text{e}}}}}\) are the diamagnetic drift frequencies of ions and electrons, ρTi and ρTe are the cyclotron radii of ions and electrons calculated in accordance with their thermal velocities [22].

Depending on the conditions of specific magnetic configurations, the parameters of the modes and the direction of their propagation can differ significantly from those given above. The boundary between the ranges of ion and electron modes is not always distinct, since their overlap is possible [23].

It can be seen from the analysis of reflectometry measurements that the Doppler shift estimated from the maximum of the spectra changes by almost an order of magnitude at the steady stage of the discharge. This may indicate that the Doppler frequency shift is determined not only by the poloidal rotation velocity, but also by its addition to the phase velocities of turbulent fluctuations (provided that the velocities are on the same order of magnitude). It was found that the ITG and ETG modes can be observed simultaneously in the L-2M stellarator [24].

It is shown theoretically that the development of ETG and ITG instabilities at the stellarator plasma boundary for modes with high heating power and the formation of a dip in the density profile at the center of the plasma column (pump-out effect) are possible. Local measurements of spectra using Doppler reflectometry diagnostics make it possible not only to measure the plasma rotation velocity, but also to describe the development of low-frequency plasma instabilities.

The characteristic frequencies of various perturbations obtained analytically are as follows:

1. Estimate of the characteristic GAM frequency by the formula \({{\omega }_{{{\text{GAM}}}}} = (2\gamma p{\text{/}}\rho R_{0}^{2})(1 + {{\mu }^{2}}{\text{/}}2)\), where p and ρ are, respectively, plasma pressure and density, gives f  ≈ 11 kHz (at the adiabatic exponents γ = 1 for electrons and γ = 5/3 for ions, Te + Ti = 20 eV, μ = 0.7) [15].

2. Two-fluid MHD taking into account the diamagnetic drift gives the perturbation frequencies \({{f}_{{2d}}} = {{k}_{{ll2}}}{{V}_{{Ti}}}{\text{/}}2\pi \) ≈ 9 kHz and \({{f}_{{3d}}} = {{k}_{{ll3}}}{{V}_{{Ti}}}{\text{/}}2\pi \) ≈ 136 kHz, where \({{k}_{{ll2}}}\) and \({{k}_{{ll3}}}\) correspond to toroidal and helical satellites of the main mode.

3. Estimate of the characteristic frequencies from reflectometry measurements in the L-2M stellarator is about 1 MHz for the ETG instability and about 100 kHz for the ITG instability [22].

4. In stellarators, a three-dimensional acoustic mode with \(\omega \sim {{\omega }_{{{\text{GAM}}}}}N\mu {\text{/}}l\), where N is the number of periods of the magnetic field, l is the number of starts of a stellarator, may exist along with the GAM on the average curvature. The estimate for L-2M gives f ≈ 110 kHz.

For magnetic measurements, two characteristic fluctuation frequencies were observed: 75 and 32 kHz. The value closest to that measured at the characteristic frequency f ~ 75 kHz is obtained if we calculate the characteristic frequency of the geodesic acoustic mode, where the mean curvature of the magnetic field is used as the curvature: \(f = {{C}_{{\text{s}}}}\sqrt {2 + {{\mu }^{2}}} ~\) ~ 64 kHz, μ ≈ 0.47 is the angle of rotational transformation, where the speed of sound \({{C}_{{\text{s}}}} = \sqrt {\gamma p{\text{/}}\rho } \) ≈ 2.7 × 107 cm/s.

In the frequency range f ~ 30 kHz, simple analytics did not allow us to obtain estimates; perhaps, the consideration of a combination of characteristic processes is required. As for the possible effect of tearing instabilities, it was found in [25] that the profile of ne to a depth of 1 cm from the separatrix is monotonic, which may indicate the absence of magnetic islands in this region (this does not exclude their formation in a deeper region). However, a more detailed theoretical analysis is required.

In future works, it is planned to present an analytical study of kinetic instabilities, and the possibility of their combination with MHD (peeling and tearing) modes.

5 CONCLUSIONS

The evolution of fluctuating signals of electrostatic and magnetic probes and Doppler reflectometry in plasma on the L-2M stellarator is presented. The L‑2M facility is a toroidal magnetic trap in which plasma is created and heated by high-power pulsed microwave radiation. Pulses with transitions to improved confinement modes accompanied by an increase in plasma density and pressure and restructuring of the peripheral electric field were analyzed. Spectral analysis of signals is carried out using Fourier analysis and various wavelets. The possible influence of magnetohydrodynamic (MHD) and kinetic instabilities on the development of transient processes is considered, and the characteristic frequencies of various disturbances in modes with transport processes are estimated. It is shown that the peeling mode can be the origin of the fast transport transition (FTT), and to explain the other type of transitions, it may be necessary to involve models of kinetic instabilities or their combination with local MHD modes. This theoretical study is planned to be carried out in the future.