Review of Advanced Implementation of Doppler Backscattering Method in Globus-M

Abstract

Doppler backscattering (DBS) is a microwave diagnostics method typically used to study the plasma rotation velocity. Apart from conventional techniques, more advanced forms of DBS implementation were suggested on Globus-M. More specifically the study of a variety of oscillating processes was performed using DBS. In this review we present a detailed description of all of the methods and techniques employed in Globus-M alongside results obtained using DBS in all the years up until the shutdown of the tokamak. These include research similar to that done on other devices into the properties of such phenomena like geodesic acoustic modes or limit cycle oscillations, along with innovative works regarding the detection and investigation of Alfven eigenmodes and filaments that were the first of their kind and that provided important and novel results. Apart from that, the specific aspects of DBS application on a spherical tokamak are discussed. An in-depth look into the gradual change and improvement of the DBS diagnostics on Globus-M is also presented in this paper.

1. Introduction

Doppler backscattering (DBS) is a microwave diagnostics method applied on many magnetic confinement fusion devices typically with the aim to study the plasma rotation velocity [1,2,3,4,5,6,7,8,9]. Since the L-H transition was shown to be caused by the suppression of turbulent plasma perturbations by the shear of the plasma drift velocity in the radial electric field (E × B velocity) [10], DBS was successfully applied to study this transition. In the tokamak ASDEX Upgrade, data for the analysis of L-H transitions with low density were obtained using this method [11]. A comparison of neoclassical calculations for the radial electric field and DBS data was made, which showed a good agreement between the two methods for different modes of operation of the tokamak. The TCV tokamak used a heterodyne V-band Doppler backscattering diagnostic system [12]. The first results show that the perpendicular rotational velocities obtained with DBS are consistent with the estimates of the poloidal rotation obtained from the charge exchange recombination spectroscopy diagnostics. A radial profile of perpendicular velocity was successfully obtained on the LHD tokamak using a Doppler reflectometer system [13]. The radial electric field was extracted from measurements of perpendicular velocity. In the work [14] the rotation characteristics of the plasma in the tokamak HL-2A before and after supersonic molecular beam injection were also analyzed. It was shown that an SMBI pulse can reduce the Doppler shift frequency, which corresponds to the flattening of the electric field. In a series of experiments on the EAST tokamak using two separate DBS systems, one for the Q-band (33–50 GHz) and the other for the V-band (50–75 GHz), Doppler shifted signals were obtained and radial profiles of perpendicular velocities were computed for the L- and H-mode [15]. In the Globus-M2 tokamak, the first results of the Doppler backscattering diagnostics during a discharge with the L-H transition show that the transition process is linked to the deformation of the poloidal rotation velocity profile [16]. This diagnostic made it possible to conduct studies in the hotter and more central plasma regions, which is a useful addition to the velocity and field data obtained by other methods.

Apart from these conventional techniques more advanced forms of DBS implementation were suggested. More specifically the study of a variety of oscillating processes was performed using DBS. For example, for the investigation of the high-frequency branch of zonal flows—the geodesic acoustic mode—the DBS diagnostics were used on JET [17,18], ASDEX Upgrade [19], HL-2A [20], EAST [21], DIII-D [22], TUMAN-3M [23], and other devices [24]. Doppler backscattering has also proved useful for the study of limit cycle oscillations whose typical frequency is significantly lower than that of the geodesic acoustic mode frequency. Such research was carried out in the ASDEX Upgrade tokamak [25], DIII-D [26,27], and HL-2A [28,29].

In this review we aim to present a detailed description of all the methods and techniques employed in Globus-M in regard to DBS diagnostics used to study all the various phenomena that occur in plasma during a discharge. A multitude of results that had been obtained using DBS in the last 10 years up until the moments of the tokamak’s ultimate shutdown in 2017 are revisited and described. These include the investigation of such phenomena like geodesic acoustic modes or limit cycle oscillations using similar research practices that had been previously adopted on other devices. Apart from that, innovative works are reexamined regarding Alfven eigenmodes and filaments that were the first of their kind and had provided important and novel results for the plasma physics community. Furthermore, the specific aspects of DBS application on a spherical tokamak are discussed and highlighted. An in-depth look into the gradual change and improvement of the DBS systems on Globus-M from a one-frequency homodyne detection system to several multi-frequency systems can also be found in this review.

2. Doppler Backscattering on Globus-M

2.1. DBS Systems

Doppler backscattering (DBS), also known as Doppler reflectometry, is a type of Thomson’s collective microwave scattering. The method is based on recording the backscattered microwave radiation with an oblique incidence of the microwave beam. Scattering mainly occurs near the cutoff of the microwave beam on plasma fluctuations with a selected wave vector k_⊥, which satisfies the Bragg condition for backscattering—k_⊥ = 2 k_i. Here k_i is the wave vector of the incident wave in the cutoff region and it is oriented in the direction of the electron or ion diamagnetic drift. When scattering fluctuations move in the diamagnetic direction with a certain velocity V_⊥, a Doppler frequency shift of backscattered radiation appears—Δω_D = k_⊥ V_⊥. The Doppler frequency shift Δω_D needed for the calculation of the perpendicular velocity V_⊥ can be obtained as the position of the ‘center of gravity’ of the complex signal I(t) + iQ(t) spectral density, or the derivative of the phase of the complex signal [30]. The sign and magnitude of the shift allow us to determine the sign and magnitude of the velocity of fluctuations in the diamagnetic direction V_⊥.

The first implementation of DBS on a spherical tokamak was carried out on Globus-M [31]. There are specific requirements for DBS application in a spherical tokamak that is characterized by a significant pitch angle (more than 20°) at the low magnetic field side. To satisfy the Bragg condition, the antenna has to be tilted not only poloidally but also toroidally. To evaluate the required antenna tilt angles, ray tracing of the incident beam was performed for the actual three-dimensional (3D) geometry of the Globus-M flux surfaces [32]. To determine the investigated area of each DBS channel, experimental electron density profiles obtained by Thomson diagnostics and magnetic surface reconstruction data obtained using the EFIT code were used. An example of the performed ray tracing is presented in Figure 1. The locations of the ports available for the installation of the DBS systems are also indicated. The first Globus-M DBS system was based on a monostatic antenna scheme, which allowed one to probe the plasma by O-mode microwaves in the frequency band 18–26 GHz (system #1 in

Table 1. DBS systems on Globus-M.

DBS Systems	#1	#2	#3
Number of systems	1	2	1
Number of frequencies	1	1	4
Frequency values, GHz	18–26	27–38	20, 29, 39, 48
Waveguide	WR-42	WR-28	WR-42, WR-28
Wave propagation	O-mode	O-mode	O-mode
Poloidal tilt angles, °	6–9	6–9	6–9
Toroidal tilt angles, °	2–5	2–5	2–5
Probing beam angle, °	−30, −45	45, −30, −45	−30
Position	port #2,3	port #1,2,3	port #2
Cut-off radii R, m	~0.55–0.58	~0.53–0.58	~0.51–0.59
Field of study	geodesic acoustic modes, filaments	geodesic acoustic modes, filaments, Alfven eigenmodes	geodesic acoustic modes, limit cycle oscillations, filaments, Alfven eigenmodes, turbulence

). It was possible to change the incident frequency from shot to shot. The tilt angles of the incident beam were chosen as 7° in the poloidal and 3° in the toroidal direction. The relevant wavelength of the scattering plasma fluctuations was in the range of 1.1–1.8 cm. The cutoff position was in the vicinity of the separatrix at high q-values. The radial resolution of the method was estimated to be about 0.5 cm. Dual homodyne detection was employed to receive the backscattered radiation [1].

The DBS system #1 was used successfully to study peripheral transport processes as well as to obtain measurements near the separatrix. Due to the usefulness of the diagnostics, the physics research program on the Globus-M tokamak required the expansion of the measured region into the area of the formation of the pedestal. Accordingly, an additional DBS system was developed so as to probe and record the scattered signal at larger radii (system #2 in Table 1). The microwave circuit of system #2 was similar to the one in system #1 with the exception of a higher frequency probing range of 27–38 GHz. The same antenna that was used for system #1 to probe the plasma was employed on system #2. The antenna was connected to this DBS system using the WR-28 to WR-42 waveguide transition. Because only one port (port #2) located in the equatorial plane was initially allocated for the DBS diagnostic, only one of the two available systems could be applied during a discharge.

Subsequently, two additional ports (ports #1 and #3) were made available for the development of Globus-M DBS diagnostic that were located at the same toroidal angle, but several centimeters, respectively, above and below the equatorial plane. Additional antenna systems were installed in these ports, and one more DBS system similar to system #2 was acquired. The simultaneous use of several DBS systems, spread out in a poloidal direction, made it possible to develop poloidal correlation Doppler reflectometry and to determine the poloidal scale of the developing plasma instabilities.

However, the use of systems #1 and #2 made it difficult to construct radial profiles of the plasma processes under investigation by means of changing the probing frequency from discharge to discharge. This method required precise repetition of plasma discharge parameters. The fluctuation of these very parameters made it difficult to accurately interpret the acquired results. Therefore, a DBS scheme with four fixed probing frequencies was developed. The 4-channel DBS system in Globus-M comprised two × two-channel DBS microwave schemes with different probing frequencies [33]. Two pairs of fixed frequencies were chosen—20, 29 GHz and 39, 48 GHz. Each frequency channel included a microwave circuit with dual homodyne detection. Two steerable antennas were used to probe the plasma by O-mode microwaves. Each of the antennas was used both as an incident and a receiving one, and were connected to a waveguide that allows only a certain range of wavelengths to pass through. One antenna was used for frequencies of 20 and 29 GHz, whereas the second one was used for the 39 and 48 GHz frequencies. The detection region covered a considerable radial interval of normalized small radii $\rho$ = 0.6–1 for a typical Globus-M discharge.

2.2. Data Analysis

Table 2. Methods applied to DBS signals to study various phenomena.

Method	Notes	Field of Study
Spectral analysis	connects the time domain and the frequency domain	geodesic acoustic modes, limit cycle oscillations, quasi-coherent fluctuations, Alfven eigenmodes, turbulences, filaments
Coherence analysis	provides information on how strongly two signals are correlated across a frequency range	geodesic acoustic modes, quasi-coherent fluctuations
Correlation analysis	information on radial correlation properties of turbulence	geodesic acoustic modes, limit cycle oscillations, filaments
Bicoherent analysis	frequencies at which nonlinear interaction is most effective	geodesic acoustic modes, limit cycle oscillations, quasi-coherent fluctuations

provides the methods of analysis of DBS data that allowed one to investigate and determine the characteristics of plasma processes such as geodesic acoustic modes, limit cycle oscillations, quasi-coherent fluctuations, Alfven eigenmodes, turbulence, and filaments.

Spectral analysis allows for the process of breaking down of any signal into its components at various frequencies, which is an important characteristic when it comes to turbulences and a large variety of instabilities. The Fourier transform (fast Fourier transform) provides this connection between the time domain and the frequency domain. There are various kinds of spectral analysis of the DBS signals available. They were applied to calculate the signal spectrum that describes the distribution of power into frequency components that the signal comprises. The fast Fourier transform is calculated in a window of a certain length that is chosen depending on the phenomenon being investigated. The window is then shifted by a time frame and the process continues. It was also possible to smooth these data so as to remove components of the background noise. Spectrograms were calculated as well in order to also have a visual representation of the spectrum as it changes in time. They are formed as the dependency of frequency on time while the power at each frequency at a given moment in time is represented by color. This is accomplished for the complex data of the DBS signals.

The complex DBS signal spectrum is calculated so as to investigate the Doppler shift and consequently the rotation velocity. The velocity spectrum reveals some oscillatory phenomena. The modulus of the complex DBS signal showcases the behavior of the fluctuation amplitude. The calculation of the coherence spectrum leads to the examination of the relation between oscillations of various DBS signals.

Correlation analysis of DBS signals can be used for obtaining information on radial correlation properties of turbulence and therefore on the turbulence spectrum with respect to radial wave numbers [34]. However for us the main goal was the detection of periodic components in the Doppler backscattering radiation, as well as the definition of the characteristics of such periodic processes. For this purpose, the calculation of cross-correlation functions (CCFs) between velocities measured at different radii or between amplitudes of DBS signals at different radii, and between the amplitude and velocity obtained at the same radius was performed. It was also possible to obtain correlation functions between the DBS signals and other diagnostic signals.

The processes of coupling between spectral components of interacting waves could be revealed via bicoherence analysis. Cross-bicoherence as a measure of phase coupling between three spectral components of signals is described by the following expressions:

$$b^2\left(f_1,f_2\right)=\frac{{\left|\langle Y_k^{*}\left(f_3\right)Y_i\left(f_1\right)Y_j\left(f_2\right)\rangle \right|}^2}{\langle {\left|Y_k\left(f_3\right)\right|}^2\rangle \langle {\left|Y_i\left(f_1\right)Y_j\left(f_2\right)\right|}^2\rangle };f_3=f_1\pm f_2$$

where Y_i(f), Y_j(f), Y_k(f) are spectra of amplitude of the complex output signal of the IQ detector or spectra of E × B velocity determined from the Doppler frequency shift.

3. Application Results

3.1. Geodesic Acoustic Mode

DBS was implemented to study geodesic acoustic modes (GAMs) on the Globus-M tokamak in the Ohmic phase of discharges at relatively low plasma density (n ≈ (2–3) × 10¹⁹ m⁻³) when the toroidal ion drift was directed away from the X-point [35,36,37]. All DBS systems (see Table 1) and in some experiments even their combinations were used to obtain information about various properties of GAMs presented in Table 3 (see the end of Section 3.1). GAMs were identified through spectral analysis of the perpendicular velocity V_⊥(t) = V_E×B(t) + V_phase(t). Since GAMs lead to oscillations of V_E×B(t), they can be detected by DBS. The measured Doppler frequency shift was obtained either from the spectrum shift of the IQ detector signal or as the phase derivative of the complex signal. GAMs manifest themselves as coherent oscillations in the frequency domain of 23–28 kHz in the case of deuterium plasma and around 36 kHz in hydrogen plasma [38,39]. These frequencies are within the large orbit drift width limit when compared with theoretical predictions [40].

The study of the temporal evolution of GAMs showed that they do not exist continually; rather, an intermittent increase and decrease of GAM amplitude was observed. There were two characteristic times for the GAM level changes [39]. The first one is the duration of a quasi-coherent GAM burst which was about 0.2 ms with an interval of 0.4 ms between bursts. This behavior corresponds to the well-known predator-prey model [41]. Another timescale is a very slow evolution of the GAM level with a period of about 5 ms. This is also the characteristic time of the evolution of the ion pressure gradient on Globus-M.

The GAMs velocity amplitude V_⊥(t) was recalculated to the radial electric field E_r assuming that the phase velocity V_phase(t) induced by GAM oscillations is relatively small. The amplitude of the electric field oscillations reached up to 3 kVm⁻¹, which sometimes exceeded the mean value of the radial electric field causing a velocity reversal at a certain phase of the GAMs oscillations [39].

Initially the GAM location was studied by varying the probing frequencies from shot to shot while all plasma parameters were kept identical. These experiments showed that GAMs exist in a very narrow layer of a few centimeters inside the separatrix [39]. The obtained radial profile of the GAM velocity amplitude is presented in Figure 2. A clear maximum and decrease in amplitude is observed in this several centimeter layer. The GAM frequency was also investigated in different locations and it was determined that no significant change of its values was present. Experiments with a four frequency DBS scheme confirmed the narrow localization of GAM [42]. Only the peripheral DBS channel could register GAM oscillations near the normalized minor radius 0.95, whereas the other channels did not detect oscillations exceeding the background level at the GAM frequency. This narrow area of GAM existence might be caused by the prevention of GAMs development by Landau damping induced by the steep q-value decrease in the edge region of Globus-M. The width of the GAMs existence layer was about 1 cm with the GAM frequency not changing significantly within this area [39]. Such observations correspond to the global GAM eigenmode nature which was predicted in work [43].

Poloidal correlation DBS using two microwave schemes with the cut-offs positioned at different poloidal angles (−30° and 45°) of the same minor cross-section was employed in Globus-M to study the poloidal structure of GAMs [32,44]. The zero phase delay between the GAM oscillations in the two simultaneously detected DBS signals was discovered by coherent spectrum analysis of the velocity oscillations, which is presented in Figure 3. The observation is consistent with an m = 0 mode structure of the GAM E × B flow as predicted by theory in [41].

Cross-bicoherence analysis of perpendicular velocity and amplitude of the complex DBS signal was successfully used to demonstrate the interaction between the GAM oscillations and plasma turbulence [32,44,45]. The results of this analysis indicate that a nonlinear interaction takes place that manifests itself in the appearance of the amplitude oscillation at the GAMs frequency f_GAM. This can be caused by the influence of turbulent Reynolds stress on zonal flows or GAMs [41].

Table 3. Properties of GAMs in Globus-M.

Investigated Property	Range of Values	Notes
Mode of operation	Ohmic heating regime	not detected in H-mode
f, kHz	23–28 (D) ~36 (H)	Gao formula, influenced by isotope effect [40]
Spectral peak width ∆f, kHz	2–6	consistent with the inverse of the GAM burst lifetime
Type	burst	Predator-prey model [41]
tlifetime, ms	0.2–0.4	determined as the time of damping of the velocity autocorrelation function
Modulation frequency fmod, Hz	300	assumption of relation to the evolution of the ion pressure gradient
Amplitude, kV m−1	up to 3	exceeds the mean radial electric field
Location R, m	0.56	related to the sharp decrease in the safety factor values followed by Landau damping [41]
Location width, cm	1	global GAM eigenmode nature ${\lambda }_{\mathrm{GAM}}={\mathrm{L}}_{\mathrm{T}}^{\frac{1}{3}}{\rho }_{\mathrm{i}}^{\frac{2}{3}}$ [43]
Poloidal structure	m = 0	Linear dynamics of zonal flow modes [41]
Non-linear interaction with turbulences	present	the influence of turbulent Reynolds stress on zonal flows or GAMs [41]

Table 3. Properties of GAMs in Globus-M.

Investigated Property	Range of Values	Notes
Mode of operation	Ohmic heating regime	not detected in H-mode
f, kHz	23–28 (D) ~36 (H)	Gao formula, influenced by isotope effect [40]
Spectral peak width ∆f, kHz	2–6	consistent with the inverse of the GAM burst lifetime
Type	burst	Predator-prey model [41]
tlifetime, ms	0.2–0.4	determined as the time of damping of the velocity autocorrelation function
Modulation frequency fmod, Hz	300	assumption of relation to the evolution of the ion pressure gradient
Amplitude, kV m−1	up to 3	exceeds the mean radial electric field
Location R, m	0.56	related to the sharp decrease in the safety factor values followed by Landau damping [41]
Location width, cm	1	global GAM eigenmode nature ${\lambda }_{\mathrm{GAM}}={\mathrm{L}}_{\mathrm{T}}^{\frac{1}{3}}{\rho }_{\mathrm{i}}^{\frac{2}{3}}$ [43]
Poloidal structure	m = 0	Linear dynamics of zonal flow modes [41]
Non-linear interaction with turbulences	present	the influence of turbulent Reynolds stress on zonal flows or GAMs [41]

3.2. Limit Cycle Oscillations

The four-frequency microwave DBS system was used to study a variety of limit cycle oscillations (LCO) characteristics (see

Table 4. Properties of LCO in Globus-M.

Investigated Property	LCO in Discharges with H-Mode	LCO in Discharges without H-Mode	Notes
Additional heating	NBI		not detected in Ohmic phase
f, kHz	~6	~8.5	the higher LCO frequencies predicted by Stringer spin-up [49]
Spectral peak width ∆f, kHz	1.5	1.5	corresponds to the calculation of spectra in a 512 ms window
Duration of LCO phase τ, ms	2–20		the duration of the LCO phase tends to be closer to the value of 2 ms in discharges with H-mode and to the values of 20 ms without H-mode
Velocity shear, s−1	order of 105–106	order of 105	predator-prey model [48]
Velocity shear behavior	a significant increase before transition to H-mode	no significant changes in behavior
Turbulence suppression	present during LCO
Turbulence modulation, %	98	81	dependence of the onset of the transition on frequency corresponds to the transition appearance model developed for the case of the appearance of zonal flows [51]
Type	bursty dynamics	standard oscillations around a certain value
Nonlinear coupling of turbulence	present		possibly, the significant coupling indicates that LCOs are dominated by zonal flows; similar results were obtained in the HL-2A tokamak in the ‘type-Y’ LCO phase [53]
Location $\mathsf{\rho }$	~0.75		2 cm inside the separatrix which is similar to other tokamaks
Location width $\mathsf{∆}\rho$	~0.4		close to the radial wavelength of the zonal flows ${\lambda }_{ZF}~\sqrt{{\rho }_{i}a}$ [41]

in the end of Section 3.2) on Globus-M. LCOs were discovered in plasma discharges with neutral beam injection (NBI) when the direction of the magnetic field was favorable for transition to H-mode (i.e., the toroidal ion drift was directed towards the X-point). LCOs were found in the form of low-frequency (5–9 kHz) oscillations in E × B drift velocity with the duration of their existence ranging from 2 to 20 ms [46]. Low frequency LCOs (~6 kHz) led to a transition to H-mode while high frequency LCOs (~8.5 kHz) resulted in the transition to L-mode or even in a disruption. The LCO frequencies in Globus-M are much higher in comparison to medium size devices [47]. A strong radial dependence of LCO frequency was not predicted by the model of limit cycle oscillations induced by a predator-prey coupling of turbulence with zonal flows [48]. However, it is in accordance with the Stringer spin-up mechanism [49]. The multichannel DBS system made it possible to estimate the velocity shear ω_s of the oscillations at LCO frequency. The amplitude of these oscillations was about 10⁵ s⁻¹, but it increased up to 10⁶ s⁻¹ just before L-H transition in the case of the low frequency LCOs. The transition to H-mode after these LCOs can be caused by an increase in the efficiency of the action of the oscillating velocity shear on the plasma turbulence while decreasing its oscillation frequency [50,51]. A periodic suppression of the turbulence power was observed as well. However, in the event of the high LCO frequency, the turbulence level would decrease not to zero, but to a certain value, after which it began to increase (standard oscillations). For the lower LCO frequency, the turbulence level decreased periodically to zero. Such bursty dynamics were also observed on TCV [52].

Auto-bicoherence analysis of the perpendicular velocity demonstrated the nonlinear coupling of the broadband turbulence and velocity oscillations at the LCOs frequency in the case of high frequency LCOs [46]. It is possible to assume that this significant coupling indicates that LCOs are dominated by zonal flows caused by Reynolds stress [41]. Similar results were obtained in the HL-2A tokamak in the ‘type-Y’ LCO phase [53].

The radial scale of the observed flow is estimated to be 4 cm [42,46]. This scale is close to the radial wavelength of the zonal flows [41]. The maximum of the amplitude profile of the velocity oscillations at LCO frequency was located at about 2 cm inside the separatrix. Moreover, the velocity profile maximum moves from the boundary to the core region during the LCO period. This was investigated by studying phase relations between the perpendicular velocity and turbulence amplitude for two radial positions, which are presented in the form of Lissajous phase diagrams in Figure 4. It was demonstrated that at R = 54.4 cm the clockwise direction in the figure corresponds to a predator-prey model [48], while at R = 56.6 cm the counterclockwise direction was associated with a transition to H-mode in work [53]. Using DBS it was possible to observe both trajectories simultaneously. The calculated autocorrelation and cross-correlation functions of amplitude and perpendicular velocity lead to the estimation of the velocity of the movement to be about 3 km s⁻¹. Apart from that, it was shown that the amplitude oscillations that occur at different cutoff positions are approximately in-phase. From all the obtained data we could conclude that it is impossible to apply the known 0D predator-prey models to explain the observed differences at different radii.

Quasi-Coherent Fluctuations

Quasi-coherent fluctuations (QCFs) were also discovered in the spectra of the DBS amplitude fluctuations. Their main properties are presented in

Table 5. Properties of QCFs in Globus-M.

Investigated Property	Range of Values	Notes
f, kHz	110	similar to values on other tokamaks like T-10 [55]
Spectral peak width ∆f, kHz	80
Type	burst
tlifetime, ms	0.1–0.2	modulated by LCO
Location $\mathsf{\rho }$	0.65	in the QCF location there is an observed increase in diffusion coefficient caused by ITG
Location width $\mathsf{∆}\rho$	0.1
Mode of operation	I-phase	present only during a certain phase of LCO
Non-linear interaction with turbulences	present	a peak of summed bicoherence at the QCFs frequency demonstrates the existence of a relationship with turbulent fluctuations
Coherence of velocity fluctuations and DBS amplitude	0.8 near QCFs frequencies
Coherence of DBS amplitude and magnetic field fluctuations	less than 0.3 near QCFs frequencies	not of a magnetic nature

. QCFs appear in the form of a spectral peak at the 110 kHz frequency with a spectral width of Δf_QC = 80 kHz [54]. QC fluctuations with similar parameters were observed on the T-10 tokamak [55]. Unlike in T-10, the fluctuations in Globus-M were observed during the I-phase discharge with LCOs. QCFs were strongly modulated by LCO. They were observed by two high-frequency DBS channels of 39 and 48 GHz at once. It corresponds to the normalized minor radii values from $\rho$= 0.6 to $\rho$= 0.7. ASTRA 7 and GENE simulations showed an increase in diffusion coefficient caused by the ion temperature gradient (ITG) instability in the QCF location. Auto-bicoherence analysis of the DBS amplitude fluctuations demonstrated nonlinear coupling of the broadband turbulence and QCFs. High level of cross-coherence of velocity fluctuations and DBS amplitude (about 0.8 at QCFs frequencies) indicates the presence QCFs in velocity. Low level of coherence of DBS amplitude and magnetic field fluctuations (<0.3) demonstrates that QCFs are not of a magnetic nature.

3.3. Filaments

For the first time DBS diagnostics was implemented to study filaments [56]. Such an implementation is possible when the poloidal size of the filaments is close to π/k_⊥ (1–3 cm in the case of DBS on Globus-M), where k_⊥ is the wave vector of the incident wave in the cutoff region. Such filaments were discovered using DBS inside the separatrix in ELMy H-mode with NBI [31,57] and in Ohmic H-mode with high MHD activity [58,59]. Filaments were observed as a sequence of bursts of quasi-coherent fluctuations (BQFs) in the IQ DBS signals. The frequency of IQ signal oscillations during a burst was used to calculate the poloidal velocity of the filaments. The distance between filaments in the perpendicular direction was calculated by measuring the temporal delay between adjacent bursts. The four frequency DBS scheme allows one to estimate the radial size of filaments. The reconstruction of the location and spatial distribution of filaments in the Globus-M tokamak is presented in Figure 5.

In total, three types of filaments were discovered. One of them is ELM filaments. They were observed during ELMs triggered by sawtooth oscillations [60] in discharges with H-mode initiated by NBI when plasma density was relatively small n_e < 1.2 × 10¹⁹ m⁻³ [61]. If the plasma density was higher, the waveform during ELMs ceased to resemble quasi-coherent oscillations, although the signal level increased greatly during ELMs. Apparently, this is due to the fact that the poloidal size of the filament depends on the density and, when the critical value is exceeded, it greatly differs from π/k_⊥. Measured parameters of ELM filaments can be found in

Table 6. Properties of filaments in Globus-M.

Investigated Property	ELM Filament	Inter-ELM Filament	MHD Induced Filament
Radial size, cm	2–6	<2	~1
Poloidal size, cm	~1–3	~1–3	~1–3
Distance between filaments, cm	~14	~7	~10
Velocity, km s−1	~11–14	~6–12	~12
Quasi-toroidal mode numbers	~12	~25	~15
Critical density ne, 1019 m−3	<1.2	1.5	~2–3.5
Location R, cm	55	54.8	57
Dα activity	present	not present	not present
Langmuir probe activity	present	not present	not present
Magnetic probe activity	present	not present	present

.

The second filament type is inter-ELM filaments. Such filaments were discovered between ELMs in discharges with H-mode initiated by NBI when plasma density was higher than the critical value n_e = 1.5 × 10¹⁹ m⁻³ [61]. Unlike radially elongated ELM filaments, inter-ELM filaments were located in a very narrow layer of <2 cm. Additionally, inter-ELM filaments were characterized by a smaller distance between each other and moved at lower velocities in comparison with ELM filaments (see Table 6). Unlike inter-ELM filaments, ELM filaments were accompanied by magnetic probe activity, which presumably indicates their peeling ballooning nature [62]. Inter-ELM filaments were also not accompanied by Langmuir probe activity. This is because of their smaller radial size and deeper location inside the separatrix, which makes them different from ELM filaments whose greater elongation reaches the separatrix where the Langmuir probe was situated.

The last type is filaments induced by neoclassical tearing modes. This type of filament was observed in Ohmic H-mode discharges with rather high densities n_e = 2–3.5 × 10¹⁹ m⁻³ [59]. The main properties are similar to inter-ELM filaments (see Table 6). Such filaments occur in a certain phase of MHD oscillations, namely, when the magnetic island is located on the low field side of the torus (opposite the antenna), where the plasma pressure gradient increases and conditions are suitable for the development of the peeling-ballooning mode. Their location is demonstrated in the reconstruction in Figure 5. It was constructed using the DBS measurements of the filament velocity and recorded time intervals between adjacent bursts, which allowed one to determine the distance between a series of filaments. Their localization in both the poloidal and toroidal directions along with the direction of the movement is clearly highlighted.

Full-wave modeling was carried out to obtain information on the behavior of the DBS signal during the transition from linear to nonlinear backscattering from filaments [63,64,65]. First, modeling showed that similar BQFs are formed in case of scattering from filament in both linear or nonlinear regimes if the filament size does not significantly exceed the characteristic value of π/k_⊥. In such an event, the Doppler shift can be determined by the expression, which is also valid in the Born approximation in slab geometry. Secondly, with an increase in the amplitude and size of the filament, the BQF deforms, forming the double-maximum BQF.

3.4. Alfven Mode

For the first time, the novel idea to use DBS for the detection of Alfven eigenmodes (AEs) was proposed on Globus-M. This is possible due to the nature of the Alfven oscillations being that of electromagnetic waves. These oscillations of the magnetic field lead to the fluctuations of the E × B velocity that is measured using DBS, which means that the amplitude of the Alfven perpendicular velocity, radial electric field can be extracted from the measurements. Studies were conducted in the Globus-M tokamak with early neutral beam injection at the stage of growth of the plasma current [66,67]. Initially, the detection of toroidal Alfven eigenmodes (TAE) using DBS was confirmed by comparing the experimentally obtained frequency of the fluctuations presented in Table 7 to the Alfven continuum frequency gaps $f_{TAE}=v_A/4\pi qR$, where $v_A=B/{\mu }_0{\rho }_D$ is the Alfven velocity (B-magnetic field, ${\rho }_D$-mass density of plasma ions), q is the safety factor, and R is the major radius [68]. The results of the calculations were similar to those acquired in experiments. Apart from that, the calculated spectrograms of rotation velocity determined using the DBS method highlighted the fact that the frequency of the TAE decreases with time, which corresponds to the concept of the slowing down of the Alfven wave. These properties were also observed in the spectrograms of the magnetic probe signals that confirmed the observation of Alfven eigenmodes. The nature of the TAE varied depending on the characteristics of the injected beam (it was shown that the isotope of the injected particles had an effect on what type of Aflven mode could develop) as well as the plasma density. The instability was observed either in the form of a short burst lasting from 0.1 up to 0.5 ms or as continuous oscillations of several ms. The length of the TAE burst apparently can be defined through a predator-prey model [69].

The DBS measurements were used to calculate various properties of the Alfven eigenmodes such as the amplitude of the radial electric field and accordingly their magnetic field. The obtained values are presented in Table 7 as well as works [70,71,72]. It is worthy of note that the magnetic field amplitude determined using DBS was compared with the measurements of the magnetic probes $B^{MP}$ located on the wall of the vacuum chamber. It was observed that the $B^{MP}$ values were generally significantly smaller than those calculated from the DBS measurements, which seems to indicate the spatial damping of the instability [71].

While several DBS systems were used to study the location of the Alfven eigenmodes, the four-frequency system had yielded the best results. The toriodal number n = 1 TAE was detected in the area with major radii values of 0.50–0.56 m. It was noted that a clear maximum of the amplitude for this n was not determined. However, the location of TAE with higher toriodal number values (n = 2, 3) was determined to be closer to the periphery with R ~ 0.51–0.59 m. The examples of profiles obtained for a variety of Alfven eigenmodes is presented in Figure 6. It clearly highlights that there is a possibility of the existence of the toroidal Alfven eigenmodes in the core plasma regions. The shift of the main TAE harmonic (i.e., with n = 1) to the inner plasma regions appears to be related to the increasing plasma current [73].

Table 7. Properties of Alfven eigenmodes in Globus-M.

Investigated Property	Range of Values	Notes
Additional heating	NBI	only observed during NBI
f, kHz	~99–160	Alfven continuum frequency gaps $f_{\mathrm{TAE}}=\frac{v_A}{4\pi qR}$ [68]
Spectral peak width ∆f, kHz	80	the decrease of the TAE frequency is related to the slowing down of the Alfven wave
Type	continuous or burst	dependent on the injected beam parameters [69]
tlifetime (for burst type), ms	~0.1–0.5	predator-prey model [69]
Radial electric field amplitude, kV m−1	~0.5–3	the oscillations of the Alfven magnetic field are accompanied by the oscillations of the measured electric field according to Maxwell’s equation $\widetilde{B_{\theta }}=\frac{{\tilde{E}}_r}{v_A}$ [73]
Magnetic field amplitude, 10−4 T	~6–25
Location R, m	~0.50–0.56	shift of TAE to the inner plasma regions is related to the increasing plasma current
Toroidal number, n	1–3	different toroidal numbers correspond to different TAE harmonics

Table 7. Properties of Alfven eigenmodes in Globus-M.

Investigated Property	Range of Values	Notes
Additional heating	NBI	only observed during NBI
f, kHz	~99–160	Alfven continuum frequency gaps $f_{\mathrm{TAE}}=\frac{v_A}{4\pi qR}$ [68]
Spectral peak width ∆f, kHz	80	the decrease of the TAE frequency is related to the slowing down of the Alfven wave
Type	continuous or burst	dependent on the injected beam parameters [69]
tlifetime (for burst type), ms	~0.1–0.5	predator-prey model [69]
Radial electric field amplitude, kV m−1	~0.5–3	the oscillations of the Alfven magnetic field are accompanied by the oscillations of the measured electric field according to Maxwell’s equation $\widetilde{B_{\theta }}=\frac{{\tilde{E}}_r}{v_A}$ [73]
Magnetic field amplitude, 10−4 T	~6–25
Location R, m	~0.50–0.56	shift of TAE to the inner plasma regions is related to the increasing plasma current
Toroidal number, n	1–3	different toroidal numbers correspond to different TAE harmonics

3.5. Turbulence

The four frequency DBS system was used to study turbulence in H-mode with and without ELMs. The experiments were performed in the Globus-M spherical tokamak operated in H-mode initiated by NBI. Plasma density fluctuations as well as plasma velocity fluctuations were investigated during the transition to the ELM-free H-mode. Studies showed a decrease in both plasma density and plasma velocity fluctuation near the separatrix in the absence of ELMs [74,75].

A more in-depth study of turbulence during ELMs allowed us to identify two types of ELMs and two corresponding types of transition to the ELM-free H-mode [76]. Low-frequency ELMs in the Globus-M tokamak are characterized by the transition to the ELM-free regime during plasma current decrease alongside a similar change in the current density profile. This transition is associated with a decrease in the intensity of sawtooth oscillations. The level of turbulence at the periphery in this case varies only slightly, which was demonstrated by analyzing the amplitude of the complex DBS signal as demonstrated in Figure 7a. It can be assumed that the transport does not change, which is indicated by the constancy of the D_α level and average density.

High-frequency ELMs are characterized by a spontaneous transition to the ELM-free regime. The peripheral turbulence amplitude decreases during this transition (as seen in Figure 7b), which was accompanied by the typical features of peripheral transport suppression: the electron density increases and D_α emission drops. A study of the turbulence amplitude spectra during this regime led to the discovery of quasi-coherent (QC) fluctuations in the ELM-free H-mode in the highest frequency (48 GHz) channel with a the cutoff radius of $\rho$ = 0.6 [77]. The properties of these fluctuations were similar to the QCFs observed in I-phase (see Table 5), except for the fact that the QC fluctuations in ELM-free H-mode were not modulated by anything. They existed continuously during the period of ELM-free H-mode. Such QCFs were not detected in the spectra of the complex IQ signal. This, apparently, indicates that the poloidal scale of the fluctuations is much larger than the method resolution of π/k_⊥.

4. Conclusions

In summary, the DBS diagnostics was successfully used to study peripheral plasma processes in the Globus-M tokamak. The application of this method was implemented on a spherical tokamak for the first time. Subsequently, DBS was implemented on the spherical tokamak MAST [8].

The simultaneous application of several DBS systems located at different poloidal angles and with different probing frequencies made it possible to determine the spatial structure of different plasma processes. As a result, geodesic acoustic modes, limit cycle oscillations, quasi-coherent fluctuations, filaments, Alfven eigenmodes, and plasma turbulence were investigated.

Two types of systems were used to study the radial distribution of GAMs: (1) two single-frequency systems with the possibility to change the probing frequency in the ranges of 18–26 GHz and 27–38 GHz, (2) a four-frequency system with fixed probing frequencies of 20, 29, 39, and 48 GHz. It turned out that the very narrow localization of GAMs near the periphery could only be studied in detail with the first type of system. It is worth noting, however, that discharge repetition in this case required much effort and additional diagnostics of various plasma parameters. It was not possible to construct a radial profile of the GAM oscillation amplitude using a four-frequency system, as the cut-offs for the selected frequencies were spaced much farther apart than the size of the GAM localization layer. Therefore, GAMs were only detected on the peripheral channel of the four-frequency system, whereas on other channels they were not observed. It was also useful for the purpose of GAM research to use poloidally spread single-frequency DBS systems positioned in a manner so that their cut-offs were located on the same magnetic surface. Such an application of DBS diagnostics in this configuration allowed for the determination of the poloidal mode number of the GAM velocity oscillations to be m = 0.

A four-frequency DBS system was used with the intent to study LCO. In contrast to GAMs, the localization of the LCO turned out to be much wider, and all four DBS channels were able to detect oscillations at this frequency, albeit with different amplitudes. The study of fluctuations in the velocity and amplitude of turbulence at different radii had made it possible to investigate the characteristic properties of the development of LCO. The phase relationship between oscillations of velocity and turbulence amplitude near the top of the pedestal region corresponded to the interaction of zonal fluxes and turbulence observed on other devices (see [78], for example). Yet near the separatrix where both velocity and turbulence fluctuations were observed, their phase coupling changed and the velocity started to increase slightly earlier than the level of turbulence. These results contradict the known zero-dimensional model of the interaction of zonal flows and turbulence [48] and indicate that the description of LCO should be at least one-dimensional. A study of the spectral characteristics of the LCO oscillations on Globus-M showed an increase in their frequency when compared to larger devices. The dependence of the frequency of oscillations on the size of the devices cannot be predicted by a predator-prey model of coupling of zonal flows with turbulence.

DBS was first utilized for filament research. Subsequently, the detection of filaments using DBS on the ASDEX Upgrade tokamak was reported [79]. The use of various probing frequencies on Globus-M made it possible to detect them both near the separatrix and in the more inner plasma regions. The simultaneous use of several probing frequencies allowed one to study their radial size. It had been found that the filaments that develop during ELMs are radially longer than the filaments that develop during inter-ELM periods. The use of poloidally spread DBS schemes helped investigate the localization of filaments and track their movement in the poloidal direction. The poloidal rotation velocity of the filaments, determined from the phase delay between the signals of the two poloidally spread antennas of the DBS diagnostic, was equal to the one determined from the Doppler frequency shift. Full-wave simulations of scattering on the filaments later confirmed the correctness of the interpretation of the obtained experimental data.

For the first time, the DBS method had also been used to perform research into Alfven oscillations that had been detected in the phase derivative of the complex DBS signal. Alfven fluctuations in the phase of the complex IQ signal were detected in the DIII-D tokamak using reflectometry [80]. On Globus-M the application of a four-frequency probing system enabled us to study the radial distribution of the TAE amplitude. It was noted that there were no fluctuations in the probing frequencies corresponding to the regions near the separatrix. The largest oscillation amplitude values were recorded on the highest frequency channel. Such measurements demonstrated the need to develop the DBS diagnostics with a higher frequency range for the inner plasma area with the goal of TAE detection in the core region.

The study of turbulence spectra led to the discovery of quasi-coherent fluctuations. Such fluctuations were detected in the ELM-free H-mode and I-phase near the top of the pedestal. The development of the QCFs was accompanied by a sharp increase in the local diffusion rate leading to the deformation of the density profile. The appearance of the QCF is associated with the development of ITG instability.

All of the operational DBS systems are now utilized on the new and improved Globus-M2 tokamak. Moreover, the DBS diagnostic continues to evolve in response to the demand associated with the development of the scientific plans for the modernization of the tokamak. In particular, an additional six-channel DBS circuit with a frequency range of 50–75 GHz has recently been installed for core region research. In experiments that aimed to completely replicate the Globus-M working conditions that were favorable for the development of TAEs, the use of a high-frequency DBS system allowed to observe the decline in the TAE amplitude in the central region of the discharge and thus the final localization profile of these oscillations was determined [81].