Introduction

Plasma is a non-equilibrium system of strong nonlinearity that provides rich phenomena for modern physics. One such physical topic is the roles of fluctuations and turbulence in the structural formation and dynamics of plasmas. Particularly, in the research of magnetically confined plasmas, various phenomena associated with turbulence have been found and investigated, especially in the research of magnetized plasma for realizing a nuclear fusion reactor. The phenomena include the non-local transport1,2,3,4,5,6 the global linkage of the turbulent activity7,8,9turbulence spreading and avalanche10,11,12,13,14,15,16which are thought to be caused by nonlinear interactions between disparate scale fluctuations covering the entire plasma17,18. The investigation of these phenomena needs simultaneous and entire cross-sectional observation of the turbulence field that often shows symmetry breaking characteristics19,20.

Tomography is a possible candidate, amongst various plasma diagnostics, to achieve the entire measurement of plasma. To confirm the capability of tomography as a tool for turbulence field measurement, a prototype system has been installed in a cylindrical device called Plasma Assembly for Nonlinear Turbulence Analysis (PANTA). The tomography system successfully demonstrates its feasibility for the entire field observation of plasma turbulence in space and time21,22. Moreover, various analyzing tools for tomographic images have been developed to extract the two-dimensional feature of the fluctuations, such as Fourier-Bessel Function analysis, Fourier Rectangular Function (FRF) analysis23modal polarization analysis24,25 and so on26.

Recently, the advanced use of the FRF analysis has been proposed27,28and the proposed analysis have been applied to an oscillatory state dominated by mother (M) mode (m = 4 mode) in PANTA plasma29. The analysis revealed that the oscillatory state repeats transition quasi-periodically to two phases distinguished by the strength in nonlinear modal coupling of M (m = 4) mode with father (F) mode (m = 1 mode). Moreover, the analyses disclose an important role of F (m = 1) mode dynamics for the phase transitions. This article presents the advanced analysis using FRF as a tool to investigate the physics of plasma turbulence, with emphasis on the excellent capabilities of the FRF analysis in its applicability on the cylindrical system and to show intriguing results of its application as an example to reveal the fundamental processes and spatiotemporal behaviors of modal structure and couplings in an oscillatory state in a cylindrical plasma.

Fourier-rectangular function (FRF) analysis

The FRF series is used to expand a two-dimensional image of plasma emission intensity, \(\:\epsilon\:\left(r,\theta\:,t\right),\) on a cylindrical coordinate. The FRF expansion of the emission intensity is defined as follows,

$$\varepsilon \left( {r,\theta ,t} \right)=\mathop \sum \limits_{n} {a_{n,0}}\left( t \right)\frac{{{R_n}\left( r \right)}}{{\sqrt 2 }}+\mathop \sum \limits_{n} \mathop \sum \limits_{{m>0}} {R_n}\left( r \right)\left[ {{a_{n,m}}\left( t \right)\cos \left( {m\theta } \right)+{b_{n,m}}\left( t \right)\sin \left( {m\theta } \right)} \right].$$
(1)

Where \(\:n\) and \(\:m\) represent the radial position and azimuthal mode number, respectively. \(\:{R}_{n}\left(r\right)\) is the function that has a finite value in a radial range from \(\:r={r}_{n}-{\delta\:}_{r}\) to \(\:r={r}_{n}+{\delta\:}_{r}\). Supposing that the functions, \(\:{\varphi\:}_{l}\left(r,\:\theta\:\right)\), are the orthogonal bases for FRF expansion, that is, \(\:{\varphi\:}_{l}\left(r,\:\theta\:\right)=\frac{{R}_{n}\left(r\right)}{\sqrt{2}}\), \(\:{\varphi\:}_{l}\left(r,\:\theta\:\right)={R}_{n}\left(r\right)\text{cos}\left(m\theta\:\right)\) or, \(\:{\varphi\:}_{l}\left(r,\:\theta\:\right)={R}_{n}\left(r\right)\text{sin}\left(m\theta\:\right)\), where the suffix, \(\:l\), is renumbered from the set of the integers, \(\:(n,m)\), in an appropriate order, they satisfy the orthonormal relationship as follows,

$${\delta _{{l_1},{l_2}}}=\smallint {\phi _{{l_1}}}\left( {r,\theta } \right)~{\phi _{{l_2}}}\left( {r,~\theta } \right)rdrd\theta.$$
(2)

where \(\:{\delta\:}_{{l}_{1},{l}_{2}}\) represents Kronecker’s delta. The FRF coefficients are obtained using the following formula,

$$\:{a}_{n0}\left(t\right)=\int\:\epsilon\:\left(r,\theta\:,t\right)\frac{{R}_{n}\left(r\right)}{\sqrt{2}}rdrd\theta\:,\:$$
$$\:{a}_{n,m}\left(t\right)=\int\:\epsilon\:\left(r,\theta\:,t\right){R}_{n}\left(r\right)\text{cos}\left(m\theta\:\right)rdrd\theta\:,\:$$
$$\:{b}_{n,m}\left(t\right)=\int\:\epsilon\:\left(r,\theta\:,t\right){R}_{n}\left(r\right)\text{sin}\left(m\theta\:\right)rdrd\theta\:.$$

By making a complex variable of the FRF coefficients, \(\:{c}_{n,m}\left(t\right)={a}_{n,m}\left(t\right)+\sqrt{-1}{b}_{n,m}\left(t\right)\), its absolute value and argument allows us to evaluate the local power and amplitude of each azimuthal mode structure, \(\:{P}_{m}\left({r}_{n},\:t\right)={\left|{c}_{n,m}\left(t\right)\right|}^{2}\), and \(\:{A}_{m}\left({r}_{n},\:t\right)=\sqrt{{P}_{m}\left({r}_{n},\:t\right)}\), respectively, and declination of the azimuthal mode, \(\:{\theta\:}_{n,m}\left(t\right)=\text{a}\text{r}\text{g}\left({c}_{n,m}\left(t\right)\right)\), as a function of radius. Also, the FRF complex coefficients can be used to quantify the tilt of the modal structure by taking the difference of the declination between neighboring radial bands, \(\:{\Delta\:}{\theta\:}_{n,m}=\text{arg}\left({c}_{n+1,m}{c}_{n,m}^{*}\right)\), where \(\:{c}_{n,m}^{*}\left(t\right)\) represents the conjugate of the complex variable \(\:{c}_{n,m}\left(t\right)\). The FRF analysis is regarded as Fourier function series expansion in azimuthal coordinate defined on a radial band and allows easy extraction of modal properties of cylindrical plasmas at radially localized position, since the azimuthal structure is expressed using the traditional Fourier function expansion on individual radial bands. This feature of the FRF analysis is one of the advantages to other methods such as the Fourier-Bessel function expansion that treats azimuthal modes in a global property in the entire plasma region. By utilizing the properties, we propose newly extended FRF usages, namely FRF structural and modal coupling function analyses, to extract the spatiotemporal structure and dynamics from the 2D imaging data. Hereafter, the cartesian coordinate \(\:(x,\:y)\) is used to infer the local property of emission intensity or its fluctuation, and the cylindrical coordinate \(\:(r,\:\:\theta\:)\) is used to express the radial profile of azimuthally averaged emission intensity or fluctuation property.

FRF structural function: The structural function is introduced to evaluate the spatial pattern of each mode and its robustness. The structural function is defined as follows,

$${\alpha _m}\left( {{r_i},~{r_j},~\tau } \right)=\frac{{\left\langle {{c_{i,m}}\left( {t+\tau } \right)c_{{j,m}}^{*}\left( t \right)} \right\rangle }}{{\left\langle {\left| {{c_{i,m}}\left( {t+\tau } \right)} \right|} \right\rangle \left\langle {\left| {c_{{j,m}}^{*}\left( t \right)} \right|} \right\rangle }}.$$
(3)

where \(\:<>\) means the temporal average, \(\:〈a\left(t\right)〉=\frac{1}{T}{\int\:}_{{t}_{s}}^{{t}_{s}+T}a\left(t\right)dt\). The suffixes, i, j, represent the corresponding radial position \(\:{r}_{i}\) and \(\:{r}_{j}\). Here the parameter, \(\:{\alpha\:}_{m}\), is termed structural function, which can be divided into two variables: structural coherence, \(\:{\gamma\:}_{m}=\left|{\alpha\:}_{m}\right|\), and tilt angle between two positions, \(\:{\theta\:}_{i,j,m}\left({r}_{i},\:{r}_{j},\:\tau\:\right)=\text{a}\text{r}\text{g}\left({\alpha\:}_{m}\left({r}_{i},\:{r}_{j},\:\tau\:\right)\right)\). The FRF expansion is particularly powerful to analyse systems of cylindrical symmetry. Based on the FRF expansion, the structural function has an advantage to evaluate rigidness of radial structure for each azimuthal mode. In other words, how radial structure of fluctuations is stable in space and time.

FRF modal coupling function

The modal coupling function is defined as a triple product of the FRF coefficients as follows,

$${\beta _{m,n}}\left( {{r_i},~{r_j},~\tau } \right)=\frac{{\left\langle {{c_{i,m}}\left( {t+\tau } \right){c_{i,n}}\left( {t+\tau } \right)c_{{j,m+n}}^{*}\left( t \right)} \right\rangle }}{{\left\langle {\left| {{c_{i,m}}\left( {t+\tau } \right){c_{i,n}}\left( {t+\tau } \right)} \right|} \right\rangle \left\langle {\left| {c_{{j,m+n}}^{*}\left( t \right)} \right|} \right\rangle }}.$$
(4)

where \(\:\left|{\beta\:}_{m,n}\right|\) and \(\:\text{a}\text{r}\text{g}\left({\beta\:}_{m,n}\right)\) are termed the coupling power and the coupling angle, respectively, correspondingly to the squared bicoherence and bi-phase in Fourier analysis. The modal coupling function analysis provides a method to clarify the nonlinear coupling in time, different from the bicoherence analysis performed on frequency domain. Thus, the analysis can show the temporal evolution on real space, or spatial propagation of modal couplings in real space and time. The presented analysis result on the oscillatory state below is a good example which succeeds in revealing hidden processes occurring in its phase transitions, with the hidden dynamics of F (m = 1) mode.

Experimental results

Overall characteristics of discharge and existence of family modes

The target plasma is produced in PANTA device for the present experiment under the following condition: plasma radius of 5 cm, filling Ar gas pressure of 400 Pa (3 mTorr), strength of the axial magnetic field of 0.13 T, and RF heating power of 6 kW. As is already reported in a previous paper of probe measurements30the oscillations are dominated by the azimuthal mode of m = 4 at the frequency of 8.9 kHz, termed here mother (M) mode. The obtained tomography data is processed using a set of the FRF expansion analysis. Figure 1 (a), (b) and (c) show the 2D profile of temporally averaged ArII emission intensity or emission of Ar+ ions, fluctuation component, and the waveforms of the quasi-periodic oscillations in local emission. Note that the emission is a function of Ar+ density, electron density and temperature. Also, the readers should note that Fig. 1 (b) depict the snapshot of the fluctuation component of emission intensity, namely \(\:\stackrel{\sim}{\epsilon\:}\left(r,\:\:\theta\:,\:\:t\right)=\epsilon\:\left(r,\:\theta\:,t\right)-\stackrel{-}{\epsilon\:}\left(r,\:\theta\:\right)\). Here, \(\:\stackrel{-}{\epsilon\:}\left(r,\:\theta\:\right)\)represents the temporally averaged local emission intensity.

Figure 2(a, b) shows a dispersion relation, the fluctuation power spectrum on the azimuthal mode and frequency plane, calculated at r = 4 cm using the FRF coefficients. Figure 2(b) can be obtained by calculating the Fourier power spectrum, \(\:{{P}_{n}\left(m,\omega\:\right)=\left|{\widehat{c}}_{n,m}\left(\omega\:\right)\right|}^{2}\) at r = 4 cm, where \(\:{\widehat{c}}_{n,m}\left(\omega\:\right)\) represents the Fourier coefficient of FRF coefficient, \(\:{c}_{n,m}\left(t\right)\). The dispersion relation indicates the presence of elemental modes at (m, f )=(1, 1.9 kHz), (3, 7.0 kHz), and (5, 10.8 kHz), which are termed father (F) mode and child (C) modes, respectively, as well as M (m = 4) mode at (4, 8.9 kHz). These modes satisfy matching conditions in both azimuthal mode numbers and frequencies. The significance of the nonlinear couplings is also confirmed by the usual Fourier bicoherence analysis applied to the temporal series data of the local emission intensity. As is shown in the dispersion diagram, the power of C (m = 5) mode is one order smaller than C (m = 3) mode. Moreover, the azimuthal resolution of C (m = 5) mode is worse particularly in the core region (see ‘Calculation of FRF coefficients’ in Method), thus, the further analyses are concentrated on C (m = 3) mode.

Figure 3 shows the properties of M (m = 4) and C (m = 3) mode: the temporal evolution of their amplitude profile with snapshots of the 2D spatial structure. The temporal evolution indicates that the amplitude of M (m = 4) mode is almost constant in time, while the amplitude of C (m = 3) mode shows temporal variation, with its maximum location kept at r = 2.0–2.5 cm.

Fig. 1
figure 1

An overview of the target plasma which is dominated by the oscillation of mother (m = 4) mode phenomenon. (a) Mean emission profile (b)two-dimensional structure of fluctuation at 300.01, 300.06, 300.11, 300.16 ms, respectively, (c) time evolution of the local emission intensity at point A (x = 0.0 cm, y = 3.2 cm) and B (x = 2.0 cm, y = 2.0 cm) denoted on panel (a).

Full size image
Fig. 2
figure 2

(a) Power spectrum of fluctuation at r = 4 cm and (b) the dispersion relationship of the target discharge at the same radial position.

Full size image
Fig. 3
figure 3

(a) Snapshots of the family modes, C1 and M, at \(\:{\varvec{\tau\:}}_{\varvec{c}}=\)285.7 ms, (b) time evolution of the amplitude radial profile of the mother, and child modes, and (c) time average of the radial profile of the amplitude of each mode.

Full size image
Fig. 4
figure 4

(a) The temporal and radial profile of the amplitude and tilt, and those at r = 2 cm of the F mode. (b) The amplitude dependence on the tilt angle at r = 2.0 cm. (c) The mean and snapshot of the F mode amplitude at 285.67 ms and 287.00 ms, together with the snapshot of the two-dimensional image of the F mode at the same time.

Full size image

In contrast, the temporal evolution of F (m = 1) mode, as is shown in Fig. 4, is found dynamic. Figure 4 reveals that the amplitude of F (m = 1) mode propagates quasi-periodically with a cycle of 1–2 ms from the plasma center to the edge. Moreover, the temporal evolutions show that the tilt angle varies periodically with the amplitude: more explicitly, the tilt angle takes its maxima when the amplitude at r ∼1.6–2.0 cm is at its minima (see the bottom plot in Fig. 4(a)): the radial position is termed here the interface layer. Note that similar phenomenon, propagation of the amplitude and formation of spiral structure, is found in a simulation study on a cylindrical magnetized plasma31. Figure 4(b) also confirms that the tilt angle stays around zero when the interface amplitude becomes closer to the maxima. Figure 4(c) shows two typical profiles and images at a maximum (t1 = 285.67 ms) and a minimum (t2 = 287.00 ms) of the amplitudes at the interface, indicating the clear difference between them. The profile exhibits simple shape without any tilting at the maxima, while it shows twisted shape across the interface layer around the minima. The F (m = 1) mode profile shows two different features during the amplitude propagation of F-mode.

Structural function analyses on the family modes

The structural function analysis reveals the detailed characteristics of the family modes. Figure 5 (a-b) show the result of structural function analysis applied on C (m = 3) and M (m = 4) modes, respectively, when the reference position is taken at \(\:{r}_{ref}=\)1.1 cm. For these modes, the characteristics are quite simple; the structural coherence is constantly high around \(\:\tau\:\)=0 independently of radial location, while it monotonically decreases as delay time, \(\:\tau\:\), becomes larger. The obtained phase differences in the structural function analysis, namely the tilt angle \(\:{\theta\:}_{i,j,m}\left({r}_{i},\:{r}_{j},\tau\:\right)\), verifies that the spatial structures of C (m = 3) and M (m = 4) modes rigidly rotate in an electron diamagnetic (or the clockwise) direction.

In its application to F (m = 1) mode, in the case of the reference position at \(\:r=\)1.1 cm (see Fig. 5(c)), the structural coherence is high in the central region of the plasma and rapidly decreases in the outer region beyond the interface, especially when \(\:\tau\:=0\:\text{m}\text{s}\). The temporal evolution of the tilt angle \(\:{\theta\:}_{i,j,m=1}\left({r}_{i},\:{r}_{j},\:\tau\:\right)\) shows that the tilt angle at r=2.0 cm changes in accordance with that at r=1.1 cm, while the tilt angle at r=3.4 cm deviates from that of the reference point. In contrast, in the case of the reference position at \(\:r=\)3.4 cm (see Fig. 5(d)), the structural coherence is high in the outer region, but it decreases rapidly inward beyond the interface, especially when \(\:\tau\:=0\:\text{m}\text{s}\). The incoherent nature can also be found from the temporal variation of the tilt angle in Fig. 5(d). The rigidness of the spatial structure seen for the case of C (m=3) and M (m=4) modes are totally different from the dynamic structure of F (m=1) mode. Though the structure function analysis reveals the presence of the interface layer on the F (m=1) mode, the formation mechanism of the interface layer is still under investigation.

The result shows that the F (m = 1) mode behaves individually or incoherently, at almost the same frequency, as two different ones located in the inner and outer region. This strange behavior of F (m = 1) mode could be related to the fact that the m = 1 azimuthal mode contains fluctuations of broad frequencies around 1.9 kHz, as is shown in the dispersion of Fig. 2. Combining the characteristics of this regional independency (incoherency) and the transition in the spatial structure reveals the unique property of F (m = 1) mode. The structural function analysis clarifies that F (m = 1) mode shows two different characteristics as a rigid spatial structure (one mode) or two different modes accompanied with the amplitude propagation of F (m = 1) mode. The former is called here the regionally connected phase (C-phase), and the latter is the regionally disconnected phase (D-phase) between the inner and outer region.

Fig. 5
figure 5

The dependence of the structural coherence on the delay time \(\:\varvec{\tau\:}\) and radial position \(\:{\varvec{r}}_{\varvec{i}}\) and the dependence of the tilt angle on the delay time (a) of C (m = 3) mode when \(\:{\varvec{r}}_{\varvec{r}\varvec{e}\varvec{f}}=1.1\:\mathbf{c}\mathbf{m}\), (b) of the M (m = 4) mode when \(\:{\varvec{r}}_{\varvec{r}\varvec{e}\varvec{f}}=1.1\:\mathbf{c}\mathbf{m}\), and of the F mode when (c) \(\:{\varvec{r}}_{\varvec{r}\varvec{e}\varvec{f}}=1.1\:\mathbf{c}\mathbf{m}\) and (d) \(\:{\varvec{r}}_{\varvec{r}\varvec{e}\varvec{f}}=3.4\:\mathbf{c}\mathbf{m}\). The error bar on the bottom panels are estimated by the full-width half-maximum (FWHM) of the probability density function of the tilt angle.

Full size image

Modal coupling function analysis and transition in mode dynamics

It is confirmed from the Fourier bicoherence analysis that the M (m = 4) mode, F (m = 1) mode and C (m = 3) modes are nonlinearly coupled. Moreover, the temporal resolving ability of the modal coupling function analysis proposed here can reveal dynamic properties of the couplings between F (m = 1) and M (m = 4) mode. First, the FRF analysis is applied to investigate the effect of the F (m = 1) mode dynamics on the nonlinear coupling by examining the amplitude change of each mode, as the indicator of the coupling strength. Figure 6 shows the amplitude variation of each mode, \(\:{\stackrel{\sim}{A}}_{m}\), which is defined as follows,

$${\tilde {A}_m}\left( {{r_i},t} \right)={A_m}\left( {{r_i},~t} \right) - {A_m}\left( {{r_i},~t} \right).$$
(5)

Note that the amplitude variations in Fig. 6 (a) are normalized by their maximum value within each panel. It is obvious that the amplitude variation of C (m = 3) goes back and forth between two phases that show larger and smaller values, termed active and quiet child phase. In the outer region the increase (or decrease) in the C (m = 3) mode amplitude is synchronized with the decrease (or increase) in the M (m = 4) mode amplitude, that is, an antiphase correlation is found between M (m = 4) and C (m = 3) modes, while the F (m = 1) mode behaves differently. Figure 6 (b) shows the histogram of the durations of these two phases. The histogram shows that the occurrence of the transitions is quasi-periodically although the occurrence of transitions is highly statistical. In addition, the histogram confirms that the duration is longer compared to the cycles of the M (m = 4) and F (m = 1) mode, ∼ 100 µs and ∼ 500 µs, respectively, while the most probable length of the duration of the phases is approximately 1.4 ms.

For investigating the origin of two phases, the modal coupling analysis is carried out on the family modes. Figure 7 (a) and (b) confirm the clear difference in the coupling strength between the M (m = 4) and F (m = 1) mode in the two phases. Note that the error of the modal coupling function strength calculated according to the “Methods” is around 0.01 for \(\:1-2\times\:{10}^{4}\) datapoints, which is so small that the error bar is much smaller than the markers in Fig. 7 (b). The absolute values of coupling are different while the profiles of the coupling functions are similar. On the other hand, it should be emphasized that the modal coupling strength even in the quiet child phase is significant in the inner region (r < 2 cm) with the reference position being outside at r = 3.4 cm, although the structural function of the F (m = 1) mode shows almost no coherence beyond the interface layer. In addition, the regional variation of the coupling angle across the interface could be associated with the regional disconnection of the F (m = 1) mode. The significant coupling strength both in the inner and outer region is mediated by the global property of the M (m = 4) mode that covers the entire plasma region. The nonlinear couplings make the individual regional parts of F (m = 1) mode reconnected.

Fig. 6
figure 6

(a)The temporal and radial evolutions of the amplitude variation of F, C1 and M (m = 4) mode. (b) The histogram of the duration of active and quiet child phases.

Full size image
Fig. 7
figure 7

(a) The temporal and radial profile of the active and quiet child phases. The active and quiet child phases show the difference in the modal coupling strength among F, C1 and M (m = 4) mode. (b) The radial profile of the modal coupling strength when \(\:\varvec{\tau\:}=0\) and \(\:{\varvec{r}}_{\varvec{r}\varvec{e}\varvec{f}}=3.4\:\mathbf{c}\mathbf{m}\).

Full size image

Discussion

The role of F (m = 1) mode dynamics is investigated in the transitions observed in the modal coupling strength. For the purpose, a conditional average is performed to obtain typical temporal evolutions of the physics quantities of family modes and their relationships during a period. The results are summarized where the origin of clock for the conditional average is selected as the F (m = 1) mode amplitude reaches the outer edge of interface region at r = 2.5 cm. From top to bottom, the left side panels in Fig. 8 (a) show the temporal evolutions of the structural function of F (m = 1) mode, the coupling function among C (m = 3), F (m = 1) and M (m = 4) modes, the amplitude variation and the tilt angle of F (m = 1) mode, and the amplitude variations of C (m = 3) and M (m = 4) modes, while the right side of these panels show the corresponding profiles at the representative times. Figure 8(b) shows the snapshots of the spatial structure evolution of F (m = 1) mode during the cycle, indicating the transition from connected to disconnected phase in its profile. Figure 8 (c) shows the absolute changes of amplitude profile of F (m = 1), C (m = 3), and M (m = 4) modes, and the local power profiles of the family modes; here, the power is defined as \(\:{P}_{m}\left(r\right)={A}_{m}^{2}\left(r\right)\). Obviously, the amplitude profile of F (m = 1) and C (m = 3) modes clearly change, while that of M (m = 4) does show a quite small variation to its average profile. Thus, the power should be exchanged between F (m = 1) and C modes through the coupling with M (m = 4) mode. In the local power profile, on the other hand, slight changes are found in the core including the interface layer. The spatial integration of each mode power, which is defined as \(\:{P}_{m}=\int\:{A}_{m}^{2}\:dS\), suggests that the temporal variation of the total system power of the family modes is less than 10%, thus the total power could be conserved during the cycle.

The conditional averaged data in Fig. 8 demonstrates the crucial roles of F (m = 1) mode activity in the transitions in the coupling strength. The nonlinear coupling (modal coupling function among F (m = 1), M (m = 4) and C (m = 3) modes, \(\:\left|{\beta\:}_{\text{1,3}}\right|\)) propagates (or coupling becomes stronger) along the trace of the local maximum of the F (m = 1) mode amplitude variation ( \(\:\stackrel{\sim}{{A}_{1}}\)) from (r, t)=(0,-1.0) to (2.5 cm, 0 ms). The propagation velocity is approximately 25 m/s, which is one order of magnitude smaller than the phase velocity of the drift waves observed in PANTA (450–560 m/s), but similar to the ion diamagnetic velocity (46 m/s). After the F (m = 1) mode amplitude maximum reaches r = 2.5 cm across the interface, the nonlinear coupling strength increases almost instantaneously in the outer region, where the M (m = 4) mode is dominant. Consequently, C (m = 3) mode grows with decays of M (m = 4) and F (m = 1) modes in the amplitude (see 5th and 6th row of Fig. 8 (a) and related panels in Fig. 8 (b)). Simultaneously, the nonlinear coupling drops in the core, and weak nonlinear coupling starts to propagate, with the minimum of F (m = 1) mode amplitude variation, from (r, t)=(0,-0.4) to (2.0 cm,0.5 ms). Then at the time when the minimum of amplitude variation reaches the interface region, the nonlinear coupling in the outer region starts to decrease, while the nonlinear coupling in inner region starts to increase. On the other hand, F (m = 1) mode shows the feature of the connected phase during the strong coupling propagation to the interface, while it starts to show the feature of the disconnected phase during the weak coupling propagation, as is shown in the structure function profile of F (m = 1) mode in the 1st row of Fig. 8 (a). Particularly, the feature of the disconnected phase is clearly observed just after t = 0 ms.

Although the phenomenon is highly statistical or probabilistic, the results suggest a possible scenario, with emphasis on the existence of two fundamental processes in sustaining the repetitive transitions, i.e., the regional connection of F (m = 1) mode and global nonlinear coupling through M (m = 4) mode. The regional connection of F (m = 1) mode occurs at t~ -0.4 ms when the interface amplitude reaches its maximum, as the result of the outward energy transfer which can be inferred from the movement of the peak of amplitude variation of F (m = 1) mode (see Fig. 8 (c)): here, the amplitude variations could be considered as an indicator of local energy of a mode. Simultaneously, the nonlinear coupling starts to spread suddenly to outer region where M(m = 4) mode is dominant. This suggests an important role of the regional connection that could make the nonlinear coupling with M(m = 4) mode global and strong to produce C(m = 3, m = 5) modes. On the other hand, if the global nonlinear couplings with M (m = 4) mode is once realized, the coupling could connect the inner and outer region even if F (m = 1) mode is regionally disconnected for t > 0 ms. The regional disconnection could prevent the internal energy transfer of F(m = 1) to the outer region, consequently, the nonlinear coupling becomes weak in the outer region again. Instead, the energy transfer may occur from outer to inner region through the nonlinear couplings to recover F (m = 1) mode into the first stage of the oscillations (see Fig. 8 (d)). The regional disconnection may realize that F(m = 1) mode could gain or lose its energy independently in the inner (or outer) region. The competition or cooperation of the two processes should play important roles in forming the quasi-periodic transitions.

Summary

Finally, the advanced methods, the analysis of the structural function and the modal coupling function analyses, using FRF expansion, is proposed for investigating imaging data of plasma turbulence obtained　in the tomography measurement of the entire plasma cross-section in PANTA. The analyses are applied to investigate the quasi-periodic transition phenomenon. The structural function analysis finds the regional coupling of F (m = 1) mode, which appears to be unified or separated quasi-periodically. On the other hand, the modal coupling analysis reveals that the two regions can be reconnected through the nonlinear couplings with M (m = 4) mode that spreads the entire plasma region. These analyses disclose the roles of F (m = 1) mode dynamics to sustain the quasi-periodic transition in coupling strength, i.e., the cooperation of the regional and modal couplings between M (m = 4) and F (m = 1) mode. The article demonstrates that the FRF analysis has excellent abilities to extract one of the most fundamental processes occurring in plasma of strong nonlinearity.

Fig. 8
figure 8

(a) The radial and temporal evolutions of the conditional-averaged structure function of F (m = 1) mode, the modal coupling strength among F (m = 1), C (m = 3) and M (m = 4) modes, amplitude variation of the F (m = 1) mode, tilt angle of the F (m = 1) mode, the amplitude variations of C (m = 3) and M (m = 4) modes. The propagation of the F (m = 1) mode amplitude was adopted for the reference time of the conditional average. The radial profile of each value at -1.0, -0.4, 0.0, 0.4, and 0.6 ms are plotted on the right panel. The amplitude variations here are normalized by their maxima during the cycle to indicate the changes in the profile. (b) The snapshot of the F (m = 1) mode spatial structure as a function of time, and (c) the temporal evolution of the radial profile of the amplitude of the F (m = 1), C (m = 1), M (m = 4) modes, together with the total power profile, \(\:{\varvec{P}}_{\varvec{t}\varvec{o}\varvec{t}\varvec{a}\varvec{l}}\left(\varvec{r}\right)={\varvec{A}}_{1}^{2}\left(\varvec{r}\right)+{\varvec{A}}_{3}^{2}\left(\varvec{r}\right)+{\varvec{A}}_{4}^{2}\left(\varvec{r}\right)\) at the times shown in (b). Line style and color of profiles in (a) and (c) correspond to that of a time indicated by each vertical lines.

Full size image

Methods

PANTA

PANTA (Plasma Assembly for Nonlinear Turbulence Analysis) is a cylindrical-shaped magnetized plasma designed for analyzing drift wave turbulence in magnetized plasma. The plasma source of PANTA consists of a cylindrical glass tube whose diameter is 10 cm. The glass tube is wrapped by the double-loop antenna connected to the RF source. The RF source feeds 7 MHz AC into the antenna to produce helicon plasma inside the glass tube. The typical input power is 3–6 kW, and the duration is 0.5 s. Using the RF plasma source, a plasma column whose length and diameter are 4 m and 10 cm, respectively, is produced inside the vacuum chamber32. Argon gas is continuously fed into the vacuum chamber from the source region and pumped by five turbo-molecular pumps equipped at the other end of the vacuum chamber. The axial magnetic field is created by the coils equally spaced along the cylindrical vacuum chamber. The typical magnetic field strength, electron density, electron and ion temperature are 0.09–0.13 T, 1.0 × 1019 m[- [3, 3 eV and 0.36 eV, respectively33,34.

Tomography system

A tomography system is installed in the cylindrical plasma device, PANTA, to observe the 2D emission profile in a region of 10 cm x 10 cm square, including the entire cylindrical plasma. The tomography system consists of six sets of light guide arrays placed around the plasma column 30 degrees apart in an azimuthal direction. Each light guide array comprise 21 lines of sight, which are composed of cylindrical collimators whose length and inner diameter are 62.5 mm and 2 mm, respectively. The collimators are used to limit their viewing angle. After the collimators, the dichroic filters are set to transmit only the Ar II emission ranging from the wavelength of 400 to 500 nm. The emissions are led, through optical feedthrough, to optical fibers in the air, which transfer the emission to photodetectors.

In the present work, the Ar II emission profile is used for the analysis. The photodetector circuit converts emission intensity to a voltage signal. The photodetector circuit is designed to measure the fluctuations in emission up to 30 kHz with a sufficiently high signal-to-noise ratio. The digitized signal of line-integrated emission is converted to the local emission profile using the Maximum Likelihood-Expectation Maximization (MLEM) method35. Using the MLEM method, the emission profile is reconstructed to 25 × 25 matrix data. The details of the hardware are documented in reference36. Note that the emission intensity depends both on the electron density and temperature, so the observed fluctuation represents the fluctuation of electron density or temperature.

Calculation of FRF coefficients

In our tomography reconstruction, the emission distribution is initially expressed by values on square regions as

$$\:\epsilon\:\left(x,y\right)\cong\:\sum\:_{l=0}^{L}{\epsilon\:}_{l}{\delta\:}_{l}\left(x,y\right),$$

where the function \(\:{\delta\:}_{l}\left(x,y\right)\) is finite only on n-th square region with \(\:{\delta\:}_{l}\left(x,y\right)=0\) on the other regions. The local emission value, \(\:{\epsilon\:}_{l}\), is obtained with Maximum Likelihood Expectation Maximization (MLEM). Then the FRF coefficient, \(\:{a}_{nm}\), is calculated from the MLEM values, by making an inner product of a transfer vector, \(\:{{\Phi\:}}_{nm,l}\) and MLEM vector, \(\:{\epsilon\:}_{n}\), as

$$\begin{gathered} \:a_{{nm}} = \int \: \smallint \:\left( {x,y} \right)\varphi \:_{{nm}} \left( {r,\theta \:} \right)dS \cong \:\int \: _{{\Delta \:S_{n} }} \left( {\sum {\:_{{n = 0}}^{N} } \smallint \:_{l} \delta \:_{l} \left( {x,y} \right)} \right) \hfill \\ \varphi \:_{{nm}} \left( {r,\theta \:} \right)dS = \sum {\:_{{l = 0}}^{L} } \Phi \:_{{nm,l}} \smallint \:_{l} \hfill \\ \end{gathered}.$$

where \(\:{\varphi\:}_{nm}\) represents the FRF basis with radial number, n, and azimuthal number, m. The meaning of these suffices are illustrated in Fig. 9. The transfer vector is defined as

$$\:{{\Phi\:}}_{nm,l}={\int\:}_{{\Delta\:}{S}_{n}}{\varphi\:}_{nm}\left(r,\:\theta\:\right)\:{\delta\:}_{l}\left(x,y\right)dxdy,$$

.

where \(\:{\Delta\:}{S}_{l}\) indicates the area of l-th square region where \(\:{\delta\:}_{l}\left(x,y\right)\) is finite. In our analysis, the number of MLEM grids is 625, and each covers a square region with the side length of \(\:{\Delta\:}=0.4\) cm. Here it is worthy to note that the FRF coefficients to satisfy \(\:{\text{r}}_{n}<m{\Delta\:}/2{\uppi\:}\) should be uncertain since elements of the transfer matrix, \(\:{{\Phi\:}}_{nm,l}\), are fallaciously small and its absolute value, \(\:\left|{{\Phi\:}}_{nm}\right|={\left[\sum\:_{l=1}^{L}{\left|{{\Phi\:}}_{nm,l}\right|}^{2}\right]}^{1/2}\), cannot be unitary due to its sinusoidal integral over a period (see Ref 24).

Fig. 9
figure 9

Schematic diagram of the relation of the basis functions and their spatial structure. The suffix of n and m denote the radial position and azimuthal mode number.

Full size image

Error evaluation of modal coupling function

Using the same method for the noise level and error estimation of wavelet bicoherence37the noise level and error of the absolute value of modal coupling function, \(\:{\gamma\:}_{\beta\:}\left({r}_{i},\:{r}_{j},\:\tau\:\right)=\left|{\beta\:}_{m,n}\left({r}_{i},\:{r}_{j},\:\tau\:\right)\right|\), can be estimated as follows,

$$\:{\text{N}\text{o}\text{i}\text{s}\text{e}\:\text{l}\text{e}\text{v}\text{e}\text{l}\::\left(\frac{\frac{1}{2}{f}_{sample\:}}{\text{min}\left[{f}_{{c}_{j,\:m}},\:{f}_{{c}_{j,\:n}},\:{f}_{{c}_{i,\:m+n}}\right]}\frac{1}{N}\:\right)}^{\frac{1}{2}}\text{E}\text{r}\text{r}\text{o}\text{r}\::\frac{2}{\sqrt{N}}{\gamma\:}_{\beta\:}\left({r}_{i},\:{r}_{j},\:\tau\:\right).$$

Here, \(\:N\) represents the number of data points to calculate the coupling function. The modal coupling function estimated at different locations with delay time \(\:\tau\:=0\) represents the strength of nonlinear coupling of fluctuations at radially distant locations, which may indicate the strength of the nonlocal coupling.