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Statistical properties of turbulence in a toroidal magnetized ECR plasma
Physics Letters A 372 (2008) 1081–1087 www.elsevier.com/locate/pla
Statistical properties of turbulence in a toroidal magnetized ECR plasma Yi Yu ∗ , Rong-hua Lu, Zhi-jiang Wang, Yi-zhi Wen, Chang-xuan Yu, Shu-de Wan, Wan-dong Liu CAS Key Laboratory of Plasma Physics, Modern Physics Department, School of Science, University of Science and Technology of China, 230026 Hefei, People’s Republic of China Received 3 March 2007; received in revised form 26 June 2007; accepted 3 July 2007 Available online 5 July 2007 Communicated by F. Porcelli
Abstract The statistical analyses of fluctuation data measured by electrostatic-probe arrays clearly show that the self-organized criticality (SOC) avalanches are not the dominant behaviors in a toroidal ECR plasma in the SMT (Simple Magnetic Torus) mode of KT-5D device. The f −1 index region in the auto-correlation spectra of the floating potential Vf and the ion saturation current Is , which is a fingerprint of a SOC system, ranges only in a narrow frequency band. By investigating the Hurst exponents at increasingly coarse grained time series, we find that at a time scale of τ > 100 µs, there exists no or a very weak long-range correlation over two decades in τ . The difference between the PDFs of Is and Vf clearly shows a more global nature of the latter. The transport flux induced by the turbulence suggests that the natural intermittency of turbulent transport maybe independent of the avalanche induced by near criticality. The drift instability is dominant in a SMT plasma generated by means of ECR discharges. © 2007 Elsevier B.V. All rights reserved. PACS: 52.55.Fa; 52.25.Gj; 52.35.Ra Keywords: Simple magnetic torus; Self-organized criticality; Hurst exponent
1. Introduction Anomalous transport in the magnetically confined plasmas is believed to be induced by the turbulence. The self-organized criticality (SOC) [1] models have been widely used in many magnetically confined plasmas to explain the anomalous transport. 1/f noise in power spectral “fingerprint” and long-range spatial and temporal correlation through scale invariance are the main ideas of the concept of SOC, in which avalanchetype long-range transport is the result of the self-organization of the near-critical, nonlinear dynamical systems. In recent years, SOC characters such as the shapes of auto-power spectra and PDFs, and the values of Hurst exponents (H ), have been explored in the scrape-off layer (SOL) of tokamaks and in the Simple Magnetic Torus (SMTs) which have a quite good approximation to the tokamak SOL regions. It shows that * Corresponding author.
E-mail address: [email protected] (Y. Yu). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.07.006
the methods for detecting auto-power spectrum, Hurst exponents and PDFs on different time scales can be useful tools in the study of turbulent plasmas. In these experiments, all of the power spectra have three distinct frequency regions, each with a characteristic power law (f 0 , f −1 , f −4 ), which implies typical SOC systems [2–4], and a pronounced peak of several kHz is observed in different devices [5,6]. Non-Gaussian PDFs for electron pressure fluctuations p are believed to arise from a nonlinear relationship between p and the plasma potential Vp [7]. The explorations of Hurst exponents show that the values of H change in different plasmas [7–10], indicating that large scale correlations probably rely on the plasma sources, the geometrical structures of the plasmas or the positions of the observation points. Among these experiments in the SMTs, hot filament or low frequency RF discharges were used. In this Letter we are to report the statistical characteristics in a SMT plasma generated by means of ECR discharges in the device KT-5D [11–13]. The power spectra, Hurst exponents and PDFs of increasingly coarse grained time series will be shown and discussed in the following sections.
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2. Experimental setup and the principle for the data analysis KT-5D is a small double-functional toroidal device (operational as a SMT or a tokamak [11]) with a major radius of R = 32.5 cm, a minor radius of r = 12.5 cm, and a limiter radius of 9 cm. When it works in the SMT mode, a constant DC current supply of 1200 A can generate a magnetic field of BT 0 = BT (R0 ) = 1000 Gauss on the torus axis R0 . Hydrogen is chosen as the working gas in most of the experiments. Two sets of 2.5 kW microwave power systems are used to produce and sustain the plasmas. The magnetron works at the frequency of fce = 2.45 GHz (corresponds to the EC resonance field BT = 2πfce me /e ∼ = 875 Gauss, where me , e are electron mass and charge respectively). The wave is injected through either the horizontal window on the low field side or the vertical window on the top of the torus. It can produce a plasma with an electron density of ne = 2–7 × 1010 cm−3 and a plasma temperature of Te = 10–20 eV. There are several sets of electrostatic-probe arrays, movable horizontally along major radius or vertically along the perpendicular line through the center of the cross-section R = R0 , for measuring the plasma parameters point by point. All signals from the probes are isolated from the vessel wall by low drift, broad bandwidth (DC-2 MHz) optical-coupled amplifiers, and feed to the data acquisition system of 48-channel synchronous digitizers (recording with a sampling rate up to 5 MHz, 12 bits). Each channel is independently isolated, effectively eliminating cross-talks between channels. The most common plasma production mechanism in the KT5D device is by application of an O-mode wave through the horizontal window on the low field side, operated in H2 pressure 2 × 10−2 Pa and toroidal field 840 Gauss on the axis. The Fourier transform of the auto-correlation function (ACF) is used to obtain the shape of the auto-power frequency spectrum. It reveals the correlation properties of a signal. Although the Hurst exponent, which implies the long-time correlation of SOC-governed systems, could be directly calculated from the ACF, it brings considerable errors for its long time lag tail. A substitute is shown as following, which is similar with the work of Fredriksen and Riccardi [14,15]. N = 131072 samples of ion saturation current Is or floating potential Vf are gained through the data acquisition system with a sample time of τs = 2 µs. For Is or Vf , the mean of a snippet with a time lag of τ is:
the probability distribution function is defined as: PDF τ = (1/Nτ ) dNτ /dτ Is , where the dNτ is the count in a length of dτ Is . Investigating the PDFs of turbulent fluctuations was emphasized by K41 [16], in which Gaussian fit PDFs were considered random fluctuations. If the shape of PDF dose not change with the time-scale, it is customary to refer to the PDF as selfsimilar. And in this condition, there exists a power-law scaling of στ ∼ τ −α or στ 1 /στ 2 ∼ (τ1 /τ2 )−α . And the Hurst exponent then can be expressed by: H = 1 − α. 0.5 < H < 1, 0 < H < 0.5 or H = 0.5 indicates positive correlations, anticorrelations or no correlations, respectively. A Hurst value between 0.5 and 1 of ion saturation current or floating voltage indicates the existence of the long-range correlation of electrostatic fluctuation which is independent of the device. 3. Results and discussions The plasma turbulence can be identified by direct analysis of the raw signals from the probes. Fig. 1 shows a snippet of the fluctuations of ion saturation current Is and the fluctuations of the floating potential Vf at a LFS position of R = 38 cm in KT-5D. It is clearly seen that there exists a periodic component of T = 160 µs, which probably is a signature of the background fluctuations according to Ref. [17]. The auto-power spectra of Is and Vf at R = 38 cm at LFS are shown in Fig. 2. It can be seen that there are two dominant regions in the frequency ranges of below 6.5 kHz and over 13 kHz with approximate decay indices of 0 and −4 (−4 for Is , −4.7 for Vf ), respectively. Ranging only in a narrow frequency band from 8 kHz to 10 kHz, the intermediate region with an index of −1, which is predicted by the SOC theory [18–20], is clearly not as dominant as the other two. The much narrower frequency band of the −1 index comparing with the work of some of other devices [5,7,9,10,14,21] may undermine the existence of SOC-governed behaviors. There are sharper separations between ranges comparing with other experimental
t+τ/2
Is (t ) dt .
Isτ (t) = 1/τ t−τ/2
And then the variance of the signal is: T στ2
= 1/T
2 Isτ (t ) dt ,
0
where T is the acquisition time of T = N τs . By means of changing the time lag of τ , we can get a series of Isτ . Then
Fig. 1. A snippet of the fluctuations of ion saturation current Is and the fluctuations of the floating potential Vf at R = 38 cm at LFS.
Y. Yu et al. / Physics Letters A 372 (2008) 1081–1087
observations such as in HT-6M [9] and TEXTOR tokamak [5]. These three ranges indicate single events with a global scale, intermediate range events related to the overlapping of avalanche transport, and small scale events, respectively, as the frequency increases. A pronounced peak at a frequency of 6.5 kHz exists which accords with the period of 160 µs mentioned before. Spectra at different radii are also explored and similar spectrum shapes are obtained. One of these spectra of Is at the HFS radius of R = 30.5 is shown in Fig. 3. The spectrum at R = 38 cm is still shown here just for comparison. As one can see, although the decay index of −1 changes to −1.7 and the decay range of −4 is not clearly observed, the pronounced peak of 6.5 kHz still exists, which resembles other experimental observations in fusion or nonfusion devices despite of a slight difference among the absolute values of the peaked frequencies. Comparing with Dr. Xu’s work in the tokamak of TEXTOR [5], the spectra at the HFS of the SMT KT-5D are similar with those in the plasma edge in tokamak with a pronounced peak at 10 kHz and the
Fig. 2. The auto-power spectra of Is and Vf at R = 38 cm at LFS.
Fig. 3. The auto-power spectra of Is at R = 38 cm at LFS and R = 30.5 cm at HFS.
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spectra at the LFS of SMT are very similar with those in the SOL or LCFS. According to Figs. 2 and 3, the existence of the peak and the f −1 range does not conflict in the SMT, while in TEXTOR Ref. [5] it is said that coherent modes hided the intermediate f −1 range because no pronounced peak was found in the spectrum when f −1 range existed. Hurst exponents H were also explored using the method as was described in Section 3. The log–log plot between στ and τ of Is at the radius of R = 30.5 cm is shown in Fig. 4. At a time scale of τ < 100 µs, H (which can be derived from the slope of the plot) is very close to 1, indicating the small time scale events. But according to Ref. [21], this H is less meaningful for the existence of the 160 µs periodic component as was discussed before (Fig. 1). At 102 µs < τ < 104 µs, H = 0.54 means that there exists no long-range correlation at this time scale. Fig. 5 shows the στ –τ plot of Vf at the same radius. At the same time scale, H equals to 0.57, which suggests no or a very weak long-range correlation. These results are almost the same with the work in a similar SMT of Blaamann [14], but ten
Fig. 4. The log–log plot between στ and τ of Is at the radius of R = 30.5 cm.
Fig. 5. The log–log plot between στ and τ of Vf at the radius of R = 30.5 cm.
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times of sample points comparing with Ref. [14] are analyzed here to avoid the unreliable statistics as were commented by Prof. K. Rypdal [21]. However it does show differences from some of the findings at the edge of Tokamaks, such as in Tore
Fig. 6. Hurst exponent distribution at different radii in the cross section at a time scale of τ > 100 µs.
Supra according to Dr. Wang’s work [10], where Hurst exponents are well above 0.6. Fig. 6 shows the H distribution at different radii in the cross section at a time scale of τ > 100 µs. It is obvious that H is very close to 0.5 as the radius changes at the low field side (LFS), and thus no long-range correlation exists on the whole. The results seem consistent with Fig. 2 that the f −1 region is not as well pronounced as the other two regions, which may suggest the intermediate time scaled events are not dominant. At most of the radial positions of the high field side (HFS) of R < 30 cm, the values of H are distributed between 0.60 and 0.75, which shows a pronounced SOC character. The PDFs of Vf and Is under different time scales are shown for the position at R = 30.5 cm (Fig. 7). The dashed line in each figure shows the best fit by a Gaussian. At shorter time-scales of τ = 2, 4, 6, 8 µs (Fig. 7(a)), the four PDFs of Vf are so similar that it is hard to distinguish one from another and only a slight degree of non-Gaussianity is found, which shows great similarity with K. Rapdal’s work [7] but great difference from Fredriksen’s [14]. The PDFs of Is at the same four time scales display a quite different image (Fig. 7(b)). Although the PDFs superposed very well, pronounced deviations from a Gaussian are observed. The non-Gaussian distribution is an extreme value distribution (EVD) which is similar to the findings in Blaamann [7]. PDFs of Vf and Is under longer time scales of 256,
(a)
(b)
(c)
(d)
Fig. 7. The PDFs of Vf and Is under different time scales for the position at R = 30.5 cm.
Y. Yu et al. / Physics Letters A 372 (2008) 1081–1087
512, 1024, 2048 µs are shown in Fig. 7(c) and (d), respectively, which display a good Gaussian fit. The larger fluctuations result from less samples because of the increased time-scales. In the following, we focus on how this statistical approach to plasma turbulence characterizes the key parameter of turbulence, namely the turbulent flux. The radial turbulent transport flux Γr is: Γr = n˜ e v˜r = n˜ e E˜ θ /B, where the fluctuation of electron density ne and toroidal electronic field Eθ are directly deduced from the fluctuation of the ion saturation current Is , the electron temperature Te and the plasma potential of Vp = Vf + 2.5Te . Fig. 8 exhibits the time sequence of Γr at a radial position of R = 33 cm, and positive Γr means outward flux along radius in the minor cross section of plasmas. It is clearly seen that the turbulent transport is stochastic, and the integral flux is mainly from the intermittent pulses of small time scales. Using similar method, statistical properties of Γr are also investigated. Fig. 9 shows the auto-
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power spectra of Γr at three different radial positions. It is to note that an f is timed to the spectra just for convenience, and no horizontal line is found in these spectra, which suggests a non-SOC system. The στ ∼ τ −α relation of Γr at R = 38 cm is also investigated (Fig. 10). A Hurst value of 0.49 indicates that no long-range correlation exists. A spatial distribution of Hurst exponents are shown in Fig. 11. The values of H are all closed to 0.5, at least at the LFS of plasmas, which also undermines the idea of a SOC plasma. By means of investigating the statistical properties of Γr , we conclude that the transport flux induced by turbulence is intermittent, although this intermittency is not of a SOC type. To discern whether this plasma is a SOC system or not more clearly, we relate the time between bursts to the amplitude of the burst (Fig. 12) using conditional
Fig. 10. The στ ∼ τ −α relation of Γr at R = 38 cm showing a Hurst value of 0.49. Fig. 8. The time sequence of Γr at a radial position of R = 33 cm.
Fig. 9. The auto-power spectra of Γr at three different radial positions.
Fig. 11. The spatial distribution of Hurst exponents of Γr along radius at the cross section of plasmas.
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Fig. 12. The time between bursts with amplitude greater than three times the standard deviation to the amplitude of the burst.
Fig. 14. The radial distribution of phase angle between the fluctuations of p and Vp .
devices [22,23]. Using the relation of: V˜p = V˜f + 2.5T˜e , we present the cross-power spectrum and the phase angle at R = 33 cm between the fluctuations of plasma pressure p = ne Te and plasma voltage Vp (Fig. 13). A pronounced peak of about 6.5 kHz is found in the cross-power spectrum, and the relevant phase angle is about 0.09π . This small phase angle consists with the property of drift mode. A radial distribution of phase angle between the fluctuations of p and Vp is shown in Fig. 14. Obviously, the drift mode is dominant because the phase angles are near zero at most of radial positions. This result is different from Refs. [22,23], in which the conclusion of a dominant flute mode is drawn. This difference reveals the fact that the fluctuation and the gradient of electron temperature are not neglectable even in a low temperature plasma. 4. Conclusions
Fig. 13. The cross-power spectrum and the phase angle at R = 33 cm between the fluctuations of plasma pressure p and plasma voltage Vp .
analysis with a threshold of 3σ , where σ is the deviation of the amplitude of the burst. A SOC-type system would lead to a near linear relation between the time and the amplitude, but Fig. 12 shows no simple relation between them. Consequently, we deduce that the underlying mechanism of the intermittent flux is not SOC-type. As one can see from Fig. 1, the phase angle between the fluctuation of Is and Vf is around π , it is a typical character of the flute mode. This is not consistent with the fact that the gradients of the density and the magnetic field are opposite at most of the radial positions in this shot. This inconsistency comes from the unsuitable neglect of the gradients of electron temperature Te , as was done in many of other magnetically confined
The plasma under ECR discharging in KT-5D shows similarities not only to other SMT devices on different discharging manners, but also to the SOL region of tokamaks. Two dominant regions of frequency dependence (f 0 and f −4 ), a narrow frequency band of f −1 index and a pronounced peak of 6.5 kHz have been observed in the spectra of the Vf and Is . The underlying mechanism of the 6.5 kHz peak is not clear, although similar peaks are also found in most of fusion or non-fusion devices. The degree of the long-range correlation (Hurst exponent) at time scales larger than 100 µs along radius at LFS ranges around 0.5, which suggests that no or a very weak long-range correlation exists. But at HFS, the H values are well above 0.5, suggesting an existence of a pronounced self-similarity or longrange correlation. The difference of Hurst values between LFS and HFS implies a more stochastic particle transport along radius at LFS, and thus suggests that the statistical behavior changes some even in the same magnetically confined device. The difference comes from different radius observation posi-
Y. Yu et al. / Physics Letters A 372 (2008) 1081–1087
tions or, in other words, maybe the plasma conditions such as the magnetic field or the electric field. At even larger time scales such as τ > 104 µs, the exploring of Hurst exponents is not performed for the larger error bar. This also indicates that although SMTs are good approximations to the SOL region of tokamaks and it is convenient for diagnosing because of the low plasma parameters, some plasma behaviors at LFS and HFS are not the same. The plasmas at HFS show more characteristic properties that are similar to the SOC system than that at LFS, and further investigation to the plasma behavior at HFS is needed. At all observable time scales, the PDFs of Vf exhibit a well Gaussian shape. The superposed PDFs of Vf imply a self-similarity at larger time scales, while it is not the case at small time scales for the existence of a 160 µs periodic signal according to Rypdal’s theory [21]. The PDFs of Is show great difference from those of Vf . These deviations from Gaussian distributions come from the high frequency tails of the spectra which suggest small scaled turbulences. The different statistics between Is and Vf is associated with the local nature of the former and more global nature of the latter. The low Hurst values around 0.5 of the fluctuations of the Is and Vf at LFS and the stochastic turbulence revealed by the Gaussian-shape PDFs of Vf imply the non-SOC characters of the transport flux induced by turbulence. By comparing the statistical properties of the turbulent signals of Vf and Is in ECR discharging plasmas with the results from some of the other Tokamaks and SMTs, it is found that although the discharging manners and the configurations of plasmas are different, the auto-power spectra, the Hurst exponents and PDFs show some similar characters. These similarities allow us to conclude that anomalous particle transport shares some universal properties in different magnetically confined devices. Here we only conclude that the auto-power spectra, the Hurst exponents and PDFs show some similar characteristic properties or “fingerprints” of a typical SOC system. As we know, many other mechanism may lead to these properties such as the 1/f spectrum and, on the other hand, many SOC systems show different spectrum of 1/f . The statistical features of the transport flux induced by turbulence dose not show the longrange correlation what is a typical characteristic property of a SOC system. The Hurst value of 0.5 indicates a complete stochastic behavior of the flux, which affirms the non-existence of the SOC. Although it is not SOC-type, the flux is still intermittent. These results imply that the natural intermittency of turbulent transport maybe independent of the avalanche induced by near criticality, and it is an experimental evidence of Mahdizadeh’s work [24]. This is true for at least the LFS of the double-functional device KT-5D. A more detailed image of the
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intermittency of the transport flux needs further investigation, and relevant work using two-dimensional distributed Langmuir probe arrays and fast cameras is underway. By means of investigating the phase angles between the fluctuations of plasma pressure p and the plasma potential Vp , we conclude that the dominant instability in the SMT discharge is drift instability. The difference between the plasma potential Vp and the floating potential Vf indicates that it is not wise to neglect the influence of the gradient and the fluctuation of electron temperature even in a low temperature plasma. It should be noted that investigations here are only for a certain toroidal field and gas pressure. As discussed in detail in Ref. [25], the plasma properties from microwave excitation significantly rely on the discharging conditions, i.e., the toroidal field, the vertical field and the gas pressure. The relevant experiments are underway. Acknowledgements This work is supported by the National Science Foundation of China (Grant Nos. 1023 5010 and 1033 5060) and grants from the Ministry of Education of the People’s Republic of China and the Chinese Academy of Sciences. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
P. Bak, C. Tang, K. Weisenfeld, Phys. Rev. Lett. 59 (1987) 381. B.A. Carreras, et al., Phys. Rev. Lett. 80 (1998) 4438. M.A. Pedrosa, et al., Phys. Rev. Lett. 82 (1999) 3621. B.A. Carreras, et al., Phys. Plasmas 6 (1999) 1885. Y.H. Xu, et al., Phys. Plasmas 11 (2004) 5413. R. Trasarti-Battostpmo, et al., Phys. Plasmas 9 (2002) 3369. K. Rypdal, et al., Phys. Plasmas 10 (2003) 2686. M. Gilmore, et al., Phys. Plasmas 9 (2002) 1312. W.-H. Wang, et al., Chin. Phys. Lett. 18 (2001) 793. G. Wang, et al., Phys. Plasmas 7 (2000) 1181. Y. Yu, et al., Rev. Sci. Instrum. 77 (2006) 123502. Y. Yu, et al., Plasma Sci. Technol. 8 (1) (2006) 91. M. Xu, et al., Rev. Sci. Instrum. 76 (2005) 043502. A. Fredriksen, et al., Phys. Plasmas 10 (2003) 4335. R. Trasartu-Battistoni, et al., Phys. Plasmas 9 (2002) 3369. A.N. Kolmogorov, et al., Sci. URSS 30 (1941) 301. A. Fredriksen, et al., Plasma Phys. Control. Fusion 45 (2003) 721. D.E. Newman, et al., Phys. Plasmas 3 (1996) 1858. B.A. Carreras, et al., Phys. Plasmas 3 (1996) 2903. T. Hwa, et al., Phys. Rev. A 45 (1992) 7002. K. Rypdal, et al., Phys. Plasmas 11 (2004) 3679. S.V. Ratynskaia, et al., Contrib. Plasma Phys. 41 (2001) 449. S.V. Ratynskaia, et al., Phys. Plasmas 9 (2002) 4135. N. Mahdizadeh, et al., Phys. Plasmas 11 (2004) 3932. Z.-h. Zhu, et al., Plasma Sci. Technol. 9 (2007) 5.
Statistical properties of turbulence in a toroidal magnetized ECR plasma Yi Yu ∗ , Rong-hua Lu, Zhi-jiang Wang, Yi-zhi Wen, Chang-xuan Yu, Shu-de Wan, Wan-dong Liu CAS Key Laboratory of Plasma Physics, Modern Physics Department, School of Science, University of Science and Technology of China, 230026 Hefei, People’s Republic of China Received 3 March 2007; received in revised form 26 June 2007; accepted 3 July 2007 Available online 5 July 2007 Communicated by F. Porcelli
Abstract The statistical analyses of fluctuation data measured by electrostatic-probe arrays clearly show that the self-organized criticality (SOC) avalanches are not the dominant behaviors in a toroidal ECR plasma in the SMT (Simple Magnetic Torus) mode of KT-5D device. The f −1 index region in the auto-correlation spectra of the floating potential Vf and the ion saturation current Is , which is a fingerprint of a SOC system, ranges only in a narrow frequency band. By investigating the Hurst exponents at increasingly coarse grained time series, we find that at a time scale of τ > 100 µs, there exists no or a very weak long-range correlation over two decades in τ . The difference between the PDFs of Is and Vf clearly shows a more global nature of the latter. The transport flux induced by the turbulence suggests that the natural intermittency of turbulent transport maybe independent of the avalanche induced by near criticality. The drift instability is dominant in a SMT plasma generated by means of ECR discharges. © 2007 Elsevier B.V. All rights reserved. PACS: 52.55.Fa; 52.25.Gj; 52.35.Ra Keywords: Simple magnetic torus; Self-organized criticality; Hurst exponent
1. Introduction Anomalous transport in the magnetically confined plasmas is believed to be induced by the turbulence. The self-organized criticality (SOC) [1] models have been widely used in many magnetically confined plasmas to explain the anomalous transport. 1/f noise in power spectral “fingerprint” and long-range spatial and temporal correlation through scale invariance are the main ideas of the concept of SOC, in which avalanchetype long-range transport is the result of the self-organization of the near-critical, nonlinear dynamical systems. In recent years, SOC characters such as the shapes of auto-power spectra and PDFs, and the values of Hurst exponents (H ), have been explored in the scrape-off layer (SOL) of tokamaks and in the Simple Magnetic Torus (SMTs) which have a quite good approximation to the tokamak SOL regions. It shows that * Corresponding author.
E-mail address: [email protected] (Y. Yu). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.07.006
the methods for detecting auto-power spectrum, Hurst exponents and PDFs on different time scales can be useful tools in the study of turbulent plasmas. In these experiments, all of the power spectra have three distinct frequency regions, each with a characteristic power law (f 0 , f −1 , f −4 ), which implies typical SOC systems [2–4], and a pronounced peak of several kHz is observed in different devices [5,6]. Non-Gaussian PDFs for electron pressure fluctuations p are believed to arise from a nonlinear relationship between p and the plasma potential Vp [7]. The explorations of Hurst exponents show that the values of H change in different plasmas [7–10], indicating that large scale correlations probably rely on the plasma sources, the geometrical structures of the plasmas or the positions of the observation points. Among these experiments in the SMTs, hot filament or low frequency RF discharges were used. In this Letter we are to report the statistical characteristics in a SMT plasma generated by means of ECR discharges in the device KT-5D [11–13]. The power spectra, Hurst exponents and PDFs of increasingly coarse grained time series will be shown and discussed in the following sections.
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2. Experimental setup and the principle for the data analysis KT-5D is a small double-functional toroidal device (operational as a SMT or a tokamak [11]) with a major radius of R = 32.5 cm, a minor radius of r = 12.5 cm, and a limiter radius of 9 cm. When it works in the SMT mode, a constant DC current supply of 1200 A can generate a magnetic field of BT 0 = BT (R0 ) = 1000 Gauss on the torus axis R0 . Hydrogen is chosen as the working gas in most of the experiments. Two sets of 2.5 kW microwave power systems are used to produce and sustain the plasmas. The magnetron works at the frequency of fce = 2.45 GHz (corresponds to the EC resonance field BT = 2πfce me /e ∼ = 875 Gauss, where me , e are electron mass and charge respectively). The wave is injected through either the horizontal window on the low field side or the vertical window on the top of the torus. It can produce a plasma with an electron density of ne = 2–7 × 1010 cm−3 and a plasma temperature of Te = 10–20 eV. There are several sets of electrostatic-probe arrays, movable horizontally along major radius or vertically along the perpendicular line through the center of the cross-section R = R0 , for measuring the plasma parameters point by point. All signals from the probes are isolated from the vessel wall by low drift, broad bandwidth (DC-2 MHz) optical-coupled amplifiers, and feed to the data acquisition system of 48-channel synchronous digitizers (recording with a sampling rate up to 5 MHz, 12 bits). Each channel is independently isolated, effectively eliminating cross-talks between channels. The most common plasma production mechanism in the KT5D device is by application of an O-mode wave through the horizontal window on the low field side, operated in H2 pressure 2 × 10−2 Pa and toroidal field 840 Gauss on the axis. The Fourier transform of the auto-correlation function (ACF) is used to obtain the shape of the auto-power frequency spectrum. It reveals the correlation properties of a signal. Although the Hurst exponent, which implies the long-time correlation of SOC-governed systems, could be directly calculated from the ACF, it brings considerable errors for its long time lag tail. A substitute is shown as following, which is similar with the work of Fredriksen and Riccardi [14,15]. N = 131072 samples of ion saturation current Is or floating potential Vf are gained through the data acquisition system with a sample time of τs = 2 µs. For Is or Vf , the mean of a snippet with a time lag of τ is:
the probability distribution function is defined as: PDF τ = (1/Nτ ) dNτ /dτ Is , where the dNτ is the count in a length of dτ Is . Investigating the PDFs of turbulent fluctuations was emphasized by K41 [16], in which Gaussian fit PDFs were considered random fluctuations. If the shape of PDF dose not change with the time-scale, it is customary to refer to the PDF as selfsimilar. And in this condition, there exists a power-law scaling of στ ∼ τ −α or στ 1 /στ 2 ∼ (τ1 /τ2 )−α . And the Hurst exponent then can be expressed by: H = 1 − α. 0.5 < H < 1, 0 < H < 0.5 or H = 0.5 indicates positive correlations, anticorrelations or no correlations, respectively. A Hurst value between 0.5 and 1 of ion saturation current or floating voltage indicates the existence of the long-range correlation of electrostatic fluctuation which is independent of the device. 3. Results and discussions The plasma turbulence can be identified by direct analysis of the raw signals from the probes. Fig. 1 shows a snippet of the fluctuations of ion saturation current Is and the fluctuations of the floating potential Vf at a LFS position of R = 38 cm in KT-5D. It is clearly seen that there exists a periodic component of T = 160 µs, which probably is a signature of the background fluctuations according to Ref. [17]. The auto-power spectra of Is and Vf at R = 38 cm at LFS are shown in Fig. 2. It can be seen that there are two dominant regions in the frequency ranges of below 6.5 kHz and over 13 kHz with approximate decay indices of 0 and −4 (−4 for Is , −4.7 for Vf ), respectively. Ranging only in a narrow frequency band from 8 kHz to 10 kHz, the intermediate region with an index of −1, which is predicted by the SOC theory [18–20], is clearly not as dominant as the other two. The much narrower frequency band of the −1 index comparing with the work of some of other devices [5,7,9,10,14,21] may undermine the existence of SOC-governed behaviors. There are sharper separations between ranges comparing with other experimental
t+τ/2
Is (t ) dt .
Isτ (t) = 1/τ t−τ/2
And then the variance of the signal is: T στ2
= 1/T
2 Isτ (t ) dt ,
0
where T is the acquisition time of T = N τs . By means of changing the time lag of τ , we can get a series of Isτ . Then
Fig. 1. A snippet of the fluctuations of ion saturation current Is and the fluctuations of the floating potential Vf at R = 38 cm at LFS.
Y. Yu et al. / Physics Letters A 372 (2008) 1081–1087
observations such as in HT-6M [9] and TEXTOR tokamak [5]. These three ranges indicate single events with a global scale, intermediate range events related to the overlapping of avalanche transport, and small scale events, respectively, as the frequency increases. A pronounced peak at a frequency of 6.5 kHz exists which accords with the period of 160 µs mentioned before. Spectra at different radii are also explored and similar spectrum shapes are obtained. One of these spectra of Is at the HFS radius of R = 30.5 is shown in Fig. 3. The spectrum at R = 38 cm is still shown here just for comparison. As one can see, although the decay index of −1 changes to −1.7 and the decay range of −4 is not clearly observed, the pronounced peak of 6.5 kHz still exists, which resembles other experimental observations in fusion or nonfusion devices despite of a slight difference among the absolute values of the peaked frequencies. Comparing with Dr. Xu’s work in the tokamak of TEXTOR [5], the spectra at the HFS of the SMT KT-5D are similar with those in the plasma edge in tokamak with a pronounced peak at 10 kHz and the
Fig. 2. The auto-power spectra of Is and Vf at R = 38 cm at LFS.
Fig. 3. The auto-power spectra of Is at R = 38 cm at LFS and R = 30.5 cm at HFS.
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spectra at the LFS of SMT are very similar with those in the SOL or LCFS. According to Figs. 2 and 3, the existence of the peak and the f −1 range does not conflict in the SMT, while in TEXTOR Ref. [5] it is said that coherent modes hided the intermediate f −1 range because no pronounced peak was found in the spectrum when f −1 range existed. Hurst exponents H were also explored using the method as was described in Section 3. The log–log plot between στ and τ of Is at the radius of R = 30.5 cm is shown in Fig. 4. At a time scale of τ < 100 µs, H (which can be derived from the slope of the plot) is very close to 1, indicating the small time scale events. But according to Ref. [21], this H is less meaningful for the existence of the 160 µs periodic component as was discussed before (Fig. 1). At 102 µs < τ < 104 µs, H = 0.54 means that there exists no long-range correlation at this time scale. Fig. 5 shows the στ –τ plot of Vf at the same radius. At the same time scale, H equals to 0.57, which suggests no or a very weak long-range correlation. These results are almost the same with the work in a similar SMT of Blaamann [14], but ten
Fig. 4. The log–log plot between στ and τ of Is at the radius of R = 30.5 cm.
Fig. 5. The log–log plot between στ and τ of Vf at the radius of R = 30.5 cm.
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times of sample points comparing with Ref. [14] are analyzed here to avoid the unreliable statistics as were commented by Prof. K. Rypdal [21]. However it does show differences from some of the findings at the edge of Tokamaks, such as in Tore
Fig. 6. Hurst exponent distribution at different radii in the cross section at a time scale of τ > 100 µs.
Supra according to Dr. Wang’s work [10], where Hurst exponents are well above 0.6. Fig. 6 shows the H distribution at different radii in the cross section at a time scale of τ > 100 µs. It is obvious that H is very close to 0.5 as the radius changes at the low field side (LFS), and thus no long-range correlation exists on the whole. The results seem consistent with Fig. 2 that the f −1 region is not as well pronounced as the other two regions, which may suggest the intermediate time scaled events are not dominant. At most of the radial positions of the high field side (HFS) of R < 30 cm, the values of H are distributed between 0.60 and 0.75, which shows a pronounced SOC character. The PDFs of Vf and Is under different time scales are shown for the position at R = 30.5 cm (Fig. 7). The dashed line in each figure shows the best fit by a Gaussian. At shorter time-scales of τ = 2, 4, 6, 8 µs (Fig. 7(a)), the four PDFs of Vf are so similar that it is hard to distinguish one from another and only a slight degree of non-Gaussianity is found, which shows great similarity with K. Rapdal’s work [7] but great difference from Fredriksen’s [14]. The PDFs of Is at the same four time scales display a quite different image (Fig. 7(b)). Although the PDFs superposed very well, pronounced deviations from a Gaussian are observed. The non-Gaussian distribution is an extreme value distribution (EVD) which is similar to the findings in Blaamann [7]. PDFs of Vf and Is under longer time scales of 256,
(a)
(b)
(c)
(d)
Fig. 7. The PDFs of Vf and Is under different time scales for the position at R = 30.5 cm.
Y. Yu et al. / Physics Letters A 372 (2008) 1081–1087
512, 1024, 2048 µs are shown in Fig. 7(c) and (d), respectively, which display a good Gaussian fit. The larger fluctuations result from less samples because of the increased time-scales. In the following, we focus on how this statistical approach to plasma turbulence characterizes the key parameter of turbulence, namely the turbulent flux. The radial turbulent transport flux Γr is: Γr = n˜ e v˜r = n˜ e E˜ θ /B, where the fluctuation of electron density ne and toroidal electronic field Eθ are directly deduced from the fluctuation of the ion saturation current Is , the electron temperature Te and the plasma potential of Vp = Vf + 2.5Te . Fig. 8 exhibits the time sequence of Γr at a radial position of R = 33 cm, and positive Γr means outward flux along radius in the minor cross section of plasmas. It is clearly seen that the turbulent transport is stochastic, and the integral flux is mainly from the intermittent pulses of small time scales. Using similar method, statistical properties of Γr are also investigated. Fig. 9 shows the auto-
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power spectra of Γr at three different radial positions. It is to note that an f is timed to the spectra just for convenience, and no horizontal line is found in these spectra, which suggests a non-SOC system. The στ ∼ τ −α relation of Γr at R = 38 cm is also investigated (Fig. 10). A Hurst value of 0.49 indicates that no long-range correlation exists. A spatial distribution of Hurst exponents are shown in Fig. 11. The values of H are all closed to 0.5, at least at the LFS of plasmas, which also undermines the idea of a SOC plasma. By means of investigating the statistical properties of Γr , we conclude that the transport flux induced by turbulence is intermittent, although this intermittency is not of a SOC type. To discern whether this plasma is a SOC system or not more clearly, we relate the time between bursts to the amplitude of the burst (Fig. 12) using conditional
Fig. 10. The στ ∼ τ −α relation of Γr at R = 38 cm showing a Hurst value of 0.49. Fig. 8. The time sequence of Γr at a radial position of R = 33 cm.
Fig. 9. The auto-power spectra of Γr at three different radial positions.
Fig. 11. The spatial distribution of Hurst exponents of Γr along radius at the cross section of plasmas.
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Fig. 12. The time between bursts with amplitude greater than three times the standard deviation to the amplitude of the burst.
Fig. 14. The radial distribution of phase angle between the fluctuations of p and Vp .
devices [22,23]. Using the relation of: V˜p = V˜f + 2.5T˜e , we present the cross-power spectrum and the phase angle at R = 33 cm between the fluctuations of plasma pressure p = ne Te and plasma voltage Vp (Fig. 13). A pronounced peak of about 6.5 kHz is found in the cross-power spectrum, and the relevant phase angle is about 0.09π . This small phase angle consists with the property of drift mode. A radial distribution of phase angle between the fluctuations of p and Vp is shown in Fig. 14. Obviously, the drift mode is dominant because the phase angles are near zero at most of radial positions. This result is different from Refs. [22,23], in which the conclusion of a dominant flute mode is drawn. This difference reveals the fact that the fluctuation and the gradient of electron temperature are not neglectable even in a low temperature plasma. 4. Conclusions
Fig. 13. The cross-power spectrum and the phase angle at R = 33 cm between the fluctuations of plasma pressure p and plasma voltage Vp .
analysis with a threshold of 3σ , where σ is the deviation of the amplitude of the burst. A SOC-type system would lead to a near linear relation between the time and the amplitude, but Fig. 12 shows no simple relation between them. Consequently, we deduce that the underlying mechanism of the intermittent flux is not SOC-type. As one can see from Fig. 1, the phase angle between the fluctuation of Is and Vf is around π , it is a typical character of the flute mode. This is not consistent with the fact that the gradients of the density and the magnetic field are opposite at most of the radial positions in this shot. This inconsistency comes from the unsuitable neglect of the gradients of electron temperature Te , as was done in many of other magnetically confined
The plasma under ECR discharging in KT-5D shows similarities not only to other SMT devices on different discharging manners, but also to the SOL region of tokamaks. Two dominant regions of frequency dependence (f 0 and f −4 ), a narrow frequency band of f −1 index and a pronounced peak of 6.5 kHz have been observed in the spectra of the Vf and Is . The underlying mechanism of the 6.5 kHz peak is not clear, although similar peaks are also found in most of fusion or non-fusion devices. The degree of the long-range correlation (Hurst exponent) at time scales larger than 100 µs along radius at LFS ranges around 0.5, which suggests that no or a very weak long-range correlation exists. But at HFS, the H values are well above 0.5, suggesting an existence of a pronounced self-similarity or longrange correlation. The difference of Hurst values between LFS and HFS implies a more stochastic particle transport along radius at LFS, and thus suggests that the statistical behavior changes some even in the same magnetically confined device. The difference comes from different radius observation posi-
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tions or, in other words, maybe the plasma conditions such as the magnetic field or the electric field. At even larger time scales such as τ > 104 µs, the exploring of Hurst exponents is not performed for the larger error bar. This also indicates that although SMTs are good approximations to the SOL region of tokamaks and it is convenient for diagnosing because of the low plasma parameters, some plasma behaviors at LFS and HFS are not the same. The plasmas at HFS show more characteristic properties that are similar to the SOC system than that at LFS, and further investigation to the plasma behavior at HFS is needed. At all observable time scales, the PDFs of Vf exhibit a well Gaussian shape. The superposed PDFs of Vf imply a self-similarity at larger time scales, while it is not the case at small time scales for the existence of a 160 µs periodic signal according to Rypdal’s theory [21]. The PDFs of Is show great difference from those of Vf . These deviations from Gaussian distributions come from the high frequency tails of the spectra which suggest small scaled turbulences. The different statistics between Is and Vf is associated with the local nature of the former and more global nature of the latter. The low Hurst values around 0.5 of the fluctuations of the Is and Vf at LFS and the stochastic turbulence revealed by the Gaussian-shape PDFs of Vf imply the non-SOC characters of the transport flux induced by turbulence. By comparing the statistical properties of the turbulent signals of Vf and Is in ECR discharging plasmas with the results from some of the other Tokamaks and SMTs, it is found that although the discharging manners and the configurations of plasmas are different, the auto-power spectra, the Hurst exponents and PDFs show some similar characters. These similarities allow us to conclude that anomalous particle transport shares some universal properties in different magnetically confined devices. Here we only conclude that the auto-power spectra, the Hurst exponents and PDFs show some similar characteristic properties or “fingerprints” of a typical SOC system. As we know, many other mechanism may lead to these properties such as the 1/f spectrum and, on the other hand, many SOC systems show different spectrum of 1/f . The statistical features of the transport flux induced by turbulence dose not show the longrange correlation what is a typical characteristic property of a SOC system. The Hurst value of 0.5 indicates a complete stochastic behavior of the flux, which affirms the non-existence of the SOC. Although it is not SOC-type, the flux is still intermittent. These results imply that the natural intermittency of turbulent transport maybe independent of the avalanche induced by near criticality, and it is an experimental evidence of Mahdizadeh’s work [24]. This is true for at least the LFS of the double-functional device KT-5D. A more detailed image of the
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intermittency of the transport flux needs further investigation, and relevant work using two-dimensional distributed Langmuir probe arrays and fast cameras is underway. By means of investigating the phase angles between the fluctuations of plasma pressure p and the plasma potential Vp , we conclude that the dominant instability in the SMT discharge is drift instability. The difference between the plasma potential Vp and the floating potential Vf indicates that it is not wise to neglect the influence of the gradient and the fluctuation of electron temperature even in a low temperature plasma. It should be noted that investigations here are only for a certain toroidal field and gas pressure. As discussed in detail in Ref. [25], the plasma properties from microwave excitation significantly rely on the discharging conditions, i.e., the toroidal field, the vertical field and the gas pressure. The relevant experiments are underway. Acknowledgements This work is supported by the National Science Foundation of China (Grant Nos. 1023 5010 and 1033 5060) and grants from the Ministry of Education of the People’s Republic of China and the Chinese Academy of Sciences. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
P. Bak, C. Tang, K. Weisenfeld, Phys. Rev. Lett. 59 (1987) 381. B.A. Carreras, et al., Phys. Rev. Lett. 80 (1998) 4438. M.A. Pedrosa, et al., Phys. Rev. Lett. 82 (1999) 3621. B.A. Carreras, et al., Phys. Plasmas 6 (1999) 1885. Y.H. Xu, et al., Phys. Plasmas 11 (2004) 5413. R. Trasarti-Battostpmo, et al., Phys. Plasmas 9 (2002) 3369. K. Rypdal, et al., Phys. Plasmas 10 (2003) 2686. M. Gilmore, et al., Phys. Plasmas 9 (2002) 1312. W.-H. Wang, et al., Chin. Phys. Lett. 18 (2001) 793. G. Wang, et al., Phys. Plasmas 7 (2000) 1181. Y. Yu, et al., Rev. Sci. Instrum. 77 (2006) 123502. Y. Yu, et al., Plasma Sci. Technol. 8 (1) (2006) 91. M. Xu, et al., Rev. Sci. Instrum. 76 (2005) 043502. A. Fredriksen, et al., Phys. Plasmas 10 (2003) 4335. R. Trasartu-Battistoni, et al., Phys. Plasmas 9 (2002) 3369. A.N. Kolmogorov, et al., Sci. URSS 30 (1941) 301. A. Fredriksen, et al., Plasma Phys. Control. Fusion 45 (2003) 721. D.E. Newman, et al., Phys. Plasmas 3 (1996) 1858. B.A. Carreras, et al., Phys. Plasmas 3 (1996) 2903. T. Hwa, et al., Phys. Rev. A 45 (1992) 7002. K. Rypdal, et al., Phys. Plasmas 11 (2004) 3679. S.V. Ratynskaia, et al., Contrib. Plasma Phys. 41 (2001) 449. S.V. Ratynskaia, et al., Phys. Plasmas 9 (2002) 4135. N. Mahdizadeh, et al., Phys. Plasmas 11 (2004) 3932. Z.-h. Zhu, et al., Plasma Sci. Technol. 9 (2007) 5.