Multi-scale dynamics of glow discharge plasma through wavelets: Self-similar behavior to neutral turbulence and dissipation Bapun K. Giri, Chiranjit Mitra, Prasanta K. Panigrahi, and A. N. Sekar Iyengar Citation: Chaos 24, 043135 (2014); doi: 10.1063/1.4903332 View online: http://dx.doi.org/10.1063/1.4903332 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Mass dependency of turbulent parameters in stationary glow discharge plasmas Phys. Plasmas 20, 052304 (2013); 10.1063/1.4804339 Continuous wavelet transform based time-scale and multifractal analysis of the nonlinear oscillations in a hollow cathode glow discharge plasma Phys. Plasmas 16, 102307 (2009); 10.1063/1.3241694 Multifractality in plasma edge electrostatic turbulence Phys. Plasmas 15, 082311 (2008); 10.1063/1.2973175 Application of wavelet multiresolution analysis to the study of self-similarity and intermittency of plasma turbulence Rev. Sci. Instrum. 77, 083505 (2006); 10.1063/1.2336754 The scaling properties of dissipation in incompressible isotropic three-dimensional magnetohydrodynamic turbulence Phys. Plasmas 12, 022301 (2005); 10.1063/1.1842133
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CHAOS 24, 043135 (2014)
Multi-scale dynamics of glow discharge plasma through wavelets: Self-similar behavior to neutral turbulence and dissipation Bapun K. Giri,1,a) Chiranjit Mitra,1,b) Prasanta K. Panigrahi,1,c) and A. N. Sekar Iyengar2,d) 1
Department of Physical Sciences, Indian Institute of Science Education and Research (IISER), Kolkata, Mohanpur 741246, India 2 Plasma Physics Division, Saha Institute of Nuclear Physics (SINP) Kolkata, 1/AF, Bidhannagar, Kolkata 700064, India
(Received 6 May 2014; accepted 22 November 2014; published online 4 December 2014) The multiscale dynamics of glow discharge plasma is analysed through wavelet transform, whose scale dependent variable window size aptly captures both transients and non-stationary periodic behavior. The optimal time-frequency localization ability of the continuous Morlet wavelet is found to identify the scale dependent periodic modulations efficiently, as also the emergence of neutral turbulence and dissipation, whereas the discrete Daubechies basis set has been used for detrending the temporal behavior to reveal the multi-fractality of the underlying dynamics. The scaling exponents and the Hurst exponent have been estimated through wavelet based detrended fluctuation analysis, C 2014 AIP Publishing LLC. and also Fourier methods and rescale range analysis. V [http://dx.doi.org/10.1063/1.4903332] Glow discharge plasma is an ideal system for investigating various features of dynamical system in a controlled manner. Both linear and nonlinear regimes can be attained through controlled parameters like magnetic field and applied voltage. In this manuscript, we make use of both Fourier, discrete and continuous wavelets to study the multiscale dynamics of this system as well as transient and periodic dynamics. It emerges that the time-frequency localization property of wavelets is ideal to delineate dynamical structures, such as self-similarity and emergence of neutral turbulence and dissipation. The effect of applied voltage and magnetic field on the dynamics is explicated.
I. INTRODUCTION
Glow discharge plasma1,2 is a prototypical nonlinear dynamical system and an ideal test bed for observing a rich variety of nonlinear phenomena, e.g., chaos, frequency entrainment,3 intermittency,4 and turbulence.5,6 These have been investigated by nonlinear time series analysis, spectral analysis, and other methods. Turbulence and fluctuations in plasma play crucial role in the transport of charge and energy, which makes their investigation an area of active study.7 However, established techniques of nonlinear time series analysis, such as correlation dimension, Lyapunov exponent, fractal dimension,8,9 only probe the global and asymptotic properties of the system and are ineffective for unravelling the local structures. Some of these techniques limit themselves to the spectral amplitude of the signal, neglecting the phase information.10 Furthermore, in many of these approaches, the short time scale characteristics are a)
[email protected] [email protected] c) [email protected] d) [email protected] b)
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often overlooked, making it difficult to obtain the time history and local properties of the system. A number of methods have been used in the literature for non-linear time series analysis. Techniques like detrended fluctuation analysis (DFA),11 multi-fractal DFA (MF-DFA),12 Recurrence Quantification Analysis (RQA),13 etc., are helpful in characterizing features like long-range correlation, anti-correlation, and multiscale self similar behavior in the signals. In DFA, fluctuations are revealed after removing linear trends, whereas in MFDFA polynomials of different degree are used to remove local trends using variable window sizes. Both the small scale and large scale fluctuations are extracted through moments having positive and negative powers in the fluctuation functions.12 RQA and Recurrence Networks14–16 are a recently developed method, which makes the time series into a finite dimensional vector making use of the embedding dimension and time delay. This allows the study of both auto-correlation and cross-correlation. It is particularly helpful in identifying phase correlations using a network approach. For characterizing multi-fractality, we have made use of MF-DFA approach. Continuous Morlet wavelet has been used to extract stationary and time-varying periodic modulations, as well as investigating the presence of neutral turbulence and dissipation in the present dynamical system, through the behavior of wavelet power as a function of scale. For these analyses, Morlet wavelet has been known to be quite useful for estimation of local multi-scale modulations through a variable Gaussian window, with a sinusoidal modulation of appropriate frequency. The continuous Morlet wavelet17 economically identifies the periodic and structured variations in the time-frequency domain, whose nature is probed as a function of magnetic field on the floating potential. We report here, a detailed study on the nature of floating potential fluctuations in an argon DC glow discharge plasma inside a magnetic field. Keeping in mind, the time varying
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character of the potential variation, a wavelet based local approach is used to bring out the non-stationary nature of this dynamical system.18 The paper is organized as follows: Sec. II provides a brief outline of the experimental setup, after which Sec. III details the results from Fourier analysis, which reveals the periodic components and also the self-similar behavior of the dynamical system. However, it is clearly inadequate to throw light on the time-varying structures and also the multifractality of the data. Section IV uses wavelet transform to precisely capture the above dynamical features of the system. Apart from time-frequency localization of the periodic modulations, wavelet transform helps unravel phase synchronization among different modes. The continuous Morlet wavelet clearly identifies the parameter domain, where the dynamical system displays neutral turbulence and dissipation, obeying Heisenberg model. Finally, we conclude in Sec. VI, after outlining the complementary nature of different methods employed and highlighting the usefulness of local approaches. The directions of further studies are also laid.
II. EXPERIMENTAL SETUP
The experiments were performed in a hollow cathode DC glow discharge plasma device, which is schematically shown in Figure 1. It comprises of a hollow cathode tube with a central wire and a Langmuir probe for monitoring the complex temporal dynamics in the plasma floating potential time series. This assembly was mounted inside a vacuum chamber and pumped down to a pressure of about 0.001 mbar and, subsequently, filled with argon gas up to a pressure 0.077 mbar. Plasma was formed at 330 V by a DC discharge voltage power supply (which could be varied from 0 to 1000 V). A copper wire was wrapped around the cathode externally. By passing current through the copper wire, magnetic field was generated inside the cathode tube. Hence, there are three control parameters, namely, pressure, discharge voltage, and applied magnetic field. The data at 330 V have been obtained. At this voltage, the applied magnetic field was varied by changing the current in the copper wire (outside coil). The fluctuations of the plasma floating potential were recorded through a digital oscilloscope. Table I shows the DC discharge voltages and the corresponding
Chaos 24, 043135 (2014) TABLE I. Applied discharge potential and corresponding magnetic fields. Potential (V)
Magnetic field (gauss)
330
0 19 28 37 46 87 96
magnetic field parameter used for the present experiment. The sampling frequency for the present experimental setup is 5 106 samples per second. The whole data correspond to 0.001 ms. The purpose of keeping the pressure and electrode configuration constant was merely to understand how the nonlinear behavior changes under changing magnetic field. The frequencies of interest are well under the Nyquist Frequency (Nyquist-Shannon sampling theorem). The floating potential time series obtained through the Langmuir probe is the object of study here. The resulting fluctuation in floating potential is due to oscillating electrons and ions in the presence of both electric and magnetic field. In this experiment, the pressure and electrode configuration have been kept constant. This has been done in order to understand more clearly the changes in the nonlinear behavior of this dynamical system, under changing magnetic field. Charge particles exhibit drift and periodic behavior in the presence of both electric and magnetic fields. Figure 2(a) reveals a single frequency, which is also clearly seen in the phase space plot of Figure 3(a). Increasing the magnetic field, one observes a low frequency mode and a subdominant high frequency mode, as seen in Figures 2(b) and 3(b). Further increase in the magnetic field leads to more high frequency components, as is evident in Figure 2(c). The corresponding phase space plot in Figure 3(c) shows complex dynamics requiring careful analysis. In Secs. III–V. MF-DFA and wavelet methods will be used to unravel the underlying dynamics. Figure 4 depicts the phase space plot of two signals at 330 V, 96 gauss, obtained by filtering out the high frequency components. The plot illustrates the periodic behavior of the signal. We now proceed to study the Fourier domain behavior of the floating potential fluctuations in the time series to show the presence of both periodic, as well as self-similar behavior. III. FOURIER ANALYSIS
Fourier analysis is a widely used tool for testing and quantifying periodic as well as self-similar behavior in a time series. Fractality is quantified through the Hurst exponent (H),19 which can be related to the power spectrum (P(k)) of a fluctuation time series PðkÞ f a ; FIG. 1. Schematic diagram of the experimental setup.
where
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FIG. 2. Signals at 330 V and their respective power spectra, as the magnetic field is increased.
a ¼ 2H þ 1:
(1)
Here, f is the frequency and, 0 H 1, is a signature of fractal behavior.9,12 We have analysed the recorded plasma discharge potential fluctuations using discrete Fourier transform (DFT), its power spectra are depicted in Figure 2, which illustrate the transition of the system from periodic to chaotic behavior, as the magnetic field is increased. Initially, the system shows a periodic behavior at 330 V and then as the magnetic field is increased, we observe the appearance of higher frequency components. Then, at 330 V and 37 gauss magnetic field, the amplitude of high frequency component gets reduced with the emergence of a low frequency component. As mentioned
FIG. 3. Signals at 330 V and their respective phase space plots, as the magnetic field is increased. This illustrates the transition from the periodic to chaotic behavior as magnetic field is increased.
before, self-similar character of the potential fluctuations can be quantified from the behavior of the power spectrum. Figure 5 depicts the nature of the time series for 330 V. The power law behavior is evident in the frequency domain. The Fourier based methodology of estimating the fractal dimension8,9 is based on the mono-fractal hypothesis, which assumes that the scaling properties are same throughout the time series. This assumption is found to be unrealisable in the present context. As can be observed, the power law coefficient is not uniform in the entire frequency region. This behavior is an indication of the multifractal nature of the signals.20,21 Therefore, it was found to be desirable and efficient to use a more general analysis, which can extract and quantify the multi-fractality. In the process, the inadequacy of the
FIG. 4. Phase space plots of two signals (x(t)) at (a) 330 V, 96 gauss, and illustrating the periodic dynamics.
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FIG. 6. Rescale range analysis of the potential time series at 330 V and 430 V shows of positive correlation. The ordering of the abscissa is as per Table I.
for detrending purpose. Figure 7 shows the variation of fluctuation functions Fq(s)12 for various values of q.12 In the original approach, polynomial fits have been used to remove local trends and for extraction of fluctuations at different scales. IV. WAVELET BASED ANALYSIS A. Wavelet transform
P FIG. 5. (a) and (c) Cumulative sum of the fluctuations yi ¼ i ðxi hxiÞ with i ¼ 1,…, N, for 330 V, 96 gauss and 430 V, 19 gauss, respectively, and (b) and (d) the respective log-log plots of the power spectra, indicating multi-fractal character.
global approach will be pointed out, which will then lead to the wavelet based approach. Using wavelets, the signal is broken down into several subsets, which when analysed separately give the local Hurst exponent describing the local singular behavior and scaling of the signals. Before proceeding to the wavelet based approach, we make use of rescale range analysis in order to further ascertain the character of the time series. A. Rescale range analysis
Rescale range analysis is a technique for providing simultaneously, a measure of variance and the long term correlation in a time series.22 Figure 6 shows the Hurst exponents calculated using rescale range analysis. Clearly, the Hurst exponents are well above the value of 0.5 showing long range correlations. We also observe that for some threshold value of magnetic field, the Hurst exponent changes from 1 to 0.6, showing transition from an extreme fractal towards Brownian motion. For estimating multi-fractality reliably, one needs to compute other moments, apart from the variance. One can make use of MFDFA,12 for which wavelets can be effectively used
A wavelet function is characterized by a translation parameter (a) and scale parameter (s).18 The wavelet transform of a signal decomposes it into components dependent upon both position and scale. Therefore, the wavelet transform of a time series is localized both in time as well as frequency domains. Unlike the Fourier transform, which gives information solely of the frequency components, the wavelet transform specifies where it occurs in the time series. B. Continuous wavelet transform (CWT)
The CWT of a function f(t), where t is time, is defined as ð 1 þ1 ta Wf ðs; aÞ ¼ pffiffi dt: (2) f ðtÞw s s 1 * Here, w(t) is the mother ta wavelet and w(t) its complex conjugate. ws;a ðtÞ ¼ w s yields all the shifted and scaled versions of w(t). Changing the wavelet scale via s and translating along the localized time via a, the amplitudes of a signal as a function of scale as well as the time can be constructed.23 The wavelet should have zero mean and must be localized in time and frequency. Explicitly, for Morlet wavew2 0
t2
let to be used in our analysis, wðtÞ ¼ p1ffiffip ðejw0 te 2 Þe 2 . It has Gaussian window, which can be varied by changing the scale. The Gaussian window is multiplied by a sinusoidal function, whose frequency also changes according to scale. As seen in Eq. (2), finding the wavelet coefficients correspond to the convolution of the signal with the Gaussian window, modulated by a sinusoidal function. The coefficients
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FIG. 8. Scalograms showing energies at different scales, clearly depicting non-stationary periodic behavior at two different scales.
FIG. 9. Wavelet power spectrum displaying the cone of influence for the signals at 330 V. The wavelet analysis clearly distinguishes the two dominant periods of the time series with the confidence level 95%.
FIG. 7. The MF-DFA fluctuation functions Fq(s) are shown versus the scale s in log-log plots for (a) 330 V, 96 gauss and (b) the q dependence of the generalized Hurst exponent h(q) variation of Hurst exponent.
are large, when the modulations in the signal are in resonance with the sinusoidal variation of the wavelet function. The wavelet power spectrum is inferred from lnjWf ðs; aÞj2 at various scales s. The scalogram plot of wavelet coefficients (at different scales) in Figure 8 shows the presence of periodicity in the time series. Figure 9 depicts the global wavelet spectrum and average variance with time. It reveals the presence of the periods corresponding to that obtained by DFT. The nonstationary character of the dominant periodic modulations is evident in the scalogram showing the advantage of timefrequency localization of the Morlet wavelets. The cone of influence shows the reliability of the wavelet coefficients in a finite size dataset. The red regions indicate high confidence level. Semi-log plot in Figure 10 of wavelet power summed over all time at different scales also shows regular periodic behavior at large scales. The presence of turbulence and dissipation can also be inferred through wavelet based approach. A turbulent power spectrum has characteristic slopes as 5/3 in the turbulent24
and 7 in the viscous dissipation regime.25,26 The Heisenberg model exhibits these power laws, where a smooth transition takes place between them. We expect the possibility of neutral turbulence and dissipation in our system, which is revealed with the Heisenberg fit of different exponents shown in Figure 11.27
FIG. 10. Semilog plot of wavelet power spectrum summed over all time at different scales, indicating dominant modulations at different scales.
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FIG. 11. The Hesisenberg fit (solid line) of different exponents. The first fit having slope 5/3, representing turbulence, while the second fit with slope 7 represents viscous dissipative regime.
The slopes obtained are 5/3 and 7, which give the turbulent and dissipative nature of the system, respectively. Some regions of the system completely match with the fitted curve. Thus, the plasma system has both the properties in suitable parameter domains. V. PHASE ANALYSIS
The presence of multi-periodic modulations raises the possibility of their synchronization. The effect of magnetic field on the same is also of deep interest, since magnetic field directly affects the phase of charge carriers. The present
section outlines the phase relations obtained from complex CWT, with the complex Morlet function given by 1 n2 p ffiffiffiffiffiffiffiffiffiffiffi ffi exp 2iF : (3) wðnÞ ¼ cn Fb ðpFb Þ Here, Fb ¼ 1 and Fc ¼ 1.5 are the values of bandwidth parameter and wavelet center frequencies, respectively. The phase angle is defined by 1 ImðwðnÞÞ : (4) wðnÞ ¼ tan ReðwðnÞÞ
FIG. 12. Wavelet phase analysis of two floating potential fluctuation signals at 330 V and magnetic fields 87 gauss and 97 gauss, respectively. Figure (a) shows the floating potential fluctuations, (b) shows the wavelet coefficients of corresponding signals reconstructed from wavelet spectral analysis, (c) phase angles of the reconstructions, and (d) shows the phase difference between the corresponding signals.
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We perform a phase analysis on the periodic modulations to study their synchronization. As is well known, magnetic field affects the phase of charge particle’s wavefunction. Here, it is worth comparing the periodic behavior of the floating potential of two different magnetic fields as function of time, as seen in the scalogram. The coherence property is extremely clear in the behavior of the phase. Antiphase behavior and inphase dynamics is clearly seen. This is further amplified in Figure 12. VI. CONCLUSION
In conclusion, we have studied the floating potential fluctuations in argon DC glow discharge plasma with the application of a magnetic field. We use Fourier and wavelet based techniques to study the non-stationarity character of the time series. The phase space shows periodic nature of the signals. However, as the potential and magnetic field is increased, the system reveals high frequency components. The non-stationary dynamics is investigated with both discrete and continuous wavelets. Discrete wavelets from the Daubechies family have been effectively used in isolating the local polynomial trends and estimating the multi-fractal nature of the time series. The continuous Morlet wavelet has been handy to identify the periodic and structured variations in the time-frequency domain, as a function of the magnetic field. We demonstrate the presence of neutral turbulence and dissipation in the plasma system by drawing correspondence with the Heisenberg model, where a slope of 5/3 corresponds to the former, while 7 to the latter. Further, the effect of magnetic field on the phases of periodic modulations has been investigated. We illustrate phase synchronization between two time series obtained at two different values of the applied magnetic field, and find that the time series which are initially out-of-phase, go into phase synchronization for an intermittent duration. Further studies of the dynamics underlying phase synchronization between the time series will be of interest. As time progresses, the system goes through a series of stages where they are either in-phase or out of phase. In this context, the phenomenon of “phase-flip” bifurcation has been recently observed, where dynamical systems coupled with delay, which are initially out-of phase later go into phase as the coupling delay gradually changes with time.28,29 It would be interesting to build a similar model for the dynamics observed here with variable delay such that it gives rise to such bursts of in-phase and anti-phase synchronization between the time series. This would provide useful insights into the presence and effects of delays on the dynamics of laboratory plasma. The exact nature of dynamics leading to turbulence and dissipation also needs closer scrutiny.
Chaos 24, 043135 (2014) 1
A. B. Stewart, “Oscillating glow discharge plasma,” J. Appl. Phys. 27, 911–916 (1956). P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics (CRC Press, 2010). 3 T. Klinger, F. Greiner, A. Rohde, A. Piel, and M. E. Koepke, “van der Pol behavior of relaxation oscillations in a periodically driven thermionic discharge,” Phys. Rev. E 52, 4316 (1995). 4 L. Sorriso-Valvo et al. “Intermittency in plasma turbulence,” Planet. Space Sci. 49, 1193 (2001). 5 R. Jha et al., “Evidence of Levy stable process in tokamak edge turbulence,” Phys. Plasmas 10, 699 (2003). 6 V. D. Shafranov, Reviews of Plasma Physics (Springer, 2001). 7 A. Fasoli et al., “Electrostatic turbulence and transport in a simple magnetized plasma,” Phys. Plasmas 13, 055902 (2006). 8 J. Feder and P. Bak, Fractals (Physics Today, 1989). 9 B. B. Mandelbrot, The fractal Geometry of Nature (Macmillan, 1983). 10 M. Nurujjaman, R. Narayanan, and A. N. S. Iyengar, “Continuous wavelet transform based time-scale and multifractal analysis of the nonlinear oscillations in a hollow cathode glow discharge plasma,” Phys. Plasmas 16, 102307 (2009). 11 C. K. Peng et al., “Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series,” Chaos: Interdisciplin. J. Nonlinear Sci. 5, 82–87 (1995). 12 J. W. Kantelhardt et al.,“Multifractal detrended fluctuation analysis of nonstationary time series,” Physica A 316, 87 (2002) and references therein. 13 N. Marwan et al.,“Recurrence plots for the analysis of complex systems,” Phys. Rep. 438, 237–329 (2007). 14 N. Marwan et al., “Complex network approach for recurrence analysis of time series,” Phys. Lett. A 373(46), 4246–4254 (2009). 15 Z. K. Gao et al., “Recurrence networks from multivariate signals for uncovering dynamic transitions of horizontal oil-water stratified flows,” Europhys. Lett. 103, 50004 (2013). 16 Z. K. Gao et al., “Multivariate recurrence network analysis for characterizing horizontal oil-water two-phase flow,” Phys. Rev. E 88, 032910 (2013). 17 A. Grossmann and J. Morlet, “Decomposition of hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723 (1984). 18 M. Vetterli, J. Kovacˇevic´, and V. K. Goyal, Fourier and Wavelet Signal Processing (Cambridge University Press, 2013). 19 H. E. Hurst, “Long-term storage capacity of reservoirs,” Trans. Am. Soc. Civ. Eng. 116, 770 (1951). 20 N. Das et al., “Probing multifractality in tissue refractive index: Prospects for precancer detection,” Opt. Lett. 38, 211 (2013). 21 S. Ghosh et al., “Differing self-similarity in light scattering spectra: A potential tool for pre-cancer detection,” Opt. Express 19, 19717 (2011). 22 J. B. Bassingthwaighte and G. M. Raymond, “Evaluating rescaled range analysis for time series,” Ann. Biomed. Eng. 22, 432 (1994). 23 C. Torrence and G. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61 (1998). 24 V. Kolmogorov and A. Nikolaevich, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 4 (1941). 25 W. Heisenberg, “Zur statistischen Theorie der Turbulenz,” Z. Phys. 124, 628 (1948). 26 M. Farge, N. Kevlahan, V. Perrier, and F. Goirand, “Wavelets and turbulence,” Proc. IEEE 84, 639 (1996). 27 U. Das et al., “Fine structure of the low-latitude mesospheric turbulence,” J. Geophys. Res. Atmos. 114, D10111 (2009). 28 B. T. Grenfell, O. N. Bjurnstad, and J. Kappey, “Travelling waves and spatial hierarchies in measles epidemics,” Nature 414, 716 (2001). 29 A. Prasad et al., “Phase-flip bifurcation induced by time delay,” Phys. Rev. E 74, 035204 (2006). 2
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CHAOS 24, 043135 (2014)
Multi-scale dynamics of glow discharge plasma through wavelets: Self-similar behavior to neutral turbulence and dissipation Bapun K. Giri,1,a) Chiranjit Mitra,1,b) Prasanta K. Panigrahi,1,c) and A. N. Sekar Iyengar2,d) 1
Department of Physical Sciences, Indian Institute of Science Education and Research (IISER), Kolkata, Mohanpur 741246, India 2 Plasma Physics Division, Saha Institute of Nuclear Physics (SINP) Kolkata, 1/AF, Bidhannagar, Kolkata 700064, India
(Received 6 May 2014; accepted 22 November 2014; published online 4 December 2014) The multiscale dynamics of glow discharge plasma is analysed through wavelet transform, whose scale dependent variable window size aptly captures both transients and non-stationary periodic behavior. The optimal time-frequency localization ability of the continuous Morlet wavelet is found to identify the scale dependent periodic modulations efficiently, as also the emergence of neutral turbulence and dissipation, whereas the discrete Daubechies basis set has been used for detrending the temporal behavior to reveal the multi-fractality of the underlying dynamics. The scaling exponents and the Hurst exponent have been estimated through wavelet based detrended fluctuation analysis, C 2014 AIP Publishing LLC. and also Fourier methods and rescale range analysis. V [http://dx.doi.org/10.1063/1.4903332] Glow discharge plasma is an ideal system for investigating various features of dynamical system in a controlled manner. Both linear and nonlinear regimes can be attained through controlled parameters like magnetic field and applied voltage. In this manuscript, we make use of both Fourier, discrete and continuous wavelets to study the multiscale dynamics of this system as well as transient and periodic dynamics. It emerges that the time-frequency localization property of wavelets is ideal to delineate dynamical structures, such as self-similarity and emergence of neutral turbulence and dissipation. The effect of applied voltage and magnetic field on the dynamics is explicated.
I. INTRODUCTION
Glow discharge plasma1,2 is a prototypical nonlinear dynamical system and an ideal test bed for observing a rich variety of nonlinear phenomena, e.g., chaos, frequency entrainment,3 intermittency,4 and turbulence.5,6 These have been investigated by nonlinear time series analysis, spectral analysis, and other methods. Turbulence and fluctuations in plasma play crucial role in the transport of charge and energy, which makes their investigation an area of active study.7 However, established techniques of nonlinear time series analysis, such as correlation dimension, Lyapunov exponent, fractal dimension,8,9 only probe the global and asymptotic properties of the system and are ineffective for unravelling the local structures. Some of these techniques limit themselves to the spectral amplitude of the signal, neglecting the phase information.10 Furthermore, in many of these approaches, the short time scale characteristics are a)
[email protected] [email protected] c) [email protected] d) [email protected] b)
1054-1500/2014/24(4)/043135/7/$30.00
often overlooked, making it difficult to obtain the time history and local properties of the system. A number of methods have been used in the literature for non-linear time series analysis. Techniques like detrended fluctuation analysis (DFA),11 multi-fractal DFA (MF-DFA),12 Recurrence Quantification Analysis (RQA),13 etc., are helpful in characterizing features like long-range correlation, anti-correlation, and multiscale self similar behavior in the signals. In DFA, fluctuations are revealed after removing linear trends, whereas in MFDFA polynomials of different degree are used to remove local trends using variable window sizes. Both the small scale and large scale fluctuations are extracted through moments having positive and negative powers in the fluctuation functions.12 RQA and Recurrence Networks14–16 are a recently developed method, which makes the time series into a finite dimensional vector making use of the embedding dimension and time delay. This allows the study of both auto-correlation and cross-correlation. It is particularly helpful in identifying phase correlations using a network approach. For characterizing multi-fractality, we have made use of MF-DFA approach. Continuous Morlet wavelet has been used to extract stationary and time-varying periodic modulations, as well as investigating the presence of neutral turbulence and dissipation in the present dynamical system, through the behavior of wavelet power as a function of scale. For these analyses, Morlet wavelet has been known to be quite useful for estimation of local multi-scale modulations through a variable Gaussian window, with a sinusoidal modulation of appropriate frequency. The continuous Morlet wavelet17 economically identifies the periodic and structured variations in the time-frequency domain, whose nature is probed as a function of magnetic field on the floating potential. We report here, a detailed study on the nature of floating potential fluctuations in an argon DC glow discharge plasma inside a magnetic field. Keeping in mind, the time varying
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character of the potential variation, a wavelet based local approach is used to bring out the non-stationary nature of this dynamical system.18 The paper is organized as follows: Sec. II provides a brief outline of the experimental setup, after which Sec. III details the results from Fourier analysis, which reveals the periodic components and also the self-similar behavior of the dynamical system. However, it is clearly inadequate to throw light on the time-varying structures and also the multifractality of the data. Section IV uses wavelet transform to precisely capture the above dynamical features of the system. Apart from time-frequency localization of the periodic modulations, wavelet transform helps unravel phase synchronization among different modes. The continuous Morlet wavelet clearly identifies the parameter domain, where the dynamical system displays neutral turbulence and dissipation, obeying Heisenberg model. Finally, we conclude in Sec. VI, after outlining the complementary nature of different methods employed and highlighting the usefulness of local approaches. The directions of further studies are also laid.
II. EXPERIMENTAL SETUP
The experiments were performed in a hollow cathode DC glow discharge plasma device, which is schematically shown in Figure 1. It comprises of a hollow cathode tube with a central wire and a Langmuir probe for monitoring the complex temporal dynamics in the plasma floating potential time series. This assembly was mounted inside a vacuum chamber and pumped down to a pressure of about 0.001 mbar and, subsequently, filled with argon gas up to a pressure 0.077 mbar. Plasma was formed at 330 V by a DC discharge voltage power supply (which could be varied from 0 to 1000 V). A copper wire was wrapped around the cathode externally. By passing current through the copper wire, magnetic field was generated inside the cathode tube. Hence, there are three control parameters, namely, pressure, discharge voltage, and applied magnetic field. The data at 330 V have been obtained. At this voltage, the applied magnetic field was varied by changing the current in the copper wire (outside coil). The fluctuations of the plasma floating potential were recorded through a digital oscilloscope. Table I shows the DC discharge voltages and the corresponding
Chaos 24, 043135 (2014) TABLE I. Applied discharge potential and corresponding magnetic fields. Potential (V)
Magnetic field (gauss)
330
0 19 28 37 46 87 96
magnetic field parameter used for the present experiment. The sampling frequency for the present experimental setup is 5 106 samples per second. The whole data correspond to 0.001 ms. The purpose of keeping the pressure and electrode configuration constant was merely to understand how the nonlinear behavior changes under changing magnetic field. The frequencies of interest are well under the Nyquist Frequency (Nyquist-Shannon sampling theorem). The floating potential time series obtained through the Langmuir probe is the object of study here. The resulting fluctuation in floating potential is due to oscillating electrons and ions in the presence of both electric and magnetic field. In this experiment, the pressure and electrode configuration have been kept constant. This has been done in order to understand more clearly the changes in the nonlinear behavior of this dynamical system, under changing magnetic field. Charge particles exhibit drift and periodic behavior in the presence of both electric and magnetic fields. Figure 2(a) reveals a single frequency, which is also clearly seen in the phase space plot of Figure 3(a). Increasing the magnetic field, one observes a low frequency mode and a subdominant high frequency mode, as seen in Figures 2(b) and 3(b). Further increase in the magnetic field leads to more high frequency components, as is evident in Figure 2(c). The corresponding phase space plot in Figure 3(c) shows complex dynamics requiring careful analysis. In Secs. III–V. MF-DFA and wavelet methods will be used to unravel the underlying dynamics. Figure 4 depicts the phase space plot of two signals at 330 V, 96 gauss, obtained by filtering out the high frequency components. The plot illustrates the periodic behavior of the signal. We now proceed to study the Fourier domain behavior of the floating potential fluctuations in the time series to show the presence of both periodic, as well as self-similar behavior. III. FOURIER ANALYSIS
Fourier analysis is a widely used tool for testing and quantifying periodic as well as self-similar behavior in a time series. Fractality is quantified through the Hurst exponent (H),19 which can be related to the power spectrum (P(k)) of a fluctuation time series PðkÞ f a ; FIG. 1. Schematic diagram of the experimental setup.
where
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FIG. 2. Signals at 330 V and their respective power spectra, as the magnetic field is increased.
a ¼ 2H þ 1:
(1)
Here, f is the frequency and, 0 H 1, is a signature of fractal behavior.9,12 We have analysed the recorded plasma discharge potential fluctuations using discrete Fourier transform (DFT), its power spectra are depicted in Figure 2, which illustrate the transition of the system from periodic to chaotic behavior, as the magnetic field is increased. Initially, the system shows a periodic behavior at 330 V and then as the magnetic field is increased, we observe the appearance of higher frequency components. Then, at 330 V and 37 gauss magnetic field, the amplitude of high frequency component gets reduced with the emergence of a low frequency component. As mentioned
FIG. 3. Signals at 330 V and their respective phase space plots, as the magnetic field is increased. This illustrates the transition from the periodic to chaotic behavior as magnetic field is increased.
before, self-similar character of the potential fluctuations can be quantified from the behavior of the power spectrum. Figure 5 depicts the nature of the time series for 330 V. The power law behavior is evident in the frequency domain. The Fourier based methodology of estimating the fractal dimension8,9 is based on the mono-fractal hypothesis, which assumes that the scaling properties are same throughout the time series. This assumption is found to be unrealisable in the present context. As can be observed, the power law coefficient is not uniform in the entire frequency region. This behavior is an indication of the multifractal nature of the signals.20,21 Therefore, it was found to be desirable and efficient to use a more general analysis, which can extract and quantify the multi-fractality. In the process, the inadequacy of the
FIG. 4. Phase space plots of two signals (x(t)) at (a) 330 V, 96 gauss, and illustrating the periodic dynamics.
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FIG. 6. Rescale range analysis of the potential time series at 330 V and 430 V shows of positive correlation. The ordering of the abscissa is as per Table I.
for detrending purpose. Figure 7 shows the variation of fluctuation functions Fq(s)12 for various values of q.12 In the original approach, polynomial fits have been used to remove local trends and for extraction of fluctuations at different scales. IV. WAVELET BASED ANALYSIS A. Wavelet transform
P FIG. 5. (a) and (c) Cumulative sum of the fluctuations yi ¼ i ðxi hxiÞ with i ¼ 1,…, N, for 330 V, 96 gauss and 430 V, 19 gauss, respectively, and (b) and (d) the respective log-log plots of the power spectra, indicating multi-fractal character.
global approach will be pointed out, which will then lead to the wavelet based approach. Using wavelets, the signal is broken down into several subsets, which when analysed separately give the local Hurst exponent describing the local singular behavior and scaling of the signals. Before proceeding to the wavelet based approach, we make use of rescale range analysis in order to further ascertain the character of the time series. A. Rescale range analysis
Rescale range analysis is a technique for providing simultaneously, a measure of variance and the long term correlation in a time series.22 Figure 6 shows the Hurst exponents calculated using rescale range analysis. Clearly, the Hurst exponents are well above the value of 0.5 showing long range correlations. We also observe that for some threshold value of magnetic field, the Hurst exponent changes from 1 to 0.6, showing transition from an extreme fractal towards Brownian motion. For estimating multi-fractality reliably, one needs to compute other moments, apart from the variance. One can make use of MFDFA,12 for which wavelets can be effectively used
A wavelet function is characterized by a translation parameter (a) and scale parameter (s).18 The wavelet transform of a signal decomposes it into components dependent upon both position and scale. Therefore, the wavelet transform of a time series is localized both in time as well as frequency domains. Unlike the Fourier transform, which gives information solely of the frequency components, the wavelet transform specifies where it occurs in the time series. B. Continuous wavelet transform (CWT)
The CWT of a function f(t), where t is time, is defined as ð 1 þ1 ta Wf ðs; aÞ ¼ pffiffi dt: (2) f ðtÞw s s 1 * Here, w(t) is the mother ta wavelet and w(t) its complex conjugate. ws;a ðtÞ ¼ w s yields all the shifted and scaled versions of w(t). Changing the wavelet scale via s and translating along the localized time via a, the amplitudes of a signal as a function of scale as well as the time can be constructed.23 The wavelet should have zero mean and must be localized in time and frequency. Explicitly, for Morlet wavew2 0
t2
let to be used in our analysis, wðtÞ ¼ p1ffiffip ðejw0 te 2 Þe 2 . It has Gaussian window, which can be varied by changing the scale. The Gaussian window is multiplied by a sinusoidal function, whose frequency also changes according to scale. As seen in Eq. (2), finding the wavelet coefficients correspond to the convolution of the signal with the Gaussian window, modulated by a sinusoidal function. The coefficients
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FIG. 8. Scalograms showing energies at different scales, clearly depicting non-stationary periodic behavior at two different scales.
FIG. 9. Wavelet power spectrum displaying the cone of influence for the signals at 330 V. The wavelet analysis clearly distinguishes the two dominant periods of the time series with the confidence level 95%.
FIG. 7. The MF-DFA fluctuation functions Fq(s) are shown versus the scale s in log-log plots for (a) 330 V, 96 gauss and (b) the q dependence of the generalized Hurst exponent h(q) variation of Hurst exponent.
are large, when the modulations in the signal are in resonance with the sinusoidal variation of the wavelet function. The wavelet power spectrum is inferred from lnjWf ðs; aÞj2 at various scales s. The scalogram plot of wavelet coefficients (at different scales) in Figure 8 shows the presence of periodicity in the time series. Figure 9 depicts the global wavelet spectrum and average variance with time. It reveals the presence of the periods corresponding to that obtained by DFT. The nonstationary character of the dominant periodic modulations is evident in the scalogram showing the advantage of timefrequency localization of the Morlet wavelets. The cone of influence shows the reliability of the wavelet coefficients in a finite size dataset. The red regions indicate high confidence level. Semi-log plot in Figure 10 of wavelet power summed over all time at different scales also shows regular periodic behavior at large scales. The presence of turbulence and dissipation can also be inferred through wavelet based approach. A turbulent power spectrum has characteristic slopes as 5/3 in the turbulent24
and 7 in the viscous dissipation regime.25,26 The Heisenberg model exhibits these power laws, where a smooth transition takes place between them. We expect the possibility of neutral turbulence and dissipation in our system, which is revealed with the Heisenberg fit of different exponents shown in Figure 11.27
FIG. 10. Semilog plot of wavelet power spectrum summed over all time at different scales, indicating dominant modulations at different scales.
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FIG. 11. The Hesisenberg fit (solid line) of different exponents. The first fit having slope 5/3, representing turbulence, while the second fit with slope 7 represents viscous dissipative regime.
The slopes obtained are 5/3 and 7, which give the turbulent and dissipative nature of the system, respectively. Some regions of the system completely match with the fitted curve. Thus, the plasma system has both the properties in suitable parameter domains. V. PHASE ANALYSIS
The presence of multi-periodic modulations raises the possibility of their synchronization. The effect of magnetic field on the same is also of deep interest, since magnetic field directly affects the phase of charge carriers. The present
section outlines the phase relations obtained from complex CWT, with the complex Morlet function given by 1 n2 p ffiffiffiffiffiffiffiffiffiffiffi ffi exp 2iF : (3) wðnÞ ¼ cn Fb ðpFb Þ Here, Fb ¼ 1 and Fc ¼ 1.5 are the values of bandwidth parameter and wavelet center frequencies, respectively. The phase angle is defined by 1 ImðwðnÞÞ : (4) wðnÞ ¼ tan ReðwðnÞÞ
FIG. 12. Wavelet phase analysis of two floating potential fluctuation signals at 330 V and magnetic fields 87 gauss and 97 gauss, respectively. Figure (a) shows the floating potential fluctuations, (b) shows the wavelet coefficients of corresponding signals reconstructed from wavelet spectral analysis, (c) phase angles of the reconstructions, and (d) shows the phase difference between the corresponding signals.
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We perform a phase analysis on the periodic modulations to study their synchronization. As is well known, magnetic field affects the phase of charge particle’s wavefunction. Here, it is worth comparing the periodic behavior of the floating potential of two different magnetic fields as function of time, as seen in the scalogram. The coherence property is extremely clear in the behavior of the phase. Antiphase behavior and inphase dynamics is clearly seen. This is further amplified in Figure 12. VI. CONCLUSION
In conclusion, we have studied the floating potential fluctuations in argon DC glow discharge plasma with the application of a magnetic field. We use Fourier and wavelet based techniques to study the non-stationarity character of the time series. The phase space shows periodic nature of the signals. However, as the potential and magnetic field is increased, the system reveals high frequency components. The non-stationary dynamics is investigated with both discrete and continuous wavelets. Discrete wavelets from the Daubechies family have been effectively used in isolating the local polynomial trends and estimating the multi-fractal nature of the time series. The continuous Morlet wavelet has been handy to identify the periodic and structured variations in the time-frequency domain, as a function of the magnetic field. We demonstrate the presence of neutral turbulence and dissipation in the plasma system by drawing correspondence with the Heisenberg model, where a slope of 5/3 corresponds to the former, while 7 to the latter. Further, the effect of magnetic field on the phases of periodic modulations has been investigated. We illustrate phase synchronization between two time series obtained at two different values of the applied magnetic field, and find that the time series which are initially out-of-phase, go into phase synchronization for an intermittent duration. Further studies of the dynamics underlying phase synchronization between the time series will be of interest. As time progresses, the system goes through a series of stages where they are either in-phase or out of phase. In this context, the phenomenon of “phase-flip” bifurcation has been recently observed, where dynamical systems coupled with delay, which are initially out-of phase later go into phase as the coupling delay gradually changes with time.28,29 It would be interesting to build a similar model for the dynamics observed here with variable delay such that it gives rise to such bursts of in-phase and anti-phase synchronization between the time series. This would provide useful insights into the presence and effects of delays on the dynamics of laboratory plasma. The exact nature of dynamics leading to turbulence and dissipation also needs closer scrutiny.
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