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Turbulence is essentially non-linear.
Turbulence is essentially non-linear.
Non-linear interactions can be detected by means of higher-order spectra (e.g. quadratic interactions can be detected through the bi-spectrum). With Fourier analysis, however, in order to achieve statistically significant values for the bi-spectrum, very long time series are necessary. This fact has mostly precluded its use in fields like plasma turbulence, since long steady-state data series are not generally available. In our work, for the first time, the bicoherence was calculated using wavelet transforms, thus making the detection of non-linear interactions with time resolution possible.
Non-linear interactions can be detected by means of higher-order spectra (e.g. quadratic interactions can be detected through the bi-spectrum). With Fourier analysis, however, in order to achieve statistically significant values for the bi-spectrum, very long time series are necessary. This fact has mostly precluded its use in fields like plasma turbulence, since long steady-state data series are not generally available. In our work, for the first time, the bicoherence was calculated using wavelet transforms, thus making the detection of non-linear interactions with time resolution possible.
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.74.395 B.Ph. van Milligen et al, Nonlinear phenomena and intermittency in plasma turbulence, Phys. Rev. Lett. '''74''', 3 (1995) 395]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.74.395 B.Ph. van Milligen et al, ''Nonlinear phenomena and intermittency in plasma turbulence'', Phys. Rev. Lett. '''74''', 3 (1995) 395]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, Wavelet bicoherence: a new turbulence analysis tool, Phys. Plasmas '''2''', 8 (1995) 3017]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>
<ref>[http://link.aip.org/link/?RSINAK/68/967/1 B.Ph. van Milligen et al, Statistically robust linear and non-linear wavelet analysis applied to plasma edge turbulence, Rev. Sci. Instrum. '''68''' (1997) 967]</ref>
<ref>[http://link.aip.org/link/?RSINAK/68/967/1 B.Ph. van Milligen et al, ''Statistically robust linear and non-linear wavelet analysis applied to plasma edge turbulence'', Rev. Sci. Instrum. '''68''' (1997) 967]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.84.4842 P.H. Diamond et al, In search of the elusive zonal flow using cross-bicoherence analysis, Phys. Rev. Lett. '''84''', 12 (2000) 4842]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.84.4842 P.H. Diamond et al, ''In search of the elusive zonal flow using cross-bicoherence analysis'', Phys. Rev. Lett. '''84''', 12 (2000) 4842]</ref>
A relation was found between confinement transitions and an increase of the bicoherence, as expected in the framework of shear flow models for turbulence stabilisation.  
A relation was found between confinement transitions and an increase of the bicoherence, as expected in the framework of shear flow models for turbulence stabilisation.  
<ref>[http://dx.doi.org/10.1088/0029-5515/48/11/115003 B.Ph. van Milligen et al, Bicoherence during confinement transitions in the TJ-II stellarator, Nucl. Fusion '''48''' (2008) 115003]</ref>
<ref>[http://dx.doi.org/10.1088/0029-5515/48/11/115003 B.Ph. van Milligen et al, ''Bicoherence during confinement transitions in the TJ-II stellarator'', Nucl. Fusion '''48''' (2008) 115003]</ref>


=== Self-similarity ===
=== Self-similarity ===


Important transport phenomena such as profile stiffness (consistency),
Important transport phenomena such as profile stiffness (consistency),
<ref>[http://www.jspf.or.jp/PFR/PFR_articles/pfr2008S1/pfr2008_03-S1070.html B.Ph. van Milligen et al, Quantifying profile stiffness, Plasma and Fusion Research, '''3''' (2008) S1070]</ref>
<ref>[http://www.jspf.or.jp/PFR/PFR_articles/pfr2008S1/pfr2008_03-S1070.html B.Ph. van Milligen et al, ''Quantifying profile stiffness'', Plasma and Fusion Research, '''3''' (2008) S1070]</ref>
power degradation, the rapid propagation of perturbations,
power degradation, the rapid propagation of perturbations,
<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen et al, Pulse propagation in a simple probabilistic transport model, Nucl. Fusion '''47''' (2007) 189]</ref>
<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen et al, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>
and the Bohm scaling of plasma confinement might be explained on the basis of profile self-regulation in the framework of the [[Self-Organised Criticality]] paradigm. This paradigm predicts that transport is regulated by avalanches, which would generate self-similar behaviour in space and time of the turbulent data.  
and the Bohm scaling of plasma confinement might be explained on the basis of profile self-regulation in the framework of the [[Self-Organised Criticality]] paradigm. This paradigm predicts that transport is regulated by avalanches, which would generate self-similar behaviour in space and time of the turbulent data.  


In order to test this hypothesis, one could determine the shape of the autocorrelation function (ACF) of turbulent signals.
In order to test this hypothesis, one could determine the shape of the autocorrelation function (ACF) of turbulent signals.
<ref>[http://link.aip.org/link/?PHPAEN/3/2664/1 B.A. Carreras et al, Fluctuation-induced flux at the plasma edge in toroidal devices, Phys. Plasmas '''3''' (7) (1996) 2664]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/3/2664/1 B.A. Carreras et al, ''Fluctuation-induced flux at the plasma edge in toroidal devices'', Phys. Plasmas '''3''' (7) (1996) 2664]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.80.4438 B.A. Carreras et al, Long-range time correlations in plasma edge turbulence, Phys. Rev. Lett. '''80''', (1998) 4438]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.80.4438 B.A. Carreras et al, ''Long-range time correlations in plasma edge turbulence'', Phys. Rev. Lett. '''80''', (1998) 4438]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.83.3653 B.A. Carreras et al, Self-similarity properties of the probability distribution function of turbulence-induced particle fluxes at the plasma edge, Phys. Rev. Lett. '''83''' (1999) 3653]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.83.3653 B.A. Carreras et al, ''Self-similarity properties of the probability distribution function of turbulence-induced particle fluxes at the plasma edge'', Phys. Rev. Lett. '''83''' (1999) 3653]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/7/3278/1 B.A. Carreras, Intermittency of plasma edge fluctuation data: Multifractal analysis, Phys. Plasmas, '''7''', 8 (2000) 3278]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/7/3278/1 B.A. Carreras, ''Intermittency of plasma edge fluctuation data: Multifractal analysis'', Phys. Plasmas, '''7''', 8 (2000) 3278]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/44/8/309 C. Hidalgo et al, Empirical similarity in the probability density function of turbulent transport in the edge plasma region in fusion plasmas, Plasma Phys. Control. Fusion '''44''' (2002) 1557]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/44/8/309 C. Hidalgo et al, ''Empirical similarity in the probability density function of turbulent transport in the edge plasma region in fusion plasmas'', Plasma Phys. Control. Fusion '''44''' (2002) 1557]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/12/052507/1 B.Ph. Van Milligen et al, Additional evidence for the universality of turbulent fluctuations and fluxes in the scrape-off layer region of fusion plasmas, Phys. Plasmas '''12''' (2005) 052507]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/12/052507/1 B.Ph. Van Milligen et al, ''Additional evidence for the universality of turbulent fluctuations and fluxes in the scrape-off layer region of fusion plasmas'', Phys. Plasmas '''12''' (2005) 052507]</ref>
Unfortunately, the most revealing information is present in the tail of the distribution (i.e., well beyond the correlation time), where statistics are generally poor.
Unfortunately, the most revealing information is present in the tail of the distribution (i.e., well beyond the correlation time), where statistics are generally poor.


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The analysis of data from Langmuir probes taken at the plasma edge in a wide variety of fusion devices reveals the existence of self-similar behaviour or long-range correlations in all devices studied. The observed variation of the Hurst exponent in the plasma edge, 0.62 < H < 0.75, is small in spite of the variety of devices.  
The analysis of data from Langmuir probes taken at the plasma edge in a wide variety of fusion devices reveals the existence of self-similar behaviour or long-range correlations in all devices studied. The observed variation of the Hurst exponent in the plasma edge, 0.62 < H < 0.75, is small in spite of the variety of devices.  
<ref>[http://link.aip.org/link/?PHPAEN/5/3632/1 B.A. Carreras et al, Self-similarity of the plasma edge fluctuations, Phys. Plasmas '''5''', 10 (1998) 3632]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/5/3632/1 B.A. Carreras et al, ''Self-similarity of the plasma edge fluctuations'', Phys. Plasmas '''5''', 10 (1998) 3632]</ref>
On the other hand, the variation of H in the scrape-off layer is much larger. In Wendelstein VII-AS, a slight decrease in H at the sheared flow layer was observed, possibly corresponding to a local decorrelation effect.
On the other hand, the variation of H in the scrape-off layer is much larger. In Wendelstein VII-AS, a slight decrease in H at the sheared flow layer was observed, possibly corresponding to a local decorrelation effect.


The repeated occurrence of values of H differing significantly from the value corresponding to random noise (H = 0.5) in all machines points to a universal aspect of the underlying turbulence. Further, the degree of self-similarity detected implies the existence of long-range correlations (with respect to the correlation time).
The repeated occurrence of values of H differing significantly from the value corresponding to random noise (H = 0.5) in all machines points to a universal aspect of the underlying turbulence. Further, the degree of self-similarity detected implies the existence of long-range correlations (with respect to the correlation time).
<ref>[http://link.aip.org/link/?PHPAEN/6/485/1 B.A. Carreras et al, Long-range time dependence in the cross-correlation function, Phys. Plasmas '''6''', 2 (1999) 485]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/6/485/1 B.A. Carreras et al, ''Long-range time dependence in the cross-correlation function'', Phys. Plasmas '''6''', 2 (1999) 485]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/6/1885/1 B.A. Carreras et al, Experimental evidence of long-range correlation and self-similarity in plasma fluctuations, Phys. Plasmas '''6''', 5 (1999) 1885]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/6/1885/1 B.A. Carreras et al, ''Experimental evidence of long-range correlation and self-similarity in plasma fluctuations'', Phys. Plasmas '''6''', 5 (1999) 1885]</ref>


In this framework, an important technique is the quiet-time analysis.
In this framework, an important technique is the quiet-time analysis.
<ref>[http://link.aps.org/doi/10.1103/PhysRevE.66.036124 R. Sánchez et al, Quiet-time statistics: A tool to probe the dynamics of self-organized-criticality systems from within the strong overlapping regime, Phys. Rev. E, '''66''' (2002) 036124]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevE.66.036124 R. Sánchez et al, ''Quiet-time statistics: A tool to probe the dynamics of self-organized-criticality systems from within the strong overlapping regime'', Phys. Rev. E, '''66''' (2002) 036124]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.90.185005 R. Sánchez et al, Quiet-time statistics of electrostatic turbulent fluxes from the JET tokamak and the W7-AS and TJ-II stellarators, Phys. Rev. Lett. '''90''', 185005 (2003)]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.90.185005 R. Sánchez et al, ''Quiet-time statistics of electrostatic turbulent fluxes from the JET tokamak and the W7-AS and TJ-II stellarators'', Phys. Rev. Lett. '''90''', 185005 (2003)]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/12/052304/1 V.E. Lynch et al, Determination of long-range correlation by quiet-time statistics, Phys. Plasmas '''12''' (2005) 052304]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/12/052304/1 V.E. Lynch et al, ''Determination of long-range correlation by quiet-time statistics'', Phys. Plasmas '''12''' (2005) 052304]</ref>


=== Turbulence classification ===
=== Turbulence classification ===


An important effort has also been made to identify and classify turbulence,
An important effort has also been made to identify and classify turbulence,
<ref>[http://link.aip.org/link/?PHPAEN/7/1408/1 E. Sánchez et al, Statistical characterization of fluctuation waveforms in the boundary region of fusion and non-fusion plasmas, Phys. Plasmas '''7''', 5 (2000) 1408]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/7/1408/1 E. Sánchez et al, ''Statistical characterization of fluctuation waveforms in the boundary region of fusion and non-fusion plasmas'', Phys. Plasmas '''7''', 5 (2000) 1408]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/42/4/302 I. García-Cortés et al, Turbulent transport studies in the JET edge plasmas in limiter configuration, Plasma Phys. Control. Fusion '''42''' (2000) 389]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/42/4/302 I. García-Cortés et al, ''Turbulent transport studies in the JET edge plasmas in limiter configuration'', Plasma Phys. Control. Fusion '''42''' (2000) 389]</ref>
<ref>C. Hidalgo et al, Intermittency and structures in edge plasma turbulence, Comptes Rendus Physique '''7''', 6 (2006) 679</ref>  
<ref>C. Hidalgo et al, ''Intermittency and structures in edge plasma turbulence'', Comptes Rendus Physique '''7''', 6 (2006) 679</ref>  
to analyse its spectra,
to analyse its spectra,
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.82.3621 M. A. Pedrosa et al, Empirical similarity of frequency spectra of the edge plasma fluctuations in toroidal magnetic confinement systems, Phys. Rev. Lett. '''82''' (1999) 3621]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.82.3621 M. A. Pedrosa et al, ''Empirical similarity of frequency spectra of the edge plasma fluctuations in toroidal magnetic confinement systems'', Phys. Rev. Lett. '''82''' (1999) 3621]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/6/4615/1 B.A. Carreras et al, Characterization of the frequency ranges of the plasma edge fluctuation spectra, Phys. Plasmas '''6''', 12 (1999) 4615]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/6/4615/1 B.A. Carreras et al, ''Characterization of the frequency ranges of the plasma edge fluctuation spectra'', Phys. Plasmas '''6''', 12 (1999) 4615]</ref>
<ref>M. A. Pedrosa et al, Studies of spectra of the edge plasma fluctuations in toroidal magnetic confinement systems, J. Plasma Fusion. Res. SERIES, '''2''' (1999) 77</ref>
<ref>M. A. Pedrosa et al, ''Studies of spectra of the edge plasma fluctuations in toroidal magnetic confinement systems'', J. Plasma Fusion. Res. SERIES, '''2''' (1999) 77</ref>
to determine its relation with local plasma parameters (rational surfaces, gradients, electric fields),
to determine its relation with local plasma parameters (rational surfaces, gradients, electric fields),
<ref>M.A. Pedrosa et al, Role of rational surfaces on fluctuations and transport in the plasma edge of the TJ-II stellarator, Czechoslovak Journal of Physics, '''50''', 12 (2000) 1463</ref>
<ref>M.A. Pedrosa et al, ''Role of rational surfaces on fluctuations and transport in the plasma edge of the TJ-II stellarator'', Czechoslovak Journal of Physics, '''50''', 12 (2000) 1463</ref>
<ref>[http://dx.doi.org/10.1088/0029-5515/42/10/305 B. Gonçalves et al, Experimental investigation of dynamical coupling between density gradients, radial electric fields and turbulent transport in the JET plasma boundary region, Nucl. Fusion '''42''' (2002) 1205]</ref>
<ref>[http://dx.doi.org/10.1088/0029-5515/42/10/305 B. Gonçalves et al, ''Experimental investigation of dynamical coupling between density gradients, radial electric fields and turbulent transport in the JET plasma boundary region'', Nucl. Fusion '''42''' (2002) 1205]</ref>
<ref>M.A. Pedrosa et al, Edge turbulence during limiter biasing experiments in the TJ-II stellarator, Czechoslovak Journal of Physics, '''53''' (2003) 877</ref>
<ref>M.A. Pedrosa et al, ''Edge turbulence during limiter biasing experiments in the TJ-II stellarator'', Czechoslovak Journal of Physics, '''53''' (2003) 877</ref>
and to visualise turbulent structures.
and to visualise turbulent structures.
<ref>[http://www.new.ans.org/pubs/journals/fst/a_1250 J. A. Alonso et al, Two-Dimensional Turbulence Analysis Using High-Speed Visible Imaging in TJ-II Edge Plasmas, Fusion Science and Technology '''50''', 2 (2006) 301]</ref>
<ref>[http://www.new.ans.org/pubs/journals/fst/a_1250 J. A. Alonso et al, ''Two-Dimensional Turbulence Analysis Using High-Speed Visible Imaging in TJ-II Edge Plasmas'', Fusion Science and Technology '''50''', 2 (2006) 301]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/48/12B/S44 J.A. Alonso et al, Impact of different confinement regimes on the two-dimensional structure of edge turbulence, Plasma Phys. Control. Fusion '''48''' (2006) B465]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/48/12B/S44 J.A. Alonso et al, ''Impact of different confinement regimes on the two-dimensional structure of edge turbulence'', Plasma Phys. Control. Fusion '''48''' (2006) B465]</ref>
<ref>[http://dx.doi.org/10.1016/j.jnucmat.2009.01.140 D. Carralero et al, Turbulence studies by fast camera imaging experiments in the TJII stellarator, J. Nucl. Mat. '''390-391''' (2009) 457]</ref>
<ref>[http://dx.doi.org/10.1016/j.jnucmat.2009.01.140 D. Carralero et al, ''Turbulence studies by fast camera imaging experiments in the TJII stellarator'', J. Nucl. Mat. '''390-391''' (2009) 457]</ref>


== References ==
== References ==
<references />
<references />
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