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Long-range correlation: Difference between revisions
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== Coherent states == | == Coherent states == | ||
Coherent system states (regular oscillations or 'modes') lead to oscillatory behaviour of the correlation function, as is easily checked by setting ''X = sin(ωt)'' and taking, e.g., ''Y= | Coherent system states (regular oscillations or 'modes') lead to oscillatory behaviour of the correlation function, as is easily checked by setting ''X = sin(ωt)'' and taking, e.g., ''Y=sin(ωt+φ)''. | ||
Note also that the correlation function is a convolution, hence its spectrum is the product of the spectra of ''X'' and ''Y'', so that γ<sub>XY</sub> 'inherits' the spectral properties of the original time series. | The cross phase ''φ'' can be determined from the delay ''Δ'' of the maxima of the cross correlation γ (modulo 2π): ''φ = -ωΔ<sub>max</sub>''. | ||
Note also that the correlation function is a [[:Wikipedia:convolution|convolution]], hence by the [[:Wikipedia:convolution theorem|convolution theorem]] its spectrum is the product of the spectra of ''X'' and ''Y'', so that γ<sub>XY</sub> 'inherits' the spectral properties of the original time series. | |||
== Turbulence == | == Turbulence == | ||
More interesting is the typical behaviour of the correlation function for turbulent states. | More interesting is the typical behaviour of the correlation function for turbulent states. | ||
In this case, the correlation function typically decays exponentially as a function of Δ and can be characterized by a single number: the 'decorrelation time' (or length) Δ<sub>corr</sub>, calculated as the distance at which the correlation has dropped from its maximum value by a factor ''1/e''. | In this case, the correlation function typically decays exponentially as a function of Δ and can be characterized by a single number: the 'decorrelation time' (or length) Δ<sub>corr</sub>, calculated as the distance at which the correlation has dropped from its maximum value by a factor ''1/e''. | ||
Δ<sub>corr</sub> constitutes the typical ''scale length'' for turbulent dynamics (turbulent transport). | |||
== Long range effects == | == Long range effects == | ||
However, often it is observed that the correlation exhibits a slower decay for large values of the delay (or distance) Δ, namely an algebraic decay proportional to 1/Δ<sup>α</sup> (α > 0 but not too large, < 2). | |||
In this case, the correlations at large delay may be quite important to understand the global system behaviour (contrasting sharply with the exponential decay case). | |||
The particular choice of the power law as the main alternative of the exponential decay is not casual: it is motivated by the fact that [[:Wikipedia:Power law|power law distributions are self-similar]]. | |||
Particularly, an algebraic decay of the mentioned type implies that no particular ''scale length'' can be assigned to the turbulent dynamics, and all scales (up to the system size) will participate in the global description of system behaviour. | |||
This unusual, slow decay of the correlation function has important consequences, implying that the system exhibits 'memory effects' or 'non-local behaviour' (self-similarity). | This unusual, slow decay of the correlation function has important consequences, implying that the system exhibits 'memory effects' or 'non-local behaviour' (self-similarity). | ||
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These issues can be understood in the framework of [[Self-Organised Criticality]]. | These issues can be understood in the framework of [[Self-Organised Criticality]]. | ||
The mathematical modelling of such systems is based on the [[Continuous Time Random Walk]] and the Generalized Master Equation. | The mathematical modelling of such systems is based on the [[Continuous Time Random Walk]] and the Generalized Master Equation. | ||
<ref>R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', [[doi:10.1103/PhysRevE.71.011111|Phys. Rev. E '''71''' (2005) 011111]]</ref> | |||
=== Experimental determination === | === Experimental determination === | ||
It can be shown that determining the long-range behaviour of the correlation function directly from γ<sub>XY</sub> is not a good idea, due to its sensitivity to noise.<ref> | It can be shown that determining the long-range behaviour of the correlation function directly from γ<sub>XY</sub> is not a good idea, due to its sensitivity to noise.<ref>B.A. Carreras, D.E. Newman, B.Ph. van Milligen, and C. Hidalgo, ''Long-range time dependence in the cross-correlation function'', [[doi:10.1063/1.873192|Phys. Plasmas '''6''' (1999) 485]]</ref> | ||
Rather, techniques such as the [[:Wikipedia:Rescaled range|Rescaled Range]], [[: | Rather, techniques to determine the [[:Wikipedia:Hurst exponent|Hurst exponent]], such as the [[:Wikipedia:Rescaled range|Rescaled Range]]<ref>B.A. Carreras, B. van Milligen, M.A. Pedrosa, et al., ''Long-Range Time Correlations in Plasma Edge Turbulence'', [[doi:10.1103/PhysRevLett.80.4438|Phys. Rev. Lett. '''80''' (1998) 4438]]</ref><ref>B.A. Carreras, B.Ph. van Milligen, M.A. Pedrosa, et al., ''Experimental evidence of long-range correlations and self-similarity in plasma fluctuations'', [[doi:10.1063/1.873490|Phys. Plasmas 6 (1999) 1885]]</ref> or Structure Functions<ref>M. Gilmore, C.X. Yu, T.L. Rhodes, and W.A. Peebles, ''Investigation of rescaled range analysis, the Hurst exponent, and long-time correlations in plasma turbulence'', [[doi:10.1063/1.1459707|Phys. Plasmas '''9''' (2002) 1312]]</ref> should be used to determine long-range correlations in data series. | ||
In practice, long-range correlations may have various origins, and proper techniques are required to distinguish between those. | |||
<ref>B.Ph. van Milligen et al, ''Parallel and perpendicular turbulence correlation length in the TJ-II Stellarator'', [[doi:10.1088/0029-5515/53/9/093025|Nucl. Fusion '''53''' (2013) 093025]]</ref> | |||
This is particularly important when trying to separate Zonal Flow contributions to the long-range correlation from other effects. | |||
<ref>B.Ph. van Milligen, E. Sánchez, A. Alonso, M.A. Pedrosa, C. Hidalgo, A. Martín de Aguilera, A. López Fraguas, ''The use of the Biorthogonal Decomposition for the identification of zonal flows at TJ-II'', [[doi:10.1088/0741-3335/57/2/025005|Plasma Phys. Control. Fusion '''57''', 2 (2015) 025005]]</ref> | |||
== See also == | == See also == | ||