Stochastic Models of Structural Plasma Turbulence 9783110936032

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MODERN PROBABILITY AND STATISTICS

Stochastic Models of Structural Plasma Turbulence

ALSO AVAILABLE IN MODERN PROBABILITY AND STATISTICS: Probabilistic Applications of Tauberian Theorems A.L. Yakimiv Generalized Poisson Models and their Applications in Insurance and Finance V.E. Bening and V.Yu. Korolev Robustness in Data Analysis: criteria and methods G.L. Shevlyakov and N.O. Vilchevski Asymptotic Theory of Testing Statistical Hypotheses: Efficient Statistics, Optimality, Power Loss and Deficiency V.E. Bening Selected Topics in Characteristic Functions N.G. Ushakov Chance and Stability. Stable Distributions and their Applcations V.M. Zolotarev and V.V. Uchaikin Normal Approximation: New Results, Methods and Problems V.V. Senatov Modern Theory of Summation of Random Variables V.M. Zolotarev Modern Probability and Statistics: Queueing Theory P.P. Bocharov, C. D'Apice, A.V. Pechinkin and S. Salerno

MODERN PROBABILITY AND STATISTICS

Stochastic Models of Structural Plasma Turbulence V. Yu. Korolev and N.N. Skvortsova

///VSP/// LEIDEN · BOSTON, 2 0 0 6

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ISBN-13: 978-90-6764-449-5 ISBN-10: 90-6764-449-8

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CONTENTS ν

Contents Preface

vu

Ion-acoustic structural turbulence in low-temperature magnetised plasma Ν. N. Skvortsova, A. E. Petrov, K. A. Sarksyan, and Ν. K. Kharchev Structural plasma turbulence and anomalous non-Brownian diffusion Ν. N. Skvortsova, G. M. Batanov, A. E. Petrov, A. A. Pshenichnikov, K. A. Sarksyan, Ν. K. Kharchev, Yu. V. Kholnov, V. E. Bening, V. Yu. Korolev, V. V. Saenko, and V. V. Uchaikin

37

Low-frequency structural plasma turbulence in stellarators Ν. N. Skvortsova, G. M. Batanov, L. V. Kolik, A. E. Petrov, A. A. Pshenichnikov, K. A. Sarksyan, Ν. K. Kharchev, Yu. V. Kholnov, J. Sanchez, T. Estrada, B. van Miliigen, Κ. Ohkubo, Τ. Shimozuma, Y. Yoshimura, and S. Kubo

63

New possibilities for the mathematical modelling of turbulent transport processes in plasma Ν. N. Skvortsova, G. M. Batanov, A. E. Petrov, A. A. Pshenichnikov, K. A. Sarksyan, Ν. K. Kharchev, V. Yu. Korolev, T. A. Maravina, J. Sanchez, and S. Kubo

87

Multi-fractal statistics of edge plasma turbulence in fusion devices V. P. Budaev

115

Analysis of experimental edge turbulence characteristic by simulation with stochastic numerical model A. O. Urazbaev, V. A. Vershkov, D. A. Shelukhin, and S. V. Soldatov

141

Monte-Carlo simulations of resonance radiation transport in plasma V. V. Uchaikin and A. Yu. Zakharov

167

Fractionally V. E. Bening,stable V. Yu.distributions Korolev, T. A. Sukhorukova, G. G. Gusarov, V. E. Saenko, V. V. Uchaikin, and V. N. Kolokoltsov

175

V

Statistical analysis of volatility of financial time series and turbulent plasmas by the method of moving separation of mixtures V. Yu. Korolev and M. Rey Hidden Markov models of plasma turbulence Α. V. Borisov, V. Yu. Korolev, and A. I. Stefanovich

Preface About five years ago, under the auspice of the Chair of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Moscow State University, the research seminar 'Stochastic Structures in Plasma Turbulence' started its regular monthly sessions. This seminar was initiated by intensive discussions concerning mathematical modelling of plasma turbulence between physicists from the General Physics Institute of the Russian Academy of Sciences and mathematicians from the Moscow State University. Very soon many specialists representing many Russian scientific centres, institutions and universities, e.g., General Physics Institute of the Russian Academy of Sciences, Russian Scientific Centre 'Kurchatov Institute,' Faculties of Computational Mathematics and Cybernetics, and of Mathematics and Mechanics of the Moscow State University, Saratov State University, Moscow State Technological University, Ufa Aviation Technological University, Ulyanovsk State University, were involved in these discussions. In order to turn these discussions into more or less regular regime, it was decided to start the sessions of the seminar mentioned above. The leaders of the seminar are Victor Korolev (Moscow State University) and Nina Skvortsova (General Physics Institute). The main aim of the communications and discussions at the seminar is the construction of satisfactory mathematical models of plasma turbulence which take into account the specific features of statistical regularities observed in real experiments such as, say, heavy-tailedness and leptokurtosity of empirical probability distributions and nontrivial correlation structure of the observed processes. In 2003, a collection of most interesting communications made at this seminar was published in Russian and immediately became an object of interest of many specialists in plasma physics and other fields dealing with chaotic processes, in particular, financial mathematics. Since then more results were obtained, so it seems that we finally came to some understanding (if not of the nature of plasma turbulence, but at least of each other) and the time has come to make these results available to the scientific community. So, this book was prepared. It contains both some reviews published in 2003 in the Russian version of the proceedings of the seminar 'Stochastic Structures in Plasma Turbulence' and some new papers dealing with the recent investigations. This collection of articles begins with an article devoted to the research on ionacoustic turbulence in current-carrying magnetized plasma 7on-acoustic structural turbulence in low-temperature magnetised plasma.' There are some reasons for such priority, first, the term 'strong structural turbulence1 itself was proposed in 1994 in a paper devoted to the ion-acoustic turbulence. Second, experimental investigations described in this article, which were being carried out for more than ten years in the TAU-1 linear device especially designed to study nonlinear plasma states, may be attested as most detailed in respect of the description of characteristics of the low-frequency (LF) strong structural (SS) turbulence. This is a determinate-chaotic state in which ensembles of chaotic structures exist in turbulence. The stochastic structures determine a number of spectral, correlation, and probabilistic parameters of this turbulence. The main characteristic feature of ion-sound structural turbulence is that the probability density functions (PDFs) of the fluctuating parameters differ from a normal distribution: the observed PDFs are leptokurtic and are characterized by heavier tails. It is demonstrated in the next articles that low-frequency strong structural turbulence exists in many toroidal plasma confinement systems, such as T-10 tokamak ('Analysis of experimental edge turbulence characteristics by simulation with stochastic

vii

numerical model') and L-2M, TJ-II and LHD stellarators ('Low-frequency structural plasma turbulence in stellarators'). With a continuous inflow of energy into an open plasma confinement system, a universal state with strong structural turbulence is established in it, irrespective of the type of plasma instability. Stable (non-Gaussian) PDFs were measured for plasma density fluctuations in the hot and edge regions of toroidal devices. The essential feature of the LF SS turbulence in tokamaks and stellarators are that the PDFs differ from the normal distribution by heavier tails and a higher peakedness. Non-Gaussian PDFs of stochastic plasma processes point to the non-Brownian (anomalous) motion of particles in stochastic fields. In a plasma with LF SS turbulence, the role of rare events with magnitudes far exceeding the average values substantially increases and needs to be estimated. In plasma physics, the possibility exists of studying the diffusion of ensembles of particles in plasma turbulence by direct measurements. Experimental time samples of particle fluxes in low-frequency strong structural turbulence have been analyzed in the article 'Structural plasma turbulence and anomalous nonBrownian diffusion.' Correlation and probabilistic properties of local fluxes indicate general features in the motion of particle ensembles in a magnetized plasma in LF SS turbulence, irrespective of the structure of the magnetic field of the device—a toroidal magnetic configuration with helical transform of magnetic lines in L-2M or a magnetic field of a straight solenoid in TAU-1. Diffusion processes in LF SS turbulence were analyzed on a basis of the following characteristics of local fluxes and their increments: time samples, correlation functions, PDFs. Anomalous diffusion is also the object of consideration in the paper 'Monte-Carlo simulations of resonance radiation transport in plasma.' In the paper 'Multifractal statistics of edge plasma turbulence in fusion devices,' plasma turbulence characteristics related to the fractal structure and turbulent transport in the T-10 tokamak and linear devise NAGDIS-II are investigated. It was shown that in these devices, PDFs of density fluctuations are typically non-Gaussian and positively skewed. The PDFs of the signal increments at different time scales demonstrate multifractal property of the process: they satisfy an evolution equation from 'quasiGaussian' at large scale to fat-tailed PDFs at small scales, fluctuations in plasma density have been analyzed in terms of the multifractal formalism revisited with wavelets. Plasma edge turbulence possesses multifractal statistics, i.e., scaling behaviour of absolute moments is described by a convex function with non-trivial self-similarity properties. The diffusion transport coefficient in edge plasma is not a trivial function of multifractality parameters. The process is characterized as superdiffusion. A new mathematical model is proposed for the probability distributions of the characteristics of the processes observed in turbulent plasmas ( ' N e w possibilities for the mathematical modelling of turbulent transport processes in plasma'). The model is based on formal theoretical considerations related to probabilistic limit theorems for a nonhomogeneous random walk and has the form of a finite mixture of Gaussian distributions. The reliability of the model is confirmed by the results of a statistical analysis of the experimental data on density fluctuations in high-temperature plasmas of the L-2M, LHD, and TJ-II stellarators and the local fluctuating flux in the TAU-1 linear device and in the edge plasma of the L-2M stellarator with the use of the estimation-maximization algorithm. It is shown that low-frequency structural turbulence in a magnetized plasma is related to non-Brownian transport, which is determined by the characteristic temporal and spatial scales of the ensembles of stochastic plasma structures. The paper 'Fractionally stable distributions' contains a comprehensive review of the properties of a family of heavy-tailed distributions which is often used for modelling chaotic processes. viii

However, in most cases fractionally stable distributions fail to be satisfactory probabilistic models of statistical regularities observed in plasma turbulence because they possess too heavy tails. As an alternative, in the paper 'Statistical analysis of volatility of financial time series and turbulent plasmas by the method of moving separation of mixtures' it is proposed to model the probability distributions emerging in experiments with turbulent plasmas by finite mixtures of Gaussian distributions. A 'dynamic' modification of the approach proposed in the paper 'New possibilities for the mathematical modelling of turbulent transport processes in plasma,' namely, the method of moving separation of mixtures is described. It is demonstrated that this method is a very powerful tool for the investigation of chaotic processes in turbulent plasmas and presents a principally new viewpoint on the nature of the diffusion in these processes. A clear analogy is underlined between chaotic processes in experiments with turbulent plasma and financial markets. The method of moving separation of mixtures shows that the parameters of the probability mixture distributions describing statistical regularities in plasma turbulence vary in time. In the paper 'Hidden Markov models of plasma turbulence,' a reasonable model of this variation is proposed and discussed. We have the pleasure to thank all the authors who contributed their papers for this collection. We thank all the participants of the seminar 'Stochastic Structures in Plasma Turbulence' who participated in the discussions. We especially thank Andrei Kolchin who managed to prepare all the materials in the readable form and edited the manuscript. Victor Korolev Nina Skvortsova October 7, 2005 Moscow

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Stochastic Models of Plasma Turbulence, pp. 1 - 3 6 V. Yu. Korolev and Ν. N. Skvortsova (Eds.) © Koninklijke Brill NV 2005.

Ion-acoustic structural turbulence in low-temperature magnetised plasma Ν. N. SKVORTSOVA, A. E. PETROV, K. A. SARKSYAN, N. K. KHARCHEV Prokhorov Institute of General Physics, Moscow, Russia

1.

INTRODUCTION

The term itself 'strong structural turbulence' was proposed in 1994 in a paper devoted to the ion-acoustic turbulence [1]. Second, experimental investigations described below, which were carried out for more than ten years in the TAU-1 linear device specially designed to study nonlinear plasma states, may be attested as most detailed in respect of the description of characteristics of the low-frequency (LF) strong structural (SS) turbulence. Third, in a report presented at the conference on nonlinear dynamics in 1989 [2], it was shown that the spectral and correlation characteristics of LF plasma turbulence in different devices can be described in the general form. At that time, we compared the ion-acoustic turbulence in the TAU-1 and the drift-dissipative turbulence in the L-2M stellarator. More recently, the general features of the LF SS turbulence were observed in many other devices. We believe that the experimental research on plasma turbulence evolves in much the same way as the experimental research on hydrodynamic turbulence. Let us quote here quote an excerpt from the book by Belotserkovskii and Oparin [3]: 'During the first years, these phenomena were interpreted to be completely stochastic processes (determined by the distributions of fluctuating variables). However, to date, a radical revolution has occurred in the understanding of these phenomena. It was found that turbulence also includes the ordered motion of almost coherent structures, and the interrelation between determinate and chaotic sources is being actively studied now.' Indeed, a kind of revolution has occurred in the research of plasma turbulence: a mixed 'determinate-chaotic' state has been found in the plasma turbulence. Moreover, in the plasma turbulence we always deal with structural plasma turbulence in which the appearance of structures is stochastic in character and the role of rare events increase i η importance (in the book 'Present Problems of Nonlinear Dynamics' [4], this state is termed the rigid turbulence). We mean by turbulence the state involving the stochastic interaction of a wealth of wave motions of different scale. The stochastic interaction of small-amplitude waves is usually considered in the well-developed theory of weak turbulence, which represents wave motions as a superposition of linear wave modes [5, 6], For large-amplitude wave motions, when the waves become strongly nonlinear, and their growth rate approaches characteristic frequencies, the plane wave approximation is no longer correct. In this case, we have to apply the theory of strong wave turbulence in plasma. Some experimental data indicating to the

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strong nature of LF plasma turbulence are observed in many devices [7-14], At present, this theory is far from being completed. Particular problems arise in describing the strong turbulence of magnetoactive plasma because of generation of numerous oscillatory branches. In this plasma, for simplicity the consideration can be restricted to one or other type of oscillation that gives rise to turbulence [15, 16] with definite properties. For example, at the present time, the vortex model is actively developed for describing the strong turbulence of drift and ion-acoustic modes in magnetoactive plasma. And this model in some cases is confirmed experimentally [17-22]. However, even such a simplified approach may fail to give an unambiguous notion of the development of turbulence on some particular oscillation branch. Note that there appear theoretical papers in which the low-frequency turbulence is considered as a strong turbulence with the parameters independent of the type of concrete plasma instability and concrete mechanisms of instability saturation [6, 23-27]. For many years, the development of a concept of strong LF turbulence has been hampered by the absence of a detailed experimental analysis of its structure. With the aim to perform such experimental studies, we have chosen the ion-acoustic turbulence, which is associated with the ion-acoustic instability and is amenable to observations in the TAU-1 model device. For many years the attention of investigators in plasma physics has been drawn to the study of the nature of ion-acoustic plasma turbulence, and individual chapters in many books have been devoted to the theoretical description of this phenomenon. As an example, we mention two books, published more than thirty years apart [5, 6]. Simultaneously with theoretical studies, experimental studies of ion-acoustic turbulence have also been conducted. The difficulty in performing experimental investigations lay in the need to measure many statistical parameters of turbulence (correlation, spectral, and so on) in order to compare with the theoretical description. For this reason, in the first experiments devoted to ion-acoustic turbulence, average values of some plasma variables were determined and associated with the intensity of the turbulence. Thus, in the paper of Ε. K. Zavoiskii [28], a well-known estimate of the anomalous resistance was made for the case of turbulent fluctuations. However, the statistical parameters of the turbulence could not be estimated because it was necessary to measure and analyse large sets of fluctuation data. For this reason, the study of a real structure of ion-acoustic turbulence was taken up only with computerised acquisition systems at the end of 1980s [18-21]. The most comprehensive and detailed data were obtained from measurements of ion-acoustic turbulence during the last decade in the TAU-1 model linear device. In the review presented below, we show that the features of ion-acoustic turbulence that were revealed in these experiments allow this turbulence to be classified as a 'determinate-chaotic' state of a stationary open nonlinear system—the low-frequency (LF) strong structural (SS) turbulence.

2.

DESCRIPTION OF TAU-1 AND METHODS OF STUDYING LF TURBULENT FLUCTUATIONS

The first experiment on detailed investigation of characteristics of LF SS turbulence was carried out in the TAU-1 linear device [29], This device was designed specifically for studying and modelling nonlinear processes in low-temperature plasma. At first, in this device, we performed investigations of averaged parameters (intensity, Fourier spectral) of LF plasma fluctuations [30-33]; the use of numerical analysis methods was in fact a natural extension of these investigations.

Ion-acoustic

structural

3

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Mobile L a n g m u i r probes Interferometer

R F generator I RF antenna ntenna

field coils

Cathode

field coils

I

•

/ί

Bl HIS

Magnetic

Magnetic

.

Plane probe

\

ί

Z-coordinate probe

Μ

RF antenna

Amplifiers R F generator

Figure 1. Schematic of the TAU-1 model device

A schematic of the TAU-1 device is shown in Fig. 1. Argon plasma in the form of a cylindrical column of diameter 4 cm and length 100 cm was produced in a uniform magnetic field with strength < 800 Oe by a steady low-energy electron beam of energy Eb ~ 6 0 - 1 5 0 eV at argon pressure ρ = (2—4) χ 10 4 torr. The plasma density was maintained at the level η = (0.9-1.2) χ 10 1 0 c m - 3 . The electron temperature was Te = 5 - 7 eV and the ion temperature was T, % 0.17g. The plasma density and electron temperature profiles were parabolic. Two types of low-frequency plasma fluctuations spectra were identified, specifically, a discrete drift oscillation spectrum in the frequency range 10-100 kHz, which were excited due to the onset of drift instability in the plasma gradient region [30], and turbulent ion-acoustic oscillations with the continuum frequency spectrum extended to the ion Langmuir frequency ω,·, which were excited due to the onset of current-driven ion-acoustic instability [32]. The characteristic frequencies in the plasma were related as follows: υ < Ω,· < ωdr < ω5 S u>u Ω ^ , ω ^ , where Ω,·, Ω^ are the ion and electron gyrofrequencies, ω υ , ω ^ are the ion and electron plasma (Langmuir) frequencies, and ω w s are the drift and ion-acoustic wave frequencies. The characteristic frequency intervals in which plasma oscillations are observed under conditions of this particular experiment are /„· ~ 10 kHz; f i r ~ 10-60 kHz; f s ~ 0 . 1 - 5 MHz; f u ~ 4 - 5 MHz; fce, f u ~ 1-1.5 GHz. Steady-state conditions of the plasma discharge in the TAU-1 could be maintained for 5 h, which is very important to the results of measuring the statistically consistent characteristics of random processes. In the TAU-1 device, it is possible to carry out experiments in afterglow plasma and to study transient plasma states. In order to obtain the afterglow regime of the discharge, the cathode voltage could be modulated by pulsed with a duration of 2 0 - 5 0 0 /is, the leading and trailing edges being as short as 1 - 2 /is. Two different systems were designed for the production of high-frequency waves in the plasma. One of these systems was intended for excitation of two extraordinary electron

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cyclotron waves with frequencies of about 1 GHz and with the regulated difference between the frequencies [33, 34]; the length of RF pulses reached 30 /xs, the rise and fall times being about 1 ßs. Another system was used to excite two oblique Langmuir waves with frequencies of 20-30 MHz and with regulated difference between the frequencies or a single oblique Langmuir wave modulated by a low-frequency signal. The length of RF pulses reached 1 ms, the rise and fall times being about 1 /i.s [33,36]. Fluctuations in the low-temperature plasma of the TAU-1 were measured with the help of different types of Langmuir probes [37]. Note that, in the TAU-1 device, we additionally carried out test measurements of turbulent fluctuations in the range of ion-acoustic and drift plasma oscillations by the contactless optical technique [38], A coincidence of spectral characteristics allowed us to use with confidence the results of probe measurements to analyse turbulent fluctuations in low-temperature plasma. Correlation probes were used to measure the radial, poloidal, and longitudinal scale lengths of fluctuations; three-tip probes measured local turbulent particle fluxes; and double probes measured electric field fluctuations. The results of probe measurements were time samples of digitised amplitudes of fluctuating plasma parameters. Such time samples of random steady-state processes in the TAU-1 consisted of several hundred thousand points that required statistical processing for obtaining the characteristics of fluctuations. For their analysis, an integrated program package for statistical processing of time samples was created and then was used in other plasma devices. The program package includes the following programs: multidimensional spectral Fourier analysis [39-41 ]; correlation analysis [40]; maximum entropy method [42]; spectral and coherence wavelet analysis [43^15]; construction of histograms [41], computation of moments of random variables; computation of the Hurst parameter ( R / S analysis [46,47]; and auxiliary programs for smoothing, filtering, and averaging of signals. Algorithms of these programs are briefly described in introductions to the articles [1,48,54]. 3.

STEADY-STATE FOURIER SPECTRA OF LF TURBULENCE IN TAU-1

In a low-temperature plasma of the TAU-1 device, two instabilities can develop, namely, the drift-dissipative instability and ion-acoustic current-driven instability. Drift oscillations arise in a plasma due drift-dissipative instability caused by the plasma-density gradient. The Fourier spectrum of drift oscillations lies in the frequency range below 100 kHz and consists of quasi-harmonics ( A w j < (oj). A typical spectrum is shown in Fig. 2a. The spectrum of ion-acoustic plasma oscillations lies in the interval from drift frequencies to the ion plasma (Langmuir) frequency and is continuous (Fig. 2b). What are physical processes that lead to these spectra ofLF plasma turbulence? What is the kind of a structure of wave motions existing in the TAU-1? The answers to these questions are offered in the sections below. 4.

RESULTS OF STUDY OF TURBULENCE

4.1. Ion-acoustic turbulence and ensembles of solitons Fig. 3a illustrates an example of the steady-state time sample (time realization, oscillogram of ion-acoustic turbulent oscillations) in plasma of the TAU-1 device. The measurement frequency range extends from 300 kHz (is limited by a low-frequency filter to cut off driftwave frequencies) to the ion plasma (Langmuir) frequency (in this particular case, 5 MHz).

Ion-acoustic

500

structural

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1500

2000

2500

Frequency, k H z

Figure 2. Steady-state Fourier spectra (a) of the drift and (b) ion-acoustic signals measured at the plasma axis; the filter frequency is 300 kHz; the time window is 26 ms, Η = 5 0 0 G, ρ = 3 χ 1 ( Γ 4 torr, Ih = 2 5 0 mA, Uh = 1 2 0 V

The signal consists of bursts of different length, pauses between them being also of different length. As we see, the signal frequency between bursts varies, and the rise and fall times of bursts themselves are also variable. The term burst is coming increasingly into play for describing signals of this kind (for example, it is adopted in theory of vibrations). The same figure also shows the autocorrelation functions (ACF) of this signal, which were calculated on two time intervals: Δ ( = 0.02 ms (Β) and 0.002 ms (C). It is seen in these figures that the ACF contains two components: a sharp peak (other than a 3 indicate that the PDF differs from the Gaussian distribution. Hence, the shape of the structures (all of them or some of them) differs from Gaussian.

Ion-acoustic

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2 Frequency, M H z

4

Frequency, M H z

Figure 11. Transformation of the spectrum of ion-acoustic oscillations in the afterglow plasma: (a) before switching off the beam; (b) 2, (c) 6, and (d) 10 μ$ after switching off the beam; l/, = 3 0 0 mA. M O ) = 6 χ 10 1 ( l c m " 3 , Uh = - 1 5 0 V, ρ = x l 0 ~ 4 torr, and Η = 0 . 0 4 Τ

0.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

5.0

Frequency, MHz

Figure 12. Wavelet spectra of ion-acoustic oscillations in the afterglow plasma for f , = 3 0 0 mA, n e (0) = 6 χ 1 0 1 0 c m - 3 , Uh = - 1 5 0 V, ρ = 3 χ 1 0 " 4 torr. and Η = 0 . 0 4 Τ, and different values of thee wavelet parameter (one realization)

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Ν. Ν. Skvortsova et al. 127000 measurement points

Figure 13. Probability density function (PDF) of the time sample of density-fluctuation amplitudes in the ionacoustic frequency range; the time window is 25.4 ms, Η = 500 G, ρ = 3 x 10~ 4 torr, //, = 200 mA, and Uh = 120 V

In addition, it is well known that the solitons exist independently of each other. In this experiment, we observed the processes of the nonlinear coupling and decay of structures. The theory [13] predicts that there can exist nonlinear ion-acoustic solitons coupled to each other. It can be suggested that the shape of such solitons should be somewhat different from Gaussian. Therefore, we can suggest that the structures inherent in the ion-acoustic turbulence are nonlinear ion-acoustic solitons. Here, it will be remembered the theoretical paper by Petviashvili, who performed the first analysis and modelling of the ion-acoustic plasma turbulence [5], In this paper, it is pointed out tat the model (in the 1960s) describing the ion-acoustic turbulent state is created for time samples of a random process with the Gaussian probability density. It is well known that the Gaussian distribution is typical of stochastic processes in conservative (closed) systems. However, as early as 1940, Kolmogorov [51] pointed to the fact that the hydrodynamic turbulence with large vortices is a self-similar stochastic process and the probability distribution of time samples of such processes has a heavier hyperbolic tail in comparison with the Gaussian distribution. It is difficult to imagine a plasma laboratory device that would not be an open system in the thermodynamic sense. Then, one would expect that random variables (amplitudes of the density, electric field, particle flux, etc.) determined by turbulent plasma processes cannot be described by Gaussian distributions. Stationary stochastic processes for which the variance of the mean decreases as η ~ a for any a between 0 and 2 were discovered by Kolmogorov in 1941 [51]. These processes were termed self-similar. In the mid-1960s, Mandelbrot [52] introduced self-similar processes in certain statistical applications and substantiated their application in hydrology and geophysics. Mandelbrot called the self-similarity parameter (or the parameter of long-range correlation) the Hurst parameter Η = \ —a/2. This parameter was named for Hurst (1951),

Ion-acoustic

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19

who empirically described long-lived correlations between water levels at overflows of Nile (the example of the self-similar process in hydrology. This parameter can vary from 0 to 1 for stationary increments; Η = 0.5 corresponds to independence of the events in the observed process, Η > 0.5 corresponds to positive correlation, and Η < 0.5 corresponds to negative correlation. Self-similar processes X(t) on the real axis are characterised by the following property: If the time scale of variation is changed by a factor of α > 0, all finite-dimensional distributions change by the factor a H . The study of the autocorrelation functions of self-similar processes is quite complicated because the sum of small individual correlations approaches infinity, which makes it necessary to take into account the correlation between the distant past and the distant future. There exist several methods of investigating the Hurst parameter of a self-similar process: using correlograms, the spectral f unction of the variance as a function of time, the method of maximum likelihood, and Mandelrot's method. In plasma physics, Mandelbrot's method was used by Carreras et al. in 1998 [53] to analyse turbulent fluctuations in the edge plasma of toroidal confinement systems. Fig. 14 shows the logarithm of the ratio of the range R of ion-acoustic turbulent data to the standard deviation S of the incrementation process for the same data versus the logarithm of the measurement time [54]. The length of the data array is 64 000 points. The figure also shows two straight lines corresponding to R / S analysis for a regular process (Hurst parameter Η = 1) and a random process (Η — 0.5). The slope of the experimental R/S dependence is different in the three time intervals. The first time interval up to 10 ß s corresponds to the average energy of turbulent process reaching a steady state, i.e., the process in this interval is not stationary, so that a Hurst parameter cannot be defined. In the next time interval, from ~ 10 μβ to ~ 200 ßs (labelled by 1 in the figure), the logarithm of R/S is directly proportional to the logarithm of the observation time, with a Hurst parameter 0.60.7. This time interval corresponds to the characteristic lifetime of nonlinear ion-acoustic solitons, which was found in preceding experiments, or to the characteristic memory time between nonrandom appearance of these structures. In the time interval from 0.2 ms to 12 ms ('2' in the figure), the logarithm of R/S is directly proportional to the logarithm of the observation time, with a Hurst parameter 0.7-0.8. This time interval reflects the characteristic nonlinear interaction time between the structures. Among the set of measurements, was a time series in which a sharp transition occurs between these intervals, which could be caused by an intense low-frequency regular drift wave that was not properly attenuated in this measurement. Thus, the stationary structural ion-acoustic turbulent process is non-Gaussian, because the Hurst parameter exceeds 0.5 in the entire observation interval. This process can be said to be self-similar. The self-similarity parameter is determined by two time dependences: the nonrandom appearance of structures in turbulence and the nonlinear interaction between these structures. The self-similarity parameter is a = 0.6-0.8 for observation time intervals such that the law describing the nonrandom appearance of structures is the governing factor. For long observation intervals, when the appearance of structures can be regarded as random events because of the multiplicity of such events, the self-similarity parameter decreases to a — 0.4, and only the law of their nonlinear interaction makes it possible to preserve a memory of the self-similar process. Fig. 15 shows the dependence obtained by R/S analysis for ion-acoustic turbulent signals for various electron beam currents in the TAU-1 plasma. The higher the beam cur-

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105

104

103

10'

10° 0.002

0.02

0.2

2

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Lag time (ms)

Figure 14. R/S dependence for the fluctuation signal; the time sample contains 64 000 points; Η = 5 0 0 G, ρ = 3 x 1 0 ~ 4 torr, Ub = 1 2 0 V, and Ih = 5 0 mA

rent, the larger the distortion of the electron distribution is and, as noted above, the farther away the excitation of ion-acoustic turbulence is f r o m the instability threshold. T w o straight lines are also drawn for the regular process (Hurst parameter Η = 1) and r a n d o m process ( Η = 0.5). T h e slope of the experimental R/S dependence is different for different b e a m currents: as the current increases, the experimental R/S plot approaches a direct r a n d o m process. However in the experiment, w e did not observe the regime in which the structural ion-acoustic turbulence could be described by a r a n d o m stochastic process rather than by a self-similar process. The structural ion-acoustic plasma turbulence is a self-similar stochastic process. T h e self-similarity parameter is determined by two temporal laws: the random character of the appearance of nonlinear ion-acoustic solitons in the plasma and the nonlinear interaction between them. Far f r o m the threshold for ion-acoustic current instability, the r a n d o m process approaches a Gaussian r a n d o m process, but this limit was not attained in the experiment.

4.3.

Plasma drift motions. An intermittent character of drift wave packets

T h e excitation of drift oscillations in the TAU-1 device initially appeared to be a simple problem; their solution as a linear problem in the theory of plasma instabilities has long been known. In a cylindrical plasma column with appropriate macroscopic plasma parameters, drift-dissipative instability caused by the plasma density gradient gives rise to drift oscillations with fixed harmonics that should be observed in the steady-state L F drift spectrum. However, it was long ago [30] that similar narrow-band oscillation spectra in the spectrum analyser were obtained by processing oscillograms that contain r a n d o m pauses with the subsequent regeneration of oscillations. Note once more that the spectrum analyser shows the spectrum averaged over many oscillation periods (rather than the current

Ion-acoustic

10

•

1

structural

1

1

10'

r

•

ι

Regular

/

-

/

ω δ

·

ι

/

process/ /

10°

21

turbulence

1=50 mA

^ 1 = 1 0 0 mA f / J [ = 3 0 0 mA

10 /

f j ^ f s f j l s

Gaussian process

10 / f -

10"

1

,

a

'

0.002

0.02

0.2

20.0

2.0

T i m e lag ( m s )

Figure 15. R/S dependence for fluctuation signals with different emission currents; each time sample contains 64 000 points, with averaging over 5 measurements, ρ = 3 x 1 0 - 4 torr, and Uf, = 120 V

spectrum or the momentary spectrum). The frequencies and growth rates of drift motions correspond to the drift-dissipative instability in the TAU-1 plasma [56,57]; The variance also corresponds to the drift-dissipative instability: ω kv
1), one might expect a damping of the turbulence because of a sharp decrease in the instability growth rate [66] γ %-W%{(me/ffli)'/2

+ (Tef Ti)3/2

exp(—Te/2Ti)}.

However, our experiments showed that the damping decrement of the LF SS turbulence is determined not by the fast damping of individual ion-acoustic oscillations, but by a much (one order of magnitude) longer time of the transverse drift of nonlinear ion-acoustic solitons across the magnetic field into low-density plasma layers [55], Hence, the linear model is inapplicable for the estimates. Let us examine the applicability of nonlinear models. At recent years, it has become very popular to compare the decay of the intensity of harmonics in the continuous Fourier spectrum with one or other turbulent fluctuation model. In general, it is the wavenumber spectra that we must compare in this case, but these spectra are difficult to measure accurately. For this reason, one passes to analysing the frequency spectra on general assumption

Ion-acoustic

structural

turbulence

31

that, as the plasma oscillation frequency increases, the oscillation wavelength decreases in the plasma turbulence. Fig. 23 shows the Fourier spectrum up to the frequency equal to onehalf the ion plasma frequency (about 5 MHz) for a long time sample of duration 3250 /is; the sampling rate is 100 ns, the frequencies below 600 kHz are cut off. The spread of the signal amplitudes along the ordinate axis is attributed to a nonzero tail of the autocorrelation function and is determined by ion-acoustic solitons. This spectrum is compared to the superimposed curves of typical spectra of the fluctuation-noise model ( ~ \ / X ) , the Kolmogorov-Obukhov model of the energy redistribution over the spectrum (1/ X 5/ " 2 ), the Kadomtsev-Petviashvili model of an ion-acoustic turbulent spectrum { \ / X ) l n ( l / X ) , and (\/ X2). Any of these models may be relevant to the case of LF SS turbulence, to judge from the shape of the frequency Fourier spectra. Nevertheless, taking into consideration any one of other properties leads to a discrepancy between the model and the physical picture. For example, the autocorrelation function of the fluctuation noise is a delta-function as distinct from what is observed in the experiment. The Kolmogorov-Obukhov model predicts the energy redistribution in the spectrum occurs primarily from small-scale to large-scale fluctuations, but the energy transfer from large-scale to small-scale structures is equiprobable in the experiment. The Kadomtsev-Petviashvili model assumes a normal (Gaussian) probability density function, which also conflicts with the experiment. Hence, the estimates based on these models fail to describe the LF SS turbulence. There exist a model of an ion-acoustic soliton [6] as a running plasma-density maximum followed by a density minimum. The velocity of such a soliton is u — u 5 ( l + ecp/'SkTe), and its dimension is Δ = (2/3ecp/kTe)1/2. The more intense the soliton, the higher is its velocity. Fig. 24 (taken from [6]) is in good agreement with Fig. 10 of this paper. It might seem that the next step should be toward the soliton turbulence model. As was noted in [16, 67], weakly interacting non-linear solitons can exist in the plasma. Let us quote here an excerpt f r o m [67, §6]: 'The interaction of real soliton-like waves is a rather complicated process and is accompanied by the energy exchange. What this means is that, at a weak interaction of the nonlinear waves, they can be considered as ensembles in which the turbulent motion should appear if certain inequalities hold.' From this excerpt, it is clear that no physical model of a mixed determinate-chaotic state of the LF SS turbulence has thus far been presented. None of the above physical models was adequate for theoretical description (even partially) of the phenomenon of LF SS turbulence.

7.

CONCLUSION

Experimental studies carried out in low-temperature magnetised plasma of the TAU-1 linear device have revealed a new determinate-chaotic state of low-frequency turbulence, that is strong structural low-frequency turbulence. (1) LF SS turbulence arises in a steady-state magnetised plasma under conditions where dynamic equilibrium exists between the energy source and sink in the device, and different mechanisms of turbulent processes may be driven by two instabilities: ionacoustic and drift-dissipative instabilities. It should be noted that the structural ionacoustic turbulence exists over a wide range of macroscopic plasma parameters. We could obtain neither the plasma regimes in which the ion-acoustic turbulence exists as

32

Ν. Ν. Skvortsova et al.

Λ (I

1 0.5

1 1.0

1 1.5

1 2.0

Γ" 2.5

Frequency, MHz

Figure 23. Fourier spectrum of the ion-acoustic structural turbulence; the length of the time realization is 32 thousand points

Figure 24. KDW solitons: a fast (solid line) and a slower (dashed line) soliton

Ion-acoustic

structural

turbulence

33

a weak turbulence (with the emission currents slightly above the instability threshold) nor the regime of unstructured turbulence (with the emission currents significantly exceeding the instability threshold). (2) A characteristic feature of this type of turbulence is the existence of stochastic plasma structures. A considerable fraction (under various conditions, from 10% to 30% of the total energy of turbulence is concentrated in the nonlinear structures. The observed structures are nonlinear ion-acoustic solitons and drift wave packets. One of channels of plasma losses across the magnetic field was found to be a drift motion of nonlinear solitons. (3) The structural ion-acoustic turbulence is a self-similar stationary stochastic process. The self-similarity parameter is determined by two temporal events: the appearance of nonlinear ion-acoustic solitons in the plasma and the nonlinear interaction (coupling and decay) between them. When the emission current far exceeds the threshold for ion-acoustic instability, the random process tends to a Gaussian random process, which, however, was not attained in the experiment. (4) In a steady-state plasma, the drift and ion-acoustic instabilities interact through ensembles of stochastic plasma structures; the appearance and disappearance of these ensembles is observed as a synchronous cycling process. The cross-correlation coefficient between the turbulent fluctuations observed in different plasma frequency ranges attains 5 0 - 6 0 % . Thus, two different types of plasma turbulence, both being determinate-chaotic states, turn out to be interrelated. (5) The experiment on feedback control of parameters of LF SS turbulence demonstrated that drift turbulence could be modified into a quasi-regular state with a reduced energy of turbulent fluctuations. The parameters of LF SS turbulence were controlled using a low-amplitude regular drift by introducing a phase shift relative to the original drift wave by some value depending on techniques used (beat of waves, or modulation waves). Mathematically, a stable non-Gaussian stochastic process modelled by the probability density function with a heavy self-similar tail closely corresponds to the LF SS turbulence. To create a mathematical model that would describe the experiment adequately is a very complicated problem, which to date has not been solved. It may appear that there is little sense in great affords toward this problem if we deal with a particular experiment on LF SS turbulence in a particular linear device with low-temperature plasma. However, in the following reviews presented in this book, it will be shown that do the low-frequency strong structural turbulence exists in many toroidal plasma confinement systems, such as tokamaks and stellarators, and future results may justify the efforts directed toward a physical model of the LF SS turbulent state.

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Stochastic Models of Plasma Turbulence, pp. 3 7 - 6 2 V. Yu. Korolev and Ν. N. Skvortsova (Eds.) © Koninklijke Brill N V 2005.

Structural plasma turbulence and anomalous non-Brownian diffusion N. N. SKVORTSOVA, G. M. BATANOV, A. E. PETROV, A. A. PSHENICHNIKOV, K. A. SARKSYAN, N. K. KHARCHEV, YU. V. KHOLNOV Prokhorov Institute of General Physics, Moscow, Russia

V. E. BENING, V. YU. KOROLEV Moscow State University, Moscow, Russia

V. V. SAENKO, V. V. UCHAIKIN Ulyanovsk State University, Ulyanovsk, Russia

1.

INTRODUCTION

Research into structural plasma turbulence is closely connected with the problems of anomalous heating and particle transport in closed magnetic confinement systems. It is clear that heat transport in the edge plasma of toroidal magnetic confinement systems is largely governed by turbulent fluctuations in the charged particle density and the electric and magnetic fields [1, 2]. The correlation between transport processes and turbulent fluctuations manifests itself most clearly during transitions to the improved confinement mode (Η-mode) and the generation of transport barriers both at the plasma edge and in the plasma core [3, 4]. A general feature of such transitions is the change in the gradients of the plasma parameters and the shear of the poloidal rotation velocity, as well as variations in the local characteristics of turbulent fluctuations. A transition from one plasma confinement mode to another can be induced in different ways, in particular, by varying the plasma heating power or the particle flux from the toroidal chamber wall [5-7]. However, both of the mechanisms by which local turbulent fluctuations affect the plasma as a whole and the influence of these fluctuations on the features of the processes occurring in plasma still remain unclear. Several reviews in this book are devoted to a description of low-frequency (LF) strong structural (SS) turbulence in a magnetised plasma. Indeed, in recent studies, a mixed 'determinate-chaotic' state was observed in plasma turbulence. Moreover, structural plasma turbulence seems to be peculiar to all plasmas. In such turbulence, the formation of structures is random in character and the role of rare events increases (in [8], such a state was called rigid turbulence). A specific feature of this kind of turbulence is the presence of stochastic structures in the plasma that comprise a considerable fraction of the energy of turbulence. For example, it was shown that LF SS turbulence is characteristic of the plasma of the L-2M stellarator: at the edge, it is produced by ensembles of M H D structures, whereas, in the plasma core, it is produced by ensembles of vortex drift structures [9, 10]. Probability

38

Ν. Ν. Skvortsova et al.

density functions of time samples of plasma parameters in the LF SS turbulence differ from a Gaussian distribution by heavy tails and higher peakedness and are described by stable probability densities. As yet, theoretical modelling of particle diffusion processes in LF SS turbulence, which are described by non-Gaussian statistics, have not been worked out in full measure. This modelling is closely allied to the problem of describing non-Brownian particle motion in mathematical statistics. The key question is to which extent the classical Fokker-PlanckKolmogorov (FPK) equation is applicable to analysing the motion of ensembles of plasma particles in stochastic plasma fields. As is well known, the FPK equation was derived from the stochastic differential equation for only the stochastic term that represents a random Gaussian process [12]. The diffusion coefficient in this approximation has a form familiar to physicians working in the field of plasma physics [18]. The analytical solution to the stochastic differential equation for the stochastic non-Gaussian term of the general form is as yet unknown. Consequently, there is no universe recipe of what corrections should be made in the FPK equations and to which terms and factors they must be introduced. The problem on the solution of the stochastic equation in the physics of plasma turbulence is simplified by a limited class of stable processes for its description. A description of particle diffusion in fractionally stable fields and fractional Brownian diffusion is presented in the reviews in this collection. In plasma physics, the possibility exists of studying the diffusion of ensembles of particles in plasma turbulence by direct measurements. Experimentally, a set of Langmuir probes is used to measure the local fluctuating particle flux across the magnetic field, which is defined as Γ = 4 h~£~, Β where Β is the longitudinal magnetic field, η " is the density fluctuation, E . . is the fluctuation of the poloidal electric fields, and c is the speed of light (this measurement technique will be described in the next section) [14]. Measurements of local fluxes are carried out in the edge plasma of many tokamaks and stellarators. Thus, measurements of a total turbulent particle flux (through the entire closed magnetic surface) were performed in the FT-2 tokamak [48]. It is shown that anomalous transport in the edge plasma is largely dependent of this flux [15, 16], On the other hand, a time series of measured values of the local flux constitutes a time sample of a stochastic diffusion process, and this sample, just as any other time sample of a random process, can be analysed by methods of spectral and probabilistic analysis. The probability density function (PDF) of a local turbulent flux in all measurements differs from Gaussian by heavier tails and higher peakedness [14, 17]. It is well known that the PDF of a process that is a product of two Gaussian processes is non-Gaussian. As regards the LF SS turbulence, the probability densities of two basic quantities for calculating the local flux, namely, the plasma density and the electric field, differ from Gaussian distributions by heavy tails [18], The conclusion that the PDF differs from the Gaussian (normal) distribution is usually deduced from the shape of the histogram constructed from the values of a local fluctuation particle flux at equidistant times. However, such a construction of the PDF is obviously incorrect. It can be easily shown (see the review due to V. Yu. Korolev in this collection) that, even for a discrete temporal sequence constructed for the classical process

Anomalous non-Brownian

diffusion

39

of Brownian motion, this approach gives an excess value of Μ 4 ~ 4, whereas Μ 4 = 3 for the Gaussian law. In this paper, we consider time samples of local particle fluxes in the edge plasma of the L-2M stellarator and in the low-temperature plasma of the TAU-1 linear device. An analysis and modelling of the process of particle diffusion in the L F SS turbulence is based on calculated probabilistic, correlation, and spectral characteristics of local particle fluxes measured experimentally in these devices.

2.

DESCRIPTION OF DEVICES AND TECHNIQUES FOR MEASURING LOCAL FLUXES

We studied times samples of local particle fluxes measured in the edge plasma of the L-2M stellarator and the low-temperature plasma of the TAU-1 model device. The parameters of the devices and the plasma are presented in [11, 19]. The L-2M stellarator has two helical windings of different polarity (/ = 2). The major radius of the torus is R = 100 cm, and the mean minor plasma radius is {r) = 11.5 cm. The plasma was produced and heated by ECR heating at the second harmonic of the electron gyrofrequency. The magnetic field at the axis of the plasma column was Β = 1.3-1.4 Τ, the gyrotron power was Ρ ο = 150-200 kW, and the microwave pulse duration was 10-12 ms. Measurements were performed at the mean plasma density (η) = (1.3-1.8) χ 10 1 3 c m - 3 . The central electron temperature was in the range Te(Q) = 0 . 6 - 1 . 0 keV. The working gas was hydrogen. In the edge plasma at the radius r/ rs — 0.9 (where rs is the separatrix radius), the density was at a level of n(r) = ( 1 - 2 ) χ 10 1 2 c m - 3 and the electron temperature was Te(r) = 3 0 - 4 0 eV. The relative level of density fluctuations was ( 8 n / n ) o u t = 0 . 2 - 0 . 2 5 in the outer plasma region and ( & n / n ) m = 0.1 in the inner plasma region. In the TAU-1 device, a cylindrical argon plasma column 4 c m in diameter and 100 cm in length was produced in a uniform magnetic field with an induction of < 800 G by a steady low-energy electron beam with the energy of Eo — 6 0 - 1 5 0 eV at the argon pressure of ρ — (2-A) χ 10~ 4 torr. The plasma density was maintained at a level of η = (0.9-1.2) χ 1 0 1 0 c m - 3 . The electron temperature was 5 - 7 eV, and the ion temperature was T, % 0 A Te. The main difference between these devices are the magnetic field topology (the L-2M is a toroidal magnetic configuration with a rotational transform of magnetic lines, whereas the TAU-1 field is produced by a straight solenoid) and the plasma temperature (Γ,,(Ο) = 0 . 6 - 1 . 0 keV in L-2M and Te = 5 - 7 eV in TAU-1). On the other hand, as was observed in previous experiments [19], the spectral and statistical characteristics of plasma turbulence in both devices are very similar (broadband continuous frequency spectra, autocorrelation functions with non-vanishing oscillating tails, etc.). In our measurements, the local fluctuating particle flux was defined as Γ = (8n e • 8vr) [14, 17], where 8ne is the plasma density fluctuation, Svr = c8E@/B is the radial velocity fluctuation, 8E@ = (δψ\ — 8φ2)/ΔΘ/is the fluctuation of the poloidal electric field, δψ is the fluctuation of the plasma floating potential, Θ is the poloidal angular coordinate, and r is the mean radius of the magnetic surface. In our experiments, variations in the local fluxes were measured with probe arrays, each consisting of three individual cylindrical probes. The probes measured the plasma density fluctuations δη (in the regime in which the ion saturation current Is is measured, 8IS ~ δη) and floating potential fluctuations δφ. These measurements were then used to calculate the radial fluctuating particle flux. The sampling rate (i.e., the reciprocal of the time interval between successive points) was up to 5 M H z ,

40

Ν. Ν. Skvortsova et al.

and the file length reached 128 kB.

3.

ANALYSIS OF THE PROBABILITY PARAMETERS OF INCREMENTS OF LOCAL FLUXES

Obviously, the local flux is governed by turbulence resulting from a variety of instabilities existing in the plasmas of these devices. The description of the local flux is simplified if its origin is associated with a limited set of plasma instabilities. In experiments carried out in the TAU-1 device, the local flux is governed by the drift-dissipative instability [20]. In the L-2M stellarator, the interchange resistive ballooning instability is dominant in the edge plasma, whereas the drift-dissipative instability prevails in the plasma core [21]. The frequencies of turbulent fluctuation spectra associated with drift-dissipative instability in TAU-1 range from several kilohertz to one hundred of kilohertz ( Δ f / f \ j^u-i % 0.3). The frequency spectrum of turbulence resulting from the resistive ballooning instability in the stellarator is broader and extends over several hundreds of kilohertz ( Δ / / / | L . 2 M % 1). Fig. 1 shows typical signals of the local flux Γ (Fig. la) and its increment Δ Γ = Γ, (tj) — t j _ ι ( t j - ] ) (Fig. lb) for the edge plasma of the L-2M stellarator. It is seen that both signals are intermittent and bursty [22, 23]. Fig. lc shows the autocorrelation functions (ACF) of the flux and its increment for the same signals. It is seen that, within the given time window, the ACF of Γ does not approach the noise level, whereas the ACF of Δ Γ approaches this level in several microseconds. The slow variation of the ACF of the local flux in the edge plasma demonstrates that the flux amplitudes does not represent a homogeneous independent sample, whereas the rapid drop in the A C F of Δ Γ indicates the random and independent character of the increments. Fig. 2 illustrates the time dependence of the local flux (Fig. 2a) and its increments (Fig. 2b) in the TAU-1 plasma and their autocorrelation functions (Fig. 2c). The autocorrelation function of the local flux Γ associated with drift instability in the low-temperature plasma approaches the noise level more rapidly than the local flux in the stellarator does, but the ACF of its increment Δ Γ approaches the noise level even more rapidly. Hence, the amplitudes of the local flux in TAU-1 does not represent a homogeneous independent sample; in contrast, the increments of the local flux constitute a homogeneous independent sample. Hence, when applying the conventional methods of the probability analysis, it is more correct to use the increments of the fluctuating flux instead of the flux amplitudes themselves. We describe the local particle flux determined by plasma fluctuations in the following form: Γ = Γ (tj) = Γ j ( t j ) e y ° t j , where Γ ; °(ί 7 ) is a slowly varying in time variable. Then we can consider a mathematical model of the dynamic characteristics of the process Γ , namely, yef — yjf(tj) depending on the random time t j and constituting the set of random variables y j f . Such a representation corresponds to the shape of the ACF of the local flux increments shown in Figs, l c and 2c. For the LF SS turbulence, even driven by a single plasma instability, the effective growth rate y c f depends on all the linear and nonlinear growth and damping rates that can occur under particular experimental conditions. We may suggest that the physical mechanisms responsible for the random character of the parameter y e f will determine the characteristic features of local fluctuation transport. Fig. 3 shows the PDFs for the processes of a local fluctuating flux and its increments in

Anomalous

non-Brownian

Amplitude, arb. units

41

diffusion

shot

444,

6 3

Ο -3 - 6

56.0 AC Γ

56.5

57.0

57.5

58.0

lime, ms

F i g u r e 1. Time behaviour of (a) the local particle flux Γ . (b) its increments Δ Γ . and (c) the ACFs of Γ and Δ Γ for the L-2M stellarator

L-2M (Figs. 3a, 3c) and TAU-1 (Figs. 3b, 3d). The asymmetry coefficient Μ 3 and the excess M4 of the corresponding PDFs are also shown. The P D F of the time sample of the local flux Γ (Figs. 3a, 3c) differs substantially from Gaussian. Note that the measured PDFs of the density fluctuations Sne and the poloidal electric field SEQ in L-2M are usually Gaussian (heave tails for then are unresolved because of a short length of the time sample). The PDFs for the density fluctuations 8ne and the floating potential δψ in TAU-1 are characterised by self-similar tails (power functions) and a Gaussian shape of the central peak. Even if the data from the time sample of Γ were independent and homogeneous and the variables 8 n e and δΕ® were described by normal probability densities, then the PDF of the local flux Γ obtained by multiplying two Gaussian fluctuating variables, namely, δη e and δΕ0 would differ from Gaussian by heavier tails and a sharper vertex (the proof of this assertion is given in [23]). In our case, where there are no homogeneous and independent data in the time sample of the local flux Γ , such a non-Gaussian PDF of the local flux is difficult to analyse. Hence, it is more interesting to consider the deviation of the increments of the local flux Δ Γ from the Gaussian PDF, because, as was noted above, the time sample for the flux increments consists of statistically homogeneous and independent values. Characteristically, the P D F of the process Δ Γ is symmetric, which indicates a dynamic symmetry of the flux increments. The asymmetry coefficient and excess of the P D F of the process Δ Γ in L-2M are M 3 = 0.9 and M 4 = 9.3; for TAU-1, they are Λ/ 3 = 0.08 and A/ 4 = 7.3.

42

Ν. Ν. Skvortsova et al.

Amplitude, .irh uniis

- 30

Shi« ru>, 30000

1

0

1

0.5

.

1-0

1

1.5

2.0

1. The second method consists of adjusting the experimentally obtained probability densities of local streams to the fractional stable distributions with the use of exhaustive search over their parameters. For some time samples of local streams in L-2M the fit is very good indeed. As an example we present the approximation of two local streams in consecutive discharges when a surface, on which there are measurements, is shifted by 2 mm along the radius (see Fig. 14 and Fig. 15). It was found that for discharge No. 46805 the walk obeys the law Δ ( χ ) oc t °·35/1·1 = 32 Λ , and for discharge No.46804, the law Δ ( χ ) α iO.os/i.2 = ;o.o42 T o c h e c k t h e h y _ pothesis about the goodness-of-fit of the distributions and fractional stable ones, the χ 2 test (see Appendix) was used. For local streams in discharges No. 46805 and No. 46804 for 9 degrees of freedom, they found χ2 = 21.44 and 8.69 respectively. These results do not contradict our hypothesis with significance level 1 — β = 0.01, for which the critical value = 21.7. Hence we can accept the hypothesis that the distribution density

56

Ν. Ν. Skvorlsova et al.

Figure 14. Approximation of values of local stream (dark points) by fractional stable distribution: a = β = 0 . 0 5 , θ = 0 . 0 6 , α = 70, b = 3 (solid line); discharge No. 46804

1.2,

Figure 15. Approximation of values of local stream (dark points) by fractional stable distribution: a β = 0 . 3 5 , θ = 0 . 0 5 , a = 4, b = 0 . 5 (solid line); discharge No.46805

1.1,

=

of local streams in discharge No. 46805 is determined by the fractional stable distribution q(x; 1.1, 0.35,0.05), and in discharge No. 46804, by the distribution q{x\ 1.2,0.05, 0.06). However, the difference in laws of particle walking in these discharges is very big. Both in the first and in the second case the particle undergoes the subdiffusion that is brightly expressed in the second case where the particle is practically not displaced. Displacement of a particle in discharges No. 46804 and No. 46805 for the first 10 time steps is shown in Fig. 16. In the same figure, the displacement of Brownian particle is also shown.

Anomalous

non-Brownian

57

diffusion

3

(0.5

Δ(χ)

(032

(0.0

J

I

I

I

I

L

L 8

' 10

Figure 16. Displacement of a particle under fractional stable laws in two consecutive discharges of stellarator L-2M; the discontinuous line shows the displacement for the Brownian motion ( Δ ( χ ) oc · / ] )

From Figs. 14-15 it is seen that if the central parts of distribution density of streams amplitude are managed to be well-approximated by fractional stable (FS) distribution, then the same FS distribution badly approximates the tails of distribution. It shows the necessity of addition of some other conditions for such a modelling. It should be noticed that it was not possible to describe some streams by FS densities at all. Probably, it is due to not only the impossibility of instant particle jumps between traps, but also the fact that the ensemble of plasma particles cannot be considered as homogeneous. It is known indeed that in plasma the part of particles with certain speeds is practically not displaced in strong magnetic fields with stoppers despite of fluctuations. On the other hand, it is known that cold particles from periphery of plasma move to the centre by ballistic laws, it is observed in experiments on distribution of nitrogen in stellarator TJ-II [39].

7.

CONCLUSIONS

Experimental time samples of particle fluxes in low-frequency strong structural turbulence have been analysed. Correlation and probabilistic properties of local fluxes indicate general features in the motion of particle ensembles in a magnetised turbulent plasma, irrespective of the structure of the magnetic field of the device-a toroidal magnetic configuration with helical transform of magnetic lines in L-2M or a magnetic field of a straight solenoid in TAU-1. Diffusion processes in LF SS turbulence were analysed on a basis of the following characteristics of local fluxes and their increments: time samples, correlation functions, probability density functions. A time sample of a local flux is intermittent and can be represented as a time series of individual flux events. The self-similarity Hurst parameter Η ~ 0.6 indicates a statistical correlation between events in the local flux and the existence of the 'influence' function in the time sample. The measured Hurst parameter did not correspond to indices for particle

58

Ν. Ν. Skvortsova et al.

walk in the Dreisen-Dykhne model of dimension 1-6, where a particle moves mainly along a magnetic field and intersects random transverse convective fluctuation fluxes, with a given width and variable velocity directions. We do not think that, increasing the dimension of the convective-flux model, we achieve a coincidence between experimental data and model estimates. Relatively small spatial sizes of stochastic plasma structures seemingly lead to specific additions to the motion of particles that fall inside, not only correction by stochastic drift with a constant velocity. The probability density function of the local flux always differs from Gaussian heavier tails and a higher peakedness and is described by stable probability densities. So, the asymmetry coefficient differs from zero, and the excess is above 3. Experiments in the L-2M stellarator demonstrated that the asymmetry of the probability density function of the local flux and, respectively, the sign of the asymmetry coefficient can indicate the prevailing direction of the particle flux, providing information on a quality of the chamber wall. In the case of small recycling on the wall, the asymmetry coefficient does not change its sign during a discharge, and the local flux is directed toward the chamber wall. The modelling of the probability density functions by fractionally-stable distributions by the procedure of exhaustive search of parameters showed a wide spread of the parameters. For some local fluxes, our attempts to find a fractionally-stable have failed. When we are dealing with fractionally-stable laws of particle walk, the physical model is easy to interpret if we consider a situation that a walking particle sometimes falls into a trap, stays therein for a time, after which instantaneously jumps from one trap into another. It is probable that a distinction of this model from experimental PDFs is twofold: first, instantaneous jumps of particles between stochastic structures cannot occur in the LF SS turbulence; second, a particle ensemble in a magnetised plasma cannot be considered as being uniform. Based on studies of autocorrelation functions, we have shown that, for a statistical analysis of characteristics of particle fluxes in a plasma to be correct, we must use an equidistant sample of the process, namely, a sample of the increments of the amplitudes of this process. The increments of local fluxes in the two devices are stochastic in nature. The PDFs of the increments are scale mixtures of Gaussians; for the drift instability, the PDF of the increments are described by a Laplacian distribution; for turbulence in the edge plasma, we have to use a more complicated form. In the model device, a steady state of LF SS turbulence of the open system (with a source and sink) is maintained for several hours; it is for this kind of system that we have obtained the probability density function with a maximum entropy—the Laplace distribution. We may state that a Laplacian distribution must be used to correctly calculate rare increments (events whose probability falls outside three sigma) related to LF SS turbulence in steadystate discharges in toroidal devices. The correlation and spectral analysis of values of the local flux of an equidistant sample of the increments of the flux amplitudes allows us to determine the characteristic ('dynamic') time of the process of local flux in the L-2M and TAU-1 plasmas; in both cases this time was one order of magnitude smaller than the correlation time of fluctuations in LF SS turbulence. The diffusion coefficient was estimated assuming the characteristic 'dynamic' time to be a decorrelation time and taking the experimentally measured radial length of LF SS structures for a characteristic spatial length. We may assume that the 'dynamic' time together with the characteristic spatial length of a nonlinear structure determine the plasma transport

Anomalous non-Brownian

59

diffusion

velocity across the magnetic field.

ACKNOWLEDGEMENTS This research was supported in part by the Russian Foundation for Basic Research, grants 03-02-17269, 04-02-16571, by the Program of the President of Russian Federation for support of leading scientific schools, grant 1965.2003.2, and by the Dynasty Foundation.

A.

GOODNESS-OF-FIT TEST χ 2

Let a random variable ξ, either one- or multidimensional, be given. Let X denote a set of values of the random variable ξ. We partition the set X into r pairwise non-overlapping sets X\ Xr, such that Ρ{ξ€

X j } = Pj

>0.

j =

r.

Obviously, p\ Η h pr = P{f e X} = 1. We take Ν independent realisations ξ ι , ξ/ν of the variable ξ and let v,· denote the number of values which fall into X, . The mathematical expectation Ευ, = JVp,·. As a measure of deviation of the 'true values' v, from the theoretical ones N p j it is convenient to introduce the variable (11)

The following theorem is true. THEOREM (K. Pearson's theorem). For any initial variable ξ and any partition X = Χ ι + • • • + Xr (such that all pi > 0) for any χ > 0

χ2 with m degrees of freedom

where the density pm(x) called the density of distribution [38] is expressed by the formula pm( χ ) = [ 2

m / 2

r(m/2)]-]x

m / 2

-

1

e-

x

'

2

The χ2 goodness-of-fit test is applied in statistics to testing hypotheses about the law of distribution of a random variable [41]. For this purpose, a sufficiently high probability β , called confidential probability, is fixed. The probability 1 — β is called the significance level. Theoretically, any value between 0 and 1 can play the part of β. Practically, β means that with probability β our hypothesis is accepted and with probability 1 — β , rejected. It is obvious that β should not be chosen too small. The values β = 0.95; 0.99; 0.999 are commonly used. Let a particular hypothesis be given that the distribution of a random variable ξ is described by some law f ( x ) . As a result of realization of Ν independent experiments, Ν

Ν. Ν. Skvortsova et al.

60

Pm(x)

0

0

4

8

12 x2(m,l-ß) 16

* 20

Figure 17. Density of χ 2 distributions with m = 7 degrees of freedom; The shaded area contains probability equal to 1 — ß\ the value X2(m, 1 — β) is determined from equation (12)

values , . . . , £jv (N is sufficiently large) are obtained. Do these Ν results break our hypothesis? In order to answer this question, we choose an arbitrary number r and partition the set of possible values X of the random variable ξ into r pairwise non-overlapping intervals, X = X\ + h Xr. Proceeding from our hypothesis, it is possible to calculate the probability

we assume that the partition is chosen so that /?,· > 0 for all i. Then using values ξι it is possible to calculate υ,· = η , / Ν , where «, is the number of values ξ] which fall into the interval Xj. This stage in statistics is called ranging. Further, by formula (11) the value of χ2Ν is calculated. If our hypothesis is true, this variable obeys the law χ 2 with (r — 1) degrees of freedom quite well. From the equation (12) the value x2(r — 1, β — 1) is obtained, which corresponds to the chosen significance level 1 — β (Fig. 17). If the calculated χ2Ν < x2(r — \ , β —1), then this result does not contradict our hypothesis; if χ2Ν > y2(r — \,β— 1), then the hypothesis should be rejected because it leads us to a contradiction. In actual practice, they usually use the tables of the distribution χ 2 where the values χ2 = x2(m, 1 — β) are contained which are the roots of the equation pm(x)

dx = 1 — β.

Anomalous non-Brownian

For 1 — β chosen and degrees of f r e e d o m m = corresponding value of coordinate χ

2

diffusion

61

1 — r one should find in the table the

and c o m p a r e it with the value χ2Ν obtained in the ex-

periment. If χ ^ , < χ2, then the result does not contradict our hypothesis and it is accepted. If X2n > X2 then the hypothesis is rejected. REFERENCES 1. J. W. Connor, P. Burraffi, J. G. Cordey, et al., - i t Plasma Phys. Controlled Fusion (1999) 41, 693. 2. A. Yoshizawa, S.-I. Itoh, K. Itoh, N. Yokoi, Plasma Phys. Controlled Fusion (2001) 43, R l R144. 3. U. Stroth, Κ. Itoh, S.-I. Itoh, H. Hartfuss, et al„ Phys. Rev. Lett. (2001) 86, 5910. 4. A. Fujisawa, H. Iguchi, T. Minami, et al., Phys. Rev. Lett. (1999) 82, 2669. 5. V. A. Vershkov, Proc. 30th EPS Conf. on Plasma Physics and Controlled Fusion, St. Petersburg, 2003, h t t p : / / e p s 2 0 0 3 . i o f f e . r u / p u b l i c / p d f s / 6. E. J. Synakowski, Plasma Phys. Controlled Fusion (1998) 40, 581. 7. V. V. Alikaev, et al., Fiz. Plazmy (2000) 26, 979. 8. G. G. Malinetskii and A. B. Potapov, Modern Problems of Nonlinear Dynamics. Editorial URSS, Moscow, 2000. 9. G. M. Batanov, A. E. Petrov, K. A. Sarksyan, et al., JET Ρ Lett. (1998) 67, 634. 10. G. M. Batanov, Κ. M. Likin, K. A. Sarksyan, and M. G. Shats, Sov. J. Plasma Phys. (1993) 19, 628. 11. K. A. Sarksyan, Ν. N. Skvortsova, Ν. K. Kharchev, and B. P. van Miliigen, Plasma Phys. Rep. (1999)25,312. 12. Μ. I. Gihman and Α. V. Skorohod, Stochastic Differential Equations. Springer, Berlin, 1972. 13. V. I. Klyatskii, Stochastic Equations by the Physicist's Eye. Fizmatlit, Moscow, 2001. 14. C. Hidalgo, Plasma Phys. Controlled Fusion (1995) 37, A53. 15. L. A. Esipov, I. E. Sakharov, E. O. Chechik, et al., Zh. Teor. Fiz. (1993) 67, 48. 16. S. V. Shatalin, S. I. Lashkul, L. A. Esipov, and E. O. Vekshina, Proc. 30th EPS Conf. on Plasma Physics and Controlled Fusion, St. Petersburg, 2003, http://eps2003.ioffe.ru/public/pdfs/ 17. G. M. Batanov, Ο. I. Fedyanin, Ν. K. Kharchev, et al., Plasma Phys. Controlled Fusion (1998) 40, 1241. 18. Ν. K. Kharchev, Ν. N. Skvortsova, K. A. Sarksyan, J. Math. Sei. (2001) 106, 2691. 19. G. M. Batanov, V. E. Bening, V. Yu. Korolev, et al. JETP Lett. (2003) 78, 502. 20. A. E. Petrov, K. A. Sarksyan, Ν. N. Skvortsova, and Ν. K. Kharchev, Plasma Phys. Rep. (2001) 27, 56. 21. Ν. N. Skvortsova, G. M. Batanov, V. E. Bening, et al.,7. Plasma Fusion Res. (2002) 5, 594-599. 22. G. M. Batanov, V. E. Bening, V. Yu. Korolev, et al., JETP Lett. (2001) 73, 126. 23. Ν. N. Skvortsova, G. M. Batanov, V. E. Bening, et al. J. Math. Sei. (2002) 112, 4205^1210. 24. I. M. Ryshik and I. S. Gradstein, Tables of Series, Products, and Integrals. Verlag Harri Deutsch, Frankfurt, 1981.

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25. Β. V. Gnedenko and V. Yu. Korolev, Random Summation: Limit Theorems and Applications. CRC, Boca Raton, FL, 1996. 26. V. Μ. Kruglov and V. Yu. Korolev, Limit Theorems for Random Sums. Moscow Univ. Press, 1990. 27. A. S. Kingsep, Introduction to Nonlinear Plasma Physics. Moscow Inst. Physics Techn., Moscow, 1996. 28. F. F. Asadullin, G. M. Batanov, L. V. Kolik, et al., Sov. J. Plasma Phys. (1981) 7, 226. 29. G. M. Batanov, V. E. Bening, V. Yu. Korolev, et al., Plasma Phys. Rep. (2003) 29, 128. 30. Ν. K. Khartchev, G. M. Batanov, A. E. Petrov, et al., Proc. 30th EPS Conf. on Plasma Physics and Controlled Fusion, St. Petersburg, 2003, h t t p : / / e p s 2 0 0 3 . i o f f e . r u / p u b l i c / p d f s / 31. Ν. M. Astafyeva, Usp. Fiz. Nauk (1996) 166, 1145. 32. F. R. Hampel, Ε. M. Ronchetti, P. J. Rousseeuw, and W. A. Stahel, Robust Statistics: The Approach Based on Influence Functions. Wiley, New York, 1986. 33. B. A. Carreras, B. van Miliigen, C. Hidalgo, et al., Phys. Rev. Lett. (1999) 83, 3653. 34. Yu. A. Dreisen and A. M. Dykhne, Zh. Eksp. Teor. Fiz., (1972) 63, 242. 35. O. G. Bakunin, Plasma Phys. Controlled Fusion (2003) 45, 1909. 36. V. M. Zolotarev, One-Dimensional Stable Distributions. AMS,Providence, RI, 1986. 37. V. V. Uchaikin and V. M. Zolotarev, Chance and Stability. Stable Distributions and their Applications. VSP, Utrecht, 1999. 38. L. Schmetterer, Einführung in die mathematische Statistik. Springer, Berlin, 1966. 39. B. P. van Miliigen, Ε. de la Luna, F. L. Tabares, Nucl. Fusion (2002) 42, 787-795. 40. G. M. Zaslavsky, M. Edelman, H. Weiltzner, et al., Plasma Phys. (2000) 7, 3691. 41. I. M. Sobol, A Primer for the Monte Carlo Method. CRC, Boca Raton, FL, 1994.

Stochastic Models of Plasma Turbulence, pp. 6 3 - 8 6 V. Yu. Korolev and Ν. N. Skvortsova (Eds.) © Koninklijke Brill N V 2005.

Low-frequency structural plasma turbulence in stellarators N. N. SKVORTSOVA, G. M. BATANOV, L. V. KOLIK, A. E. PETROV, A. A. PSHENICHNIKOV, K. A. SARKSYAN, N. K. KHARCHEV, YU. V. K H O L N O V Prokhorov Institute of General Physics, Moscow, Russia

J. SANCHEZ, T. ESTRADA, B. van MILLIGEN EUROATOM-CIEMAT, Madrid, Spain

K. O H K U B O , T. S H I M O Z U M A , Y. YOSHIMURA, S. K U B O National Institute of Fusion Research, Toki, Japan

1.

INTRODUCTION

Studies of the LF turbulence in closed magnetic confinement systems have been very popular in recent years. The European Conference on Plasma Physics in 2003 have shown that such experimental studies are carried out in all tokamaks and stellarators existing presently in the world: T-10, LHD, TJ-II, DIII-D, JET, etc. (see reports P-2.56, P-3.121, P-4.5, 0 - 2 . 1 A , P-120 at the Conference [1]). The reason why the studies of this plasma state attract particular attention is that there are many experimental facts pointing directly to the influence of LF turbulence on the macroscopic characteristic of plasmas in closed magnetic confinement systems. For example, the LF turbulence governs anomalous transport in the edge plasma [2, 3], the changes in its parameters correlate with observations of the internal and external transport barriers in plasma [4, 5], the statistical parameters of the turbulent particle flux in the edge plasma correlate with the gas influx from the chamber wall [6]. Another problem is the estimation of the frequency of occurrence of large-amplitude contingent events (of 'catastrophic type') in devices with long discharges and in devices with steady-state discharges [7], An important factor is, in our opinion, the progress in computerisation of experiment, which makes it possible to accumulate large arrays of statistical data (temporal and spatial samples) and to calculate the steady-state and transient characteristics (spectral, correlation, probabilistic, dimensional and other) of LF plasma turbulence. The L-2M stellarator (Prokhorov Institute of General Physics, Moscow) is one of the first toroidal devices, in which 25 years it has been proposed to study LF turbulence in the steady-state phase of discharge with the methods of numerical analysis of data. It was 1989 that we presented a report [8] on the general form of characteristics of two types of LF turbulence initiated by instabilities of different nature in the L-2M stellarator and the TAU-1 linear model device. In the course of further studies of LF plasma turbulence, based

64

Ν. Ν. Skvortsova et al.

Table 1. Device Major radius R, cm Average minor radius r, cm Magnetic field Β, Τ Input microwave power PQ, kW Average density (η), 1 0 1 3 c m - 3 Central electron temperature Te(0), eV Relative fluctuation level in the edge plasma Steady-state phase of a discharge, ms

(8n/n\dge

L-2M 100 11.5 1.3-1.4 150-200 1.0-1.3 400-800 0.2-0.25 10-12

LHD 800 80 < 3 600 — 1.0 > 1000 > 1000

TJ-11 150 10-22 < 1.2 200^100 < 1.0 500-800 0.2-0.25 200-300

on characteristics of turbulence in TAU-1, we recognised specific features of the state that in 1994 was proposed to term LF structural turbulence [9]. This is a determinate-chaotic state in which ensembles of chaotic structures exist in turbulence. The stochastic structures determine a number of spectral, correlation, and probabilistic parameters of this turbulence. The structural ion-acoustic plasma turbulence is described in the first review of this book. The present paper describes the features of LF structural plasma turbulence in the L-2M, TJ-II, and LHD stellarators.

2.

2.1.

DESCRIPTION OF DEVICES AND TURBULENCE INVESTIGATION TECHNIQUES Experimental

devices

Studies of characteristics of the low-frequency turbulent plasma fluctuations were carried out in three toroidal plasma confinement systems: the L-2M, TJ-II, and LHD stellarators. The L-2M device is an / —2 stellarator, its parameters are described in detail in [10]. The device is a modified stellarator L-2 (modification was finished in 1995) and is situated at the Prokhorov Institute of General Physics (Moscow). The values of the main parameters are given in the first column of Table 1. In the edge plasma at the radius r / a = 0.9 (a is the separatrix radius), the density is η ~ 1-2 χ 10 12 c m - 3 , and the electron temperature is close to Te ~ 3 0 ^ 0 eV. The plasma is produced and heated at the second harmonic of the electron gyrofrequency by a single 75 GHz gyrotron. The LHD stellarator is the most advanced stellarator superconducting heliac with a divertor [11], The heliac was constructed in 1998 in the National Institute of Fusion Science (NIFS, Toki, Japan). The plasma parameters of LHD are listed in the second column of Table 1. The plasma is produced and heated under electron cyclotron resonance (ECR) conditions at the fundamental and second harmonics of the electron gyrofrequency by several 168 GHz and 84 GHz gyrotrons. The TJ-II device is an I = 4 stellarator [12]. This device was constructed in 1998 in the Plasma Physics Department of the CIEMAT (Madrid, Spain). The values of the main parameters are given in the third column of Table 1. The magnetic field structure is produced by 32 coils creating a toroidal field, two coils of a vertical field, and a central conductor. In the experiment, the plasma was produced and heated by a single 53.2 GHz gyrotron at the second harmonic of the electron gyrofrequency. In complex configurations of magnetic fields typical of stellarators, the plasma has a

Structural

plasma

turbulence

65

Figure 1. Magnetic coil system of TJ-II. The plasma column inside is shown by gray colour.

variable cross section along the torus. A s an illustration, Fig. 1 shows the magnetic system of TJ-II enclosing a p l a s m a c o l u m n with a variable cross section and the average radius varying f r o m 0.1 to 0 . 2 2 m. In spite of different magnetic configurations in the L - 2 M , TJ-II. and L H D , all these stellarators have flattened radial density profiles and peaked electron temperature profiles. Fig. 2 c o m p a r e s the radial p l a s m a density and temperature profiles f o r the L - 2 M , TJ-II, and L H D stellarators. It is well k n o w n that a toroidal magnetic c o n f i n e m e n t system is an open p l a s m a system in the t h e r m o d y n a m i c sense. All diversity of steady-state p l a s m a fluctuations, which are studied u n d e r conditions of d y n a m i c equilibrium, has a source and sink of energy, and also there may be p l a s m a regions w h e r e the energy is redistributed b e t w e e n different turbulent states. T h e r e f o r e , in the e x p e r i m e n t s on studying p l a s m a fluctuations, it is important to have a possibility to c h a n g e these c o m p o n e n t s by varying certain p a r a m e t e r s of the device itself, namely, the heating power, the magnetic field configuration (the rotational t r a n s f o r m angle), and the neutral influx f r o m the c h a m b e r wall. All devices meet these requirements. In all stellarators, the heating p o w e r is varied as the input microwave p o w e r is varied. Various magnetic field configurations can be produced in the devices. T h u s , five different configurations can be produced in L - 2 M with different values of the rotational t r a n s f o r m at the magnetic axis: 0.175, 0 . 1 1 9 , 0.082, 0 . 0 6 4 , and 0.043. A variety of configurations can by produced by gradually varying the rotational t r a n s f o r m in TJ-II. A c h a n g e in properties of the first wall (the sink of energy) in the devices is achieved by various m e t h o d s of conditioning the wall (boronisation or the insertion of a limiter). In addition, f o r describing the turbulence in any m e d i u m , it is very important to maintain and study the steady state, because only in this case one might expect that ergodic conditions would be fulfilled and the results obtained would be statistically consistent. For this reason, it is significant that

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(a)

Shot no. 53910

Figure 2. Radial density and temperature profiles in (a) L-2M, (b) TJ-II, and (c) LHD in scattering experiments; experimental data are shown by points and solid lines, theoretical profiles are shown by dashed lines

measurements were made in the steady-state phase, which lasted about 10 ms in L-2M, 300 ms in TJ-II, and 1 s in LHD.

2.2.

Turbulence investigation

techniques

Plasma density fluctuations in the heating region (L-2M and LHD) were measured from scattering of the heating gyrotron radiation [13,14]. High-temperature plasma density fluctuations in regions displaced in the radial and toroidal directions from the heating region were measured from 2 mm microwave scattering (L-2M and TJ-II) [15, 16]. Density and potential fluctuations in the low-temperature plasma at the edge of the plasma column in L-2M were measured with the help of probes [17]. Magnetic field fluctuations outside the plasma column were measured with the help of pickup magnetic coils (L-2M and TJ-II) [18]. In Table 2, listed all diagnostics are listed for measuring fluctuations, along with the plasma regions where measurements were performed. A plasma column in stellarators is conventionally divided into two regions: a low-temperature edge plasma and a hightemperature core plasma. We cannot indicate distinct boundaries between these regions, but only can say that the edge plasma occupies about 1/5-1/4 of the plasma radius. Studies of the edge plasma are currently performed in all toroidal devices; however, studies in the central plasma region, in particular, where microwave heating produced are still rare. In the L-2M stellarator, we managed to measure fluctuations throughout the volume of the plasma column in several poloidal cross sections of the torus. Results of measurement in all of the devices were digital files of ations of plasma variables in the form of time samples. These time several hundred thousand points. All methods operated with a single numerical processing of time samples. To analyse the characteristics

amplitudes of fluctusamples consisted of program package for of fluctuation signals

Structural plasma

turbulence

67

Table 2. Device Measurements in the edge plasma Measurements near the mid-radius of the plasma column Measurements near the centre of the plasma column Measurements outside the plasma

L-2M Probes 2 m m scattering

LHD

TJ-11 2 m m scattering

Gyrotron radiation scattering Magnetic probes

Magnetic probes

obtained in the experiments in all of the devices, we used a common program package created for analysing random time sequences [19-23]. This program package included a multidimensional Fourier spectral analysis, correlation analysis, spectral and coherence wavelet analysis, construction of histograms, computation of moments of random values, computation of the Hurst parameter (R/S analysis), auxiliary programs for smoothing, filtering, and averaging of signals.

3.

RESULTS OF FLUCTUATION STUDIES

3.1. Studies of fluctuations in L-2M plasma LF fluctuations are naturally present throughout the plasma volume in the L-2M stellarator, from the centre to the edge. In this paper, we only consider parameters that are common to fluctuations, irrespective of their location in the plasma column. Fig. 3 shows the time behaviour of (a) density fluctuations in the high-temperature plasma in the heating region; (b) fluctuations of the potential of the low-temperature plasma at the edge of the plasma column; and (c) fluctuations of the magnetic field outside the plasma (frequencies below 1 kHz are cut off). The signals consist of bursts of different length and pauses between them. The frequencies during bursts and their rise and fall times vary in time. (At present, the term 'bursttype' is commonly used to describe such signals.) The amplitudes of fluctuations are large enough in comparison with the average values of these parameters. A minimum level of fluctuations was measured for the magnetic field and amounted several hundredth percent ( Β / Β ~ 5 χ 10~5). The relative value of density fluctuations inside the plasma column was considerably greater and varied from 20-30% (η/ η 0.2—0.3) at the edge to 10% in the centre (η/η ~ 0.1). Values of the floating potential and electric field fluctuations in the edge plasma reached 10-20% ( φ / φ ~ 0.1-0.2). Autocorrelation functions (ACFs) of time samples of LF turbulent fluctuations in L-2M consist of a finite-width peak (differing from a < k\\VTe, the variance relation is = ωΐ

where υ, = {Te/Μ)χ!2,

(1 +

andr~ l =

ps = vs/Qe,

(ii) For frequencies in the interval A: n

k±vr e psk2)Qer„' \d\nn/dr\.

< ω < k\\ vje, the variance relation is k20

0)2 =

k±

r2 .

For the second frequency interval, the instability condition is d In Te/d Inn > 2. This inequality is easily satisfied in the core plasma up to radius r / a < 0.7, because the temperature profiles are narrower than the density profiles in L-2M, LHD, and TJ-II (Fig. 19). Thus, the drift-dissipative instability can result in low-frequency

Structural plasma

turbulence

83

plasma oscillations in the observation regions in all three of the devices; however, the characteristic frequencies, which are determined by the plasma parameters of each device, will be different. In the observation region of TJ-II (r/a % 0.4-0.6), one could expect the excitation of oscillations in the frequency range 10-100 kHz. In the gyrotron-scattering region in the L-2M stellarator (r/a % 0.3), oscillations may be excited in the frequency range 10-30 kHz. In the central region of LHD (r/a ss 0.2), one could expect oscillations with frequencies of about 1 kHz. The instability driven by trapped electrons can appear only in the region where ^ ^jf- > 0. Plasma oscillations can arise at a distance r/a > 0.25 from the centre, because the instability driven by trapped electrons cannot develop if the plasma density profile has a minimum in the centre (Fig. 19). The oscillations are excited in the frequency range ω ι < v e / e h , where ve is the electron-ion collision frequency and ε/, is a small parameter defined as a function of the helical components of the magnetic field (for tokamaks, this parameter is equal to ε/, % 0.2, and for stellarators, it depends on the minor radius and varies in the range ε/, % 0.1-0.3). For all three devices, these frequencies do not exceed 30 kHz. In some configuration of the magnetic field, one might expect the development of M H D instability in the core plasma [36]. In some magnetic configurations in TJ-II in the presence of 3 / 2 and 5 / 4 resonances, we observed a rapid modification of the spectra of scattered signals and their correlation with the spectra of magnetic probe signals. (Narrow-band spectra of magnetic probe signals in such magnetic configurations were also observed in previous TJ-II experiments [37, 38].) In the edge plasma of L-2M, the buildup of turbulent fluctuations may be attributed to the resistive-ballooning M H D instability. Possible instability regions and characteristic of the above-mentioned plasma instabilities in our devices are discussed in [14, 17]. The point is that different instabilities initiate the same turbulent state, namely, LF SS turbulent state. (3) The specific feature of LF SS turbulence is the existence of stochastic plasma structures. The nonlinear structures comprise a considerable fraction of the energy of turbulence —from 10% to 30% in different plasma regions. In LF SS turbulence in L-2M, we distinguished the following stochastic plasma structures: extended M H D radial and poloidal structures in the edge plasma and drift vortices near the mid-radius of the plasma column. Stochastic structures in the central regions of L-2M, LHD, and TJ-II were not identified. It should be noted that the structure models known as 'blob' [39] and 'zonal flow' (by a zonal flow is meant a solitary stable structure whose radial dimension is much less than its poloidal and toroidal dimensions) [40] are appropriate for describing solitary long-lived structures such as drift vortices or hydrodynamic vortices on shallow water [32] rather than for describing a whole set of ensembles of stochastic structures of different spatial and temporal dimensions, which are observed experimentally in structural turbulence. The high (up to 50% for frequencies below 150 kHz) wavelet coherence coefficient was observed between time samples of amplitudes of density fluctuations in the central region and at the edge of the plasma column in L-2M. Thus, turbulent fluctuations in LF SS turbulence in the plasmas of the toroidal devices turn out to be interrelated probably due to ensembles of stochastic plasma structures. The steady state of plasma turbulence

of the same nature is observed in the plasmas

of

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three toroidal devices and is defined as the low-frequency strong structural plasma turbulence. As one of the possible scenarios of the development of plasma turbulence, we can propose a scenario similar to the formation of the steady-state low-frequency structural turbulence. Plasma oscillations are excited due to linear instabilities; after these oscillations reach a certain threshold amplitude, the processes pass into the nonlinear stage and nonlinear stochastic structures form. Then, nonlinear interaction arises among the structures of the same type (coalescence and decay) or among the structures of different types. The characteristic times of these processes are longer than the inverse linear growth rates. When describing this interaction, it is necessary to take into account all of the accompanying processes: the drift of the structures from the region where they were formed, linear decay, the suppression of instability because of the change in the local parameters (e.g., local heating), etc. With a continuous inflow of energy into an open system, a universal state with strong structural turbulence is established in it, irrespective of the type of linear instability. The essential feature of the LF SS turbulence is that the PDFs differ from the normal distribution by heavier tails and higher peakedness. Stable (non-Gaussian) PDFs were measured for plasma density fluctuations in the central regions of all three toroidal devices. Taking into account the above said, we can formulate several problems for future investigations. Do the turbulent plasma states under study belong to the systems with dynamic chaos? Can the transitions in such systems be controlled with the help of weak regular waves (the so-called regime of stochastic resonance [41])? For example, in [42], a weak signal was successfully used to affect the transition from the state with a broad drift spectrum to the single-mode state with a suppressed noise spectrum. In this case, it turned out that the controlling signal should be appropriately phased. Is the consideration of non-Brownian particle motion of much importance to the study of LF SS turbulence [34, 35, 43^15]? How do rare events in LF SS turbulence affect the macroscopic plasma characteristics in steady-state discharges in toroidal devices? In the transport theory, the question of non-Brownian motion of particles in non-Gaussian fields is of particular interest and is still far from clear, along with the necessity of taking rare events into consideration when estimating the diffusion in the steady-state plasma. Several reviews in this book are devoted to the non-Brownian particle diffusion and to attempts to estimate the coefficient of this diffusion.

ACKNOWLEDGEMENTS This research was supported in part by the Russian Foundation for Basic Research, grants 04-02-16571 and 03-02-17269, the LIME Program (Japan), and the Program of the President of the Russian Federation for support of leading scientific schools, grant 1965.2003.2. REFERENCES 1. Proc. 30th EPS Conf. on Plasma Physics and Controlled

Fusion, St. Petersburg,

2003,

http://eps2003.ioffe.ru/public/pdfs/ 2. J. W. Connor, P. Burraffi, J. G. Cordey, et al., Plasma Phys. Controlled Fusion (1999) 41, 639. 3. A. Yoshozawa, S.-I. Itoh, K. Itoh, and N. Yokoi, Plasma Phys. Controlled Fusion (2001) 43, R l .

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turbulence

4. U. Stroth, Κ. Itoh, S.-I. Itoh, et al., Phys. Rev. Lett. (2001) 86, 5910. 5. A. Fujisawa, H. Iguchi, T. Minami, et al., Phys. Rev. Lett. (1999) 82, 2669. 6. Ν. K. Khartchev, G.M. Batanov, K.A. Sarksian, et al. Proc. 30th EPS, 2003, pp. 3-176. 7. B. Saotic, Plasma Phys. Controlled Fusion (2002) 44, Β11. 8. G. M. Batanov, K. A. Sarksian, Α. V. Sapozhnikov, et al., Proc. IVInt. Conf. on Nonlinear and Turbulent Processes in Physics, 1. Kiev, 1989, p. 231. 9. V. V. Abrakov, A. E. Petrov, K. A. Sarksyan, and Ν. N. Skvortsova, Plasma Phys. Rep. (1994) 20, 959. 10. V. V. Abrakov, D. K. Akulina, E. D. Andryukhina, et al., Nucl. Fusion (1997) 37, 233. 11. O. Motojima, H. Yamada, A. (http://www.nifs.ac.jp/)

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12. C. Alejaldre, J. Alonco, I. Almoguera, et al, Plasma Phys. Controlled Fusion (1999) 41, Β1091 (http://www.ciemat.es/eng/instalacion/tj-II.html) 13. Κ. A. Sarksyan and A. E. Petrov, Microwave Scattering Technique and Methods of Determining the Parameters of Oscillation Processes in Plasma. http://www.plasmaiofan.ru/edu/lec4.htm 14. G. M. Batanov, A. E. Petrov, K. A. Sarksyan, et al., Plasma Phys. Rep. (2003) 29, 363. 15. G. M. Batanov, Κ. M. Likin, K. A. Sarksyan, and M. G. Shats, Sov. J. Plasma Phys. (1993) 19, 628.

16. Ν. K. Kharchev, G. M. Batanov, L. V. Kolik, et al., J. Math. Sei. (2002) 112, 3846. 17. G. M. Batanov, Ο. I. Fedyanin, Ν. K. Kharchev, et al., Plasma Phys. Controlled Fusion (1998) 40, 1241. 18. R. H. Huddlestone and L. L. Stanley, Eds., Plasma Diagnostic Techniques. Academic Press. New York, 1965. 19. B. P. van Miliigen, Ε. Sanchez, Τ. Estrada, et al., Phys. Plasmas (1995) 2, 3017. 20. K. A. Sarksyan, Ν. N. Skvortsova, Ν. K. Kharchev, and B. P. van Miliigen, Plasma Phys. Rep. (1999) 25,312. 21. B. A. Carreras, B. van Miliigen, Μ. A. Pedrosa, et al., Phys. Rev. Lett. (1998) 80, 4438. 22. Ν. N. Skvortsova, K. A. Sarksyan, and Ν. K. Kharchev, JETP Lett. (1999) 70, 201. 23. Ν. K. Kharchev, Ν. N. Skvortsova, and K. A. Sarksyan, J. Math. Sei. (2001) 106, 2691. 24. D. K. Akulina, G. M. Batanov, A. E. Petrov, L. V. Kolik, and K. A. Sarksyan, JETP Lett. (1999) 69, 407. 25. Ν. N. Skvortsova, G. M. Batanov, L.V. Kolik, et al, J. Plasma Fusion and Research (2002) 5, 328. 26. G. M. Batanov, L. V. Kolik, A. E. Petrov, et al., JETP Lett. (1998) 68, 585. 27. G. M. Batanov, K. A. Sarksyan, Ν. K. Kharchev, et al., JETP Lett. (2000) 72, 174. 28. G. M. Batanov, A. E. Petrov, K. A. Sarksyan, and Ν. N. Skvortsova, JETP Lett. (1998) 67. 662. 29. A. F. Aleksandrov, L. S. Bogdankevich, and A. A. Rukhadze, Fundamentals dynamics. Vysshaya Shkola, Moscow, 1978.

of Plasma Electro-

30. Β. B. Kadomtsev and O. P. Pogutse, Dokl. Akad. Nauk SSSR (1969) 18, 553. 31. L. M. Kovrizhnykh, and S. V. Shchepetov, Sov. J. Plasma Phys. (1981) 7, 229.

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32. Μ. V. Nezlin and Ε. Ν. Snezhkin, Rossby Vortices and Spiral Structures.

Nauka, Moscow, 1990.

33. A. E. Petrov, K. A. Sarksyan, Ν. N. Skvortsova, and Ν. K. Kharchev, Plasma Phys. Rep. (2001) 27, 56. 34. G. M. Batanov, V. E. Bening, V. Yu. Korolev, et. al., JETP Lett. (2001) 73, 144. 35. G. M. Batanov, V. E. Bening, V. Yu. Korolev, et al., Plasma Phys. Rep. (2002) 28, 128. 36. Yu Chanhuan, D. L. Browe, Zhao Shujin, R. V. Bravenec, et al., Nucl. Fusion (1992) 32, 15. 37. I. Garsia-Cortes, E. de la Luna, F. Castejon, et al., Nucl. Fusion (2000) 40, 1867. 38. Ν. N. Skvortsova, G. M. Batanov, A. E. Petrov, et al., Proc. (http://eps2003.ioffe.ru/public/pdfs/). 39. S. I. Krashenninikov, Proc. 30th (http://eps2003.ioffe.ru/public/pdfs/)

EPS,

30th EPS,

2003, p. 120

2003,

pp. 4-20

40. M. A. Malkov, P.H. Dyamind, and M.N. Rosenbluthl, Phys. Plasmas (2001) 8, 5073. 41. V. S. Anishchenko, Τ. E. Vladislavov, and V. V. Astakhov, Nonlinear Stochastic Systems. Saratov Univ. Press, Saratov, 1999.

Dynamics

of Chaotic

42. A. E. Petrov, K. A. Sarksyan, Ν. N. Skvortsova, et al., Plasma Phys. Rep. (1997) 23, 606. 43. G. M. Zaslavsky, M. Edelman, H. Weiltzner, et al., Plasma Phys. (2000) 7, 3691. 44. O. G. Bakunin, Plasma Phys. Controlled Fusion (2003) 45, 1909. 45. B. P. van Miliigen, Ε. de la Luna, F. L. Tabares, et al., Nucl. Fusion (2002) 42, 787.

and

Stochastic Models of Plasma Turbulence, pp. 87-113 V. Yu. Korolev and Ν. N. Skvortsova (Eds.) © Koninklijke Brill NV 2005.

New possibilities for the mathematical modelling of turbulent transport processes in plasma N . N . S K V O R T S O V A , G . M . B A T A N O V , A . E. P E T R O V , A . A . P S H E N I C H N I K O V , K. A . S A R K S Y A N , N . K. K H A R C H E V Prokhorov Institute of General Physics, Moscow, Russia V. Y U . K O R O L E V τ . A . M A R A V I N A Moscow State University, Moscow, Russia J. S A N C H E Z EUROATOM-CIEMAT, Madrid, Spain S. K U B O National Institute of Fusion Research, Toki, Japan

A b s t r a c t — A new m a t h e m a t i c a l m o d e l is proposed f o r the probability distributions of the characteristics of the processes observed in turbulent plasmas. T h e m o d e l is based on f o r m a l theoretical considerations related to probabilistic limit t h e o r e m s for a n o n - h o m o g e n e o u s r a n d o m walk and has the f o r m of a finite m i x t u r e of Gaussian distributions. T h e reliability of the m o d e l is c o n f i r m e d by the results of a statistical analysis of the experimental data on density fluctuations in high-temperature p l a s m a s of the L - 2 M , L H D , and TJ-II stellarators and the local fluctuating flux in the TAU-1 linear device and in the e d g e p l a s m a of the L - 2 M stellarator with the use of the e s t i m a t i o n - m a x i m i s a t i o n algorithm. It is shown that l o w - f r e q u e n c y structural t u r b u l e n c e in a m a g n e t i s e d plasma is related to n o n - B r o w n i a n transport, w h i c h is determined by the characteristic temporal and spatial scales of the e n s e m b l e s of stochastic p l a s m a structures. M e c h a n i s m s that can be responsible f o r the r a n d o m nature of time s a m p l e s of the local turbulent flux in TAU-1 are indicated. A new physical concept of the intermittence of p l a s m a turbulent pulsations is developed on the basis of the statistical separation of mixtures in terms of the m o d e l proposed. T h e intermittence of plasma pulsations is shown to be associated with the generation of p l a s m a structures (solitons and vortices) and their nonlinear interaction, as well as with their d a m p i n g and drift.

1.

INTRODUCTION. STRONG STRUCTURAL PLASMA

TURBULENCE

O n e o f the f u n d a m e n t a l p r o b l e m s in c r e a t i n g a c o n t r o l l e d f u s i o n r e a c t o r is p l a s m a instability r e s u l t i n g in a n o m a l o u s p a r t i c l e a n d e n e r g y l o s s e s .

F o r this r e a s o n , m u c h e f f o r t h a s b e e n

d e v o t e d t o s t u d y i n g p l a s m a i n s t a b i l i t i e s a n d s e a r c h i n g f o r m e t h o d s o f their s u p p r e s s i o n . L o w - f r e q u e n c y ( L F ) p l a s m a t u r b u l e n c e c a u s e d b y t h e s e i n s t a b i l i t i e s is an i m p o r t a n t l o s s channel.

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Studies of LF turbulence in closed magnetic confinement systems have become very popular in recent years. In [1] it is showed that such experimental studies are presently being carried out in all of the existing tokamaks and stellarators, such as T-10, LHD, TJ-II, JET, CASTOR, and FT-2 (see, e.g., Reports P-2.56, P-3.121, P-4.5, 0-2.1A, and P-179). The same trend was also noted in [2], where studies on LF turbulence in toroidal devices were supplemented with reports on experiments in linear devices and gas discharges. It is not surprising that almost one-half of the papers presented in [3] were devoted to various turbulent phenomena of this kind: blobs, streamers, zonal flows, solitons, vortices, etc. (see, e.g., Reports PL-1,1-01,1-02,1-04,1-06, 0-6, and 0-7). It should be noted that not all of these theoretical concepts have been confirmed experimentally. The studies of this plasma state attract particular attention because there are many experimental facts pointing to the influence of LF turbulence on the macroscopic plasma characteristic in closed magnetic confinement systems. For example, LF turbulence governs anomalous transport in the edge plasma [4, 5], the changes in its parameters correlate with observations of internal and external transport barriers in plasma [6-8], and the statistical parameters of the turbulent particle flux in the edge plasma correlate with the gas influx from the chamber wall [9]. Note that the question of the frequency at which large-amplitude random events of 'catastrophic' type occur in long-term and steady-state discharges still remains open [10]. An important factor is that the computerisation of experiment has made it possible to accumulate large arrays of statistical data (temporal and spatial samples) and then calculate steady-state and transient characteristics (spectral, correlational, probabilistic, dimensional, and other) of LF plasma turbulence. This stimulated experimental studies on LF turbulence and provided a great body of new information requited for its theoretical description and analysis. In many toroidal and linear devices, LF turbulence in the plasma core and at the plasma periphery has the form of strong structural (SS) turbulence [11-13]. The term strong structural turbulence means that there are ensembles of stochastic, nonlinearly interacting plasma structures against the background of well-developed steady-state plasma turbulence. SS plasma turbulence was first discovered in the TAU-1 linear device with a longitudinal magnetic field [14] and then was observed in the L-2M, LHD, and TJ-II stellarators [15]. Measurements of the fluctuation parameters in most of the toroidal devices indicate that LF SS turbulence may be present in these devices. Let us consider the characteristic features of this phenomenon using as an example the L-2M stellarator, in which turbulence has been studied over the entire plasma volume [9]. Time samples of any fluctuating plasma parameter are bursty in character. Such time samples are more adequately described by finite-duration oscillating wavelets rapidly decaying in time. The observed wavelet spectra contain quasi-harmonics, and their correlation functions have oscillating tails. The LF SS turbulence has been observed over the entire plasma volume in L-2M, although, in different plasma regions, different instabilities are responsible for its excitation: drift-dissipative instability, MHD resistive ballooning instability [9], and trapped-electron-driven instability [3]. Nonlinear structures comprise a considerable fraction (from 10 to 30% in different plasma regions) of the turbulence energy. Turbulent fluctuations in LF SS turbulence are correlated over the entire plasma volume via ensembles of stochastic plasma structures. The main characteristic feature of LF SS turbulence is that the probability density functions (PDFs) of the fluctuating parameters differ from a normal distribution: the observed PDFs are leptokurtic and are characterised by heavier tails. Non-Gaussian PDFs of stochastic plasma processes point to a non-Brownian

Turbulent transport processes in plasma

89

(anomalous) motion of particles in stochastic fields [16]. In a plasma with LF SS turbulence, the role of rare events with magnitudes far exceeding the average values substantially increases and needs to be estimated. Thus, the fundamental problem of describing the nature (state) of LF SS turbulence leads us to the applied problem of describing anomalous plasma transport in closed magnetic configurations. So far, the methods for modelling particle diffusion in a plasma with LF SS turbulence described by a non-Gaussian statistics have been poorly developed. Such a modelling is closely related to the problem of describing non-Brownian particle motion in the probability theory. The key question is whether the classical Fokker-Planck-Kolmogorov (FPK) equation can be used to analyse the motion of ensembles of plasma particles in stochastic plasma fields. It is well known [17] that the FPK equation was derived from the stochastic differential equation only for the stochastic term representing a random Gaussian process. The diffusion coefficient in this approximation has a form familiar to plasma physicists [18]. A general analytic solution to the stochastic differential equation for the stochastic non-Gaussian term is still lacking. Therefore, no universe prescription has been devised for making corrections in the FPK equation and determining the terms and factors into which they are to be introduced. In this paper, we propose a new mathematical model for probability distributions of the characteristics of the processes observed in turbulent plasmas. The model is based on formal theoretical considerations related to probabilistic limit theorems for a non-homogeneous random walk and has the form of a finite mixture of Gaussian distributions. The reliability of the model is confirmed by the results of a statistical analysis of the experimental data on plasma density fluctuations in high-temperature plasmas of the L-2M, LHD, and TJ-II stellarators and the local fluctuating flux in the TAU-1 linear device and in the edge plasma of the L-2M stellarator with the use of the estimation-maximisation (EM) algorithm. It is shown that LF SS turbulence is related to anomalous transport in a magnetised plasma. A new physical concept of the intermittence of plasma turbulent pulsations is developed on the basis of the statistical separation of mixtures in terms of the model proposed. The intermittence of plasma pulsations is shown to be associated with the generation of plasma structures and their nonlinear interaction, as well as with their damping and drift.

2.

MATHEMATICAL MODEL

2.1. Non-homogeneous random walk with a continuous time There have been many attempts to explain the observed leptokurtic PDFs. The most progress in solving this problem has been achieved with the use of the limit theorems for a homogeneous random walk with a discrete or a continuous time. According to these theorems, the so-called stable or fractionally stable PDFs characterised by power-law tails can be used as an alternative to a Gaussian distribution [19-21]. As applied to plasma turbulence, such models were considered in [22-25]. However, stable or fractionally stable models sometimes fail to provide an adequate description of plasma turbulence. First, a basic assumption underlying these models is the absence of the second moment (variance) of the distribution of random particle jumps and/or the absence of the first moment (mathematical expectation) of the distribution of time intervals between the jumps. For this assumption to be valid, it is necessary that, at the least, these random variables with positive probabil-

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ities could take arbitrary large values. It is evident that this assumption fails to be valid in practice, because the recorded processes are always limited in space and time, so the above random variables possess all the moments. Second, a statistical analysis of experimental data shows that, although the tails of the observed PDFs are heavier than those described by a Gaussian law, they are much lighter than the tails of stable non-Gaussian PDFs, which decrease as 0(x~a) with 0 < a < 2 a s x ^ - o o [16]. In contrast to traditional models, our mathematical models are based on the limit theorems for a non-homogeneous random walk with a continuous time. This walk differs from a homogeneous walk in that the distributions of the random time intervals between successive jumps of a wandering particle are, in general, different. The assumption of the non-homogeneity (different distributions of the time intervals between successive jumps) of a random walk is consistent with a concept that the rate of the coordinate increment of a particle that undergoes Brownian motion in a turbulent medium is essentially nonhomogeneous. Let N(t) be the number of jumps of a walking particle over the time interval [0, ?], where t > 0. The instants of jumps form a chaotic point random process on the time axis. However, by virtue of the above concept, this chaotic random process is non-homogeneous. As is well known [26], the most reasonable stochastic models of non-homogeneous chaotic point processes are doubly stochastic Poisson processes, which are also referred to as Cox processes. They are defined as follows. Let N\(t), t > 0, be a homogeneous Poisson process with a unit intensity, and A(t), t > 0, be a stochastic process independent of Ν\ (t) with the following properties: Λ(0) = 0 and P(A (/)) < oo) = 1 for any time t > 0 and the trajectories Λ (/) do not decrease and are continuous from the right. A doubly stochastic Poisson process (a Cox process) is defined as a superposition of Ν ι (t) and A(t): N(t) = Ni(A(t)),

t> 0.

In this case, we will say that the Cox process N(t) is controlled by the process Λ ( t ) . In particular, if the process A (t) admits the representation

t > 0, where 'k(t) is a positive stochastic process with integrable trajectories, then λ](ί) can be interpreted as an instantaneous stochastic intensity of the process N(t). For this reason, the process A(t) controlling the Cox process N(t) is called the accumulated intensity of the process N(t). The properties of Cox processes are described in detail in [27, 28], The objective of this section is to formulate the problem of the modelling of nonhomogeneous chaotic flows of events in a turbulent plasma with the use of generalised Cox processes and to demonstrate that the deviations of the observed distributions of the processes from normal can be attributed to substantial variations in the intensity of nonhomogeneous chaotic flows of events described by Cox processes. Let X\, Χ ι , • • • be identically distributed random variables. We assume that for any

Turbulent transport processes in plasma

t > 0 the random variables N(t),

X\, Χι

91

are independent of one another. The process

N(t)

S{t) = Y ^ X j ,

t > o,

(1)

j= 1 describes the coordinate of a particle that undergoes a non-homogeneous random walk at a time t; this process will further be referred to as a generalised Cox process (for definiteness, we assume that — 0). As was noted above, general processes S(t) of form (1) with a random intensity d'k(t) = A (t)/dt are adequate models of real random walks (in particular, those governed by plasma turbulence), for which the property of homogeneity is rather an exception than a rule. The parameters of turbulent plasma fluctuations observed in tokamaks and stellarators are nonhomogeneous in both space and time. Hereinafter, we will assume that the random variables {Xj}j>ι have at least first two moments. We denote EXi = a and DX\ = σ 2 , where 0 < σ 2 < oo. It will be shown below that, even under these assumptions, the limit distributions of generalised Cox processes can have arbitrary heavy tails. In this section, as an illustrative example, we will consider a case where a = 0. The reasons for this are as follows: First, our object here is to describe the principles of constructing adequate stochastic models of plasma turbulence, without going into details. Second, as will be demonstrated in the subsequent sections, the actual values of the parameter a turn out to be close to zero. We will formulate the necessary and sufficient conditions for the convergence of the one-dimensional distributions of generalised Cox processes with jumps that possess the above properties, without imposing any moment restrictions on the control process. We will demonstrate that the asymptotic behaviour of the process S(t) is completely determined by the asymptotic behaviour of the accumulated intensity A(/). Furthermore, we will see that the heavy (e.g., Pareto-type) tails of distributions that are limit for sums (1) can be caused by an extremely wide spread in the values of the control process A(?) rather than by 'poor' behaviour of summands (e.g., by the absence of their moments). In what follows, the symbol stands for convergence in distribution. A standard normal (Gaussian) distribution function will be denoted by Φ(.ν): Φ(.γ) = —L= f e~""/2du, V 2 π J-σο

- o c < .γ < oo.

The symbol Ε denotes the mathematical expectation (averaging) with respect to the probabilistic measure P. Let d(t) > 0 be an auxiliary normalisation (scaling) function increasing without bound as / — o c . THEOREM 1. Let A (t) increase without bound as t - * oo. Then for a one-dimensional distribution of a normalised generalised Cox process to converge to a distribution of some random variable Z, S(t) ;

=> Z.

t ^

OO.

σ/dü) it is necessary

and sufficient that there exist α nonnegative

random variable U such that

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Ν. Ν. Skvortsova et al.

(i) Ρ ( Ζ < χ) = ί Jo

Φ ( - ^ Λ dP(U ν ν "7

< u) = ΕΦ

—ΟΟ < Χ < 00,

- » 00.

The proof can be found in [29, 30]. Note that condition (ii) of Theorem 1 can be interpreted as the requirement for the accumulated intensity to be statistically regular: the limit of the ratio A(t)/d(t) as / —> oo can be random, but it must exist. Another interpretation of this condition is that, at large t, the distribution of the random variable A(t)/d(t) depends only slightly on /. In this case, condition (i) implies that the limit distribution of a generalised Cox process is a scale mixture of normal distributions that has heavier (in general, arbitrarily heavier) tails in comparison to a normal distribution. From Theorem 1 and the identifiability of a family of scale mixtures of normal distributions, the following corollary immediately follows. COROLLARY 1. Under the conditions of Theorem 1,

if and only if

In other words, the limit distribution of a generalised Cox process can be normal if and only if the random variable Λ (t)/d(t) is asymptotically (as 1 —> oo) nonrandom. Another corollary of Theorem 1 is a criterion for the convergence of one-dimensional distributions of generalised Cox processes with a zero average and finite variance to stable distributions. We will show that one-dimensional distributions of generalised Cox processes with the properties described above are asymptotically stable if and only if their control processes are asymptotically stable. Let Ga ß(x) be a stable distribution function with an index a and a parameter Θ. It is well known that such a distribution function is defined by its characteristic function

where oo 0, / > 1. Such a representation is possible when Λ (t) is a homogeneous process with independent increments and a generalised Cox process is observed at equidistant instants of time, i.e., Z , are the increments of the control process A(/) on time intervals between observations. In accordance w ith definition (1), we assume that ΛΓ, (Λ(η)) S(n)=

Σ

Χ

(3)

ί·

7= 1

In this case, in view of Theorem 2 of this paper and Theorem 2 of Section 35 in [31], we arrive at the following theorem. THEOREM 3. The one-dimensional distributions of a normalised generalised discrete-time Cox process S(n))/S„ are weakly converging to a strictly stable distribution Ga,ο for a certain choice of constants δ„ if and only if Ρ ( Z 1 Z x L χ^οο ρ (Ζ, > kx)

=

/2

for any k > 0. In other words, heavy tails often observed in stable distributions that are limiting for generalised Cox processes as the intensity increases can arise not only in cases where the distributions of jumps are characterised by heavy tails. As seen from Theorem 3, even for arbitrary light tails of the distributions of jumps, heavy tails of the limiting laws may arise due to heavy (Pareto) tails of the distributions of increments of the control process.

2.2.

Leptokurtocity

of scale mixtures of normal

distributions

The mixtures Ε Φ ( χ / s/TJ) are always more leptokurtic and, consequently, possess heavier tails in comparison to a normal distribution. Indeed, scale mixtures of normal distributions correspond to the product of a stochastically independent normal random variable X and nonnegative random variable Y = v u . As a numerical characteristic of leptokurtocity, we consider the excess factor κ { Ζ ) , which for a random variable Ζ such that Ε Ζ 4 < oo is defined as (Z -

EZ\4

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Ν. Ν. Skvortsova et al.

If Ρ(Χ < χ) = Φ (χ), then κ(Χ) = 3. For densities that are more leptokurtic (and, accordingly, possess heavier tails) in comparison to normal densities, we see that κ > 3, whereas for densities that are less leptokurtic, κ < 3. From the statements formulated below it follows that the scale mixtures of normal distributions are more leptokurtic as compared to a normal distribution. LEMMA 1. Let X and Y be independent random variables with finite fourth moments. If EX — 0 and Ρ (Y > 0) = 1, then κ(ΧΥ) > κ{Χ). Furthermore, κ{ΧΥ) = κ{Χ) if and only ifP{Y = const) = 1. Therefore, if X is a standard normal random variable and U is a nonnegative random variable independent of X such that EU 2 < oc, then κ ( Χ \fU) > 3; k ( X V U ) = 3 if and only if U is nonrandom. Using Jensen's inequality, we obtain another inequality that directly relates the tails of scale mixtures of normal distributions to the tails of the normal distributions themselves. LEMMA 2. Let α nonnegative EU~l/2 = 1. Then

random variable U satisfy the normalisation

χ > 0.

condition

(4)

It follows from Lemma 2 that if X is a standard normal random variable and U is a nonnegative random variable independent of X such that EU = 1, then for any χ > 0

P(\xVU\

> x ) > p f l j r i > χ) = 2[i -

φ(χ)],

(5)

i.e., the scale mixtures of normal distributions are always more leptokurtic than a normal distribution and, consequently, possess heavier tails. For more details on the properties of mixtures of normal distributions, see [32, 33].

2.3.

Specific features of the statistical analysis of scale mixtures of normal

distributions

The class of scale mixtures of normal distributions with a zero average is very extensive. In particular, it includes Cauchy, Student, and symmetric strictly stable distributions (see, e.g., [32, 33]). It follows from the above considerations that the statistical analysis of the distribution of the increments of turbulent plasma processes reduces to the separation of mixtures, i.e., to the statistical determination of the mixing distribution of the control process, which is an unknown parameter of the statistical problem under consideration. Without any additional assumptions, the parametric set of mixing distributions coincides with the set of all the distributions concentrated on the nonnegative semi-axis. The choice of an appropriate distribution is a very laborious statistical problem. Therefore, it is desirable to reduce the parametric set (i.e., the class of admissible mixing distributions) by adopting some additional assumptions. In this section, we propose one possible approach to solving this problem. According to this approach, a discrete distribution with a finite number of jumps can be used a mixing distribution. The approach is based on the following considerations.

Turbulent transport processes in plasma

95

(i) Any probability distribution concentrated at the nonnegative semi-axis can be approximated arbitrarily closely by a discrete distribution with a finite number of jumps. Therefore, there are reasons to believe that, considering a finite mixture of normal distributions, we are dealing with a convenient approximation of an actual distribution. (ii) It follows from Theorem 1 that, when the generalised Cox processes are used as models of a non-homogeneous random walk describing the observed turbulent plasma process, the form of the mixing distribution in the limiting law is completely determined by the character of the accumulated intensities and, hence, by the statistical features of changes in the instantaneous intensities of elementary processes. The instantaneous intensities of random walks with a continuous time are naturally related to the diffusion coefficients. In turn, the diffusion coefficients characterise the types of dynamic structures formed in a turbulent plasma (in other words, each type of structure is characterised by its own rate of change). In mixtures of the form

the scale parameter σ also has the meaning of the diffusion coefficient, whereas < σ) stands for the fraction of structures characterised by diffusion from a small interval [σ, σ 4- do] in the general picture of plasma turbulence. Hence, replacing the mixture ΕΦ(λ'/ \ f Ü ) by its finite discrete approximation | > Φ ( £ ) Ε Φ ( _ £ = ) ,

and statistically estimating the parameters σ \ . σ ζ , . . . , σ^ and p \ , p j />£, we c a n distinguish typical structures and describe their contributions to the general picture. Without going into analytical details, we note that, using the general limit theorems for generalised Cox processes (see, e.g., [27, 30]), it is possible to theoretically justify models for the distributions of the increments of turbulent plasma processes in the form of more general finite shift-scale (drift-diffusion) mixtures of normal distributions as

Σ>·(νΟ· where a j is the drift coefficient and Oj is the diffusion coefficient of the j th component. It is this approach that was used by us in statistical analysis of the processes observed.

3.

3.1.

EXPERIMENTAL DEVICES AND M E T H O D S TO INVESTIGATE T U R B U L E N C E Experimental

devices

Studies and modelling of LF SS plasma turbulence were performed for four devices: the L-2M, TJ-II, and LHD stellarators and the TAU-1 linear device. The main parameters of

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Ν. Ν. Skvortsova et al.

Table 1. Device Major radius R, cm Average minor radius r, cm Magnetic field Β, Τ Input microwave power Pq , kW Average density («), 1 0 1 3 c m - 3 Central electron temperature Te(0), eV Relative fluctuation level in the edge plasma

(δη/η\^ge

L-2M 100 11.5 1.3-1.4 150-200 1.0-1.3 400-800 0.2-0.25

LHD 800 60 < 3 600 -1.0 > 1000

TJ-11 150 10-22 < 1.2 200-400 < 1.0 500-800 0.2-0.25

TAU-1 2 < 0.06

0.001 4-7 0.2-0.3

these devices are listed in Table 1. The densities, temperatures, and diameters of the plasmas vary from η ~ 10 10 c m - 3 , Te ~ 5 eV, and D ~ 4 cm for a low-temperature plasma of the TAU-1 linear device to « ~ 1 0 1 3 - 1 0 1 4 c m " 3 , Te ~ 10 3 -1() 4 eV, and Ζ) ~ 150 cm for the high-temperature plasma of the most advanced LHD superconducting stellarator. Irrespective of the type of plasma device and even the kind of plasma instability, the LF turbulence in a magnetised plasma exhibits general features that allow one to compare the results obtained under so different conditions. It is well known that in order to describe the probability parameters of turbulence in any medium, it is of primary importance to study steady turbulent states, since only in this case one might expect that ergodic condition would be satisfied and the results obtained would be statistically consistent. The typical duration of the steady-state phase of a discharge is 10 ms in L-2M, 300 ms in TJ-II, 1 s in LHD (at present, the maximum discharge duration of a few minutes has been achieved in this device), and three to five hours in TAU-1. The L-2M device is an / = 2 stellarator, its parameters are described in detail in [34]. In the edge plasma, at a radius of rfa = 0.9 (here a is the separatrix radius), the plasma density is η = (1-2) χ 10 12 c m - 3 and the electron temperature is Te = 30—40 eV. The plasma is produced and heated by a 75 GHz gyrotron under the electron cyclotron resonance (ECR) conditions at the second harmonic of the electron gyrofrequency. The TJ-II device is an / = 4 stellarator [35]. The average plasma radius varies along the torus from 0.1 to 0.22 m. In the experiments under consideration, the plasma was produced and heated by a single 53.2 GHz gyrotron at the second harmonic of the electron gyrofrequency. The LHD device is the largest superconducting heliac with a divertor [36]. In the experiments under consideration, the plasma was produced and heated by several 168 GHz and 84 GHz gyrotrons under the ECR conditions at the fundamental and second harmonics of the electron gyrofrequency. The TAU-1 device was specially designed for studying and modelling nonlinear processes in a low-temperature plasma [37], In TAU-1, a cylindrical argon plasma column of diameter 4 cm and length 100 cm was produced in a uniform magnetic field of strength

bb, x\bb is the time of growth, z2bb is the time of disintegration (lifetime) of fluctuation. Their ratio is commonly taken to be equal to 0.5. Because to is a random variable, the amplitude at each moment of time is random, too. Thus, if υ is not equal to zero, the centre of fluctuation shifts along the axis (Fig. 8 shows evolution of a fluctuation). The linear superposition of a number of fluctuations forms a 'turbulent' layer. For adequate agreement with the experimental data, the fluctuations must overlap. So, for any fluctuation it is necessary that in a Δ-neighbourhood of the fluctuation centre there must be more than one fluctuation centre. The number of overlapped perturbations increased towards the plasma centre.

3.3. The total field of turbulence As the result, we obtain the function of two variables

x,t,

Nbb G(x, t) = J2 A"gc(x, t) + Σ A%b(x, t) + F, n= m= 1 where Nqc and Nbb are the total number of fluctuations for quasi-coherent fluctuations and broadband respectively. Thus, the function G(xz,t), where is the coordinate of probe

Analysis of

turbulence

153

Figure 7. Schematics of ID model e

ι-'an l

2 bb

—e

'"'On l bb

r

Figure 8. Evolution of a fluctuation in time

position, is a model equivalent of the probe signal. In the case of realization of this model on a PC, instead of continuous function there is a sequence. In the model, the average experimental value of density at the given radius consists of the constant background F and the fluctuated density level caused by a large number of superimposed positive perturbations.

3.4.

Realization of the model of turbulence on PC ' - ' O n

' - ' O n

The factor {e r2«f — e r | « r } for both types of fluctuation as / — ίο —> oo exponentially decreases, and at realization on a PC after some moment the computer works with fluctu-

154

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et al.

2,0

spectrum with 4io

1,5 1

1,0 0,5

V

v

(a)

v J t ^

0,0 spectrum with 4t


?iife.

To realize on a PC and to optimise calculations, the following scheme is applied: the PC considers some number of fluctuations much less than Nbb + Nqc. Thus, at t = 0, all to belong to the segment [0, ?iife]. If t — to„ > then to„ is associated with new to„ + z, where ζ is a random variable lying in the segment [/nfe, 'stoch], istoch is defined, and xon is associated with new xo· Then, the amplitude of fluctuations for new χ ο« and ton again grows and then exponentially decrease down to new t — to > iiife, and so on. Thus the number of involved fluctuations becomes less than one-tenth in comparison with Nbb + Nqc, and it depends on the difference between infe and istoch· The minimum of fluctuations number at the defined level of turbulence is reached at t\jfe = istoch· However stochasticity of fluctuations then disappears and in the turbulence there appears a period t Hfe (that is, perturbations occur at identical intervals) which is seen in the spectrum as the peaks at the frequencies / = 1 /t\\fe (see Fig. 9a). Clearly, if the length of the segment [xo/> xol] is large enough, then the probability for a fluctuation to fall in one and the same area and to affect on the signal of the probe is small. In the model we set i st0 ch = 2 % ε · The resulting spectrum is shown in Fig. 9b. Thus, the function G(x, t) and (in the case of realization on a PC, it is a matrix) has no zero values in the spatial interval from Xof to χ0ι + on the axis χ and the time interval from 0 to /count + 'life- It is obvious that characteristics of the turbulence in the area t > ?coum

155

Analysis of turbulence

•

•*

>-

·

·

30

•

Η

-i

^

20 ΙΛ 3.

.. »..

si Ü) -o

delay tim e inserted in 1D stochastic model delay tim e obtained from sirr ulated signals in ID sto .to/ + u?iife, fluctuations are not born, they get to this area due to the velocity along the axis x. In the area .x < x0/ + vtufe, just the opposite situation occurs: fluctuations may be born here, but because of moving along the axis χ fluctuations born in other places cannot get here. The case ν < 0 can be completely resolved by interchanging χ ο/ and x 0 / and then reverting the sign of v. So we arrive at the following criterion: Xo/ + u/],fe > χ > xoi ~ υ'life is the area where the properties of the turbulence are in the equilibrium state in space. So, the probe can be placed on distance no less than Wnfe away from each side. As the result, for any υ the zone of calculation is limited: Xq/ + vt\\h > χ > λ*0; — and < t < fount· From the first experiments on modelling, it was revealed that the speed obtained by standard procedures (inclination of a cross-phase and finding the maximum of the crosscorrelation functions) is over-estimated. Moreover, the value can differ by two-three orders from that fed into the program. The graph of speed function found as the maximum of the cross-correlation function of lifetime is given in Fig. 10. The circle-labelled line shows the value of speed fed into the program. Obviously, if the lifetime value increases, the speed approximates the true value. As our experiments show, the value of lifetime varies from 5 to 20 ms. Thus, the value of speed found by a standard method can differ from the true speed by the factor of 1.5-2. The reason is that the fluctuation amplitude changes in time. Upon passing through the probes, there is a change of amplitude of the fluctuation and as a result the probe maximum is displaced from the true position. This is the cause of the correction

156

Α. Ο. Urazbaev et al.

to the speed that has been found from modelling. The parameters of turbulence are found by fitting and comparing the spectra of the model and the actual experiment proceeding as follows: the width of the Doppler components of the spectrum is proportional to the ratio of turbulence velocity to its dimension. The spectrum extends upon decreasing the lifetime of fluctuations, too. The Doppler half-width of the autocorrelation function is equal to Δ / υ whereas the contribution of the time components is small. The probability distribution function of amplitude, further PDF, is fitted by the assumption that the experimental density on the given radius consists of non-fluctuating background F and the addition Σ ^ Γ ι AgC(x, i) + E m = i t) caused by a large number of positive fluctuations. The matter is that overlapping of large number of positive fluctuation leads to a non-zero level after averaging Ση=\ ^gc(x> 0 + E m = i t) in time, which may even be comparable with F . It is possible to consider the case of formation of plasma density structure by overlapping of fluctuations without a non-fluctuating background F, but this requires considering a large numbers of fluctuations N q c and Nbb which leads to a significant growth of calculation time. Thus, some uncertainty arises: the same PDF can be fitted either by increasing amplitudes of individual fluctuations and the amount of fluctuations but reducing the background F or by reducing the amount and amplitude of fluctuations but increasing the background F . However, in practice, it is possible to achieve the maximum conformity of the form of the PDF for moderate number of fluctuations. In Fig. 11, signals are shown from Langmuir probe saturation current, their spectra and the PDF for different radii of the tokamak. We see that at the periphery of the plasma the signals look like stochastic single splashes close to zero level and the PDF differs from the Gauss distribution. In the high temperature region of plasma, the signals look like noise band at non-zero level and PDF approaches the Gauss function. So we see that a turbulence generally can be simulated by superposition of positive functions. From the other hand, it is naturally noticed that with the increase of the overlapped perturbations the resulting form of fluctuations approaches the noise and the PDF became more symmetric and similar to Gaussian. This just coincides with the experimental tendency where the probe shifts into the high temperature plasma regions. So only positive functions are enough to model the plasma turbulence both at the periphery and the hot region of the tokamak. With constant perturbation amplitude, in the PDF of functions G(xz,t) equidistant maxima are seen corresponding to the amplitude A bb (the number of quasi-coherent fluctuations is much less than of broadband fluctuations, therefore they have no significant contribution to the PDF), the doubled amplitude and the tripled amplitude of the fluctuation. In the experimental signal, such maxima are not observed, one can say that it is more correct to vary the amplitude of fluctuation in limits from zero to the actual amplitude. The results of the simulation and experiment comparison are presented in Fig. 12 for the radius of probes 30 cm. The fitting is very good indeed. The comparison of model and experimental results for the radius 29 cm is presented in Fig. 13. They are interesting because of presence of quasi-coherent fluctuations in the spectrum.

3.5.

Results of modelling Langmuir probes experimental

data

The graph of the turbulence parameters as a function of the minor tokamak radius obtained by means of ID modelling and their approximation are presented in Fig. 14. Experimental measurements are carried out from radius of 29 cm up to 32 with the step 0.5 cm. As the

157

Analysis of turbulencc

^ probe = 33 cm

! 4 6 810l2t4l6l820

60 50 40 30 20 10

Äprobe = 31 cm

0

0 2 4 6 8101214161 SO

80 if probe = 30 Cm

60 40 20

0,

02468101214)61820

= 2 9 cm

0 2 4 6 8101214161820

650651 652653 654 655 time, its

frequency, kHz

density, x 1 0 1 2 cm

3

Figure 11. Form change of fluctuations of saturation current, amplitude spectra and PDFs for several Langmuir probe set

result, for each radius the values of parameters are obtained. Approximating the data, it is possible to find functional dependences between t i j j ( r ) , t2bb(r) (hbb(r) = /ifcfc(>')/2), r r Dbb( )A0bb( )< Nbbir), u(r), and F(r). In the experiment, quasi-coherent fluctuations are observed by Langmuir probes only on radius 29 cm. Therefore, parameters of quasicoherent fluctuations have no radial dependence and are taken as constants. For radii smaller than 29 cm, Nq is set equal to zero. Apparently, the amplitude of single fluctuation varies slightly, whereas the density of fluctuations increases three times and its profile resembles the profile of the density. The poloidal dimension function has a minimum at 30 cm; the lifetime function monotonously decreases to the centre of the plasma flux. The velocity correction found by means of modelling is significant only in the velocity shear and the edge regions.

158

Α. Ο. Urazbaev et al.

ω Λ (0

jC —, Ο. "Ό « 2 0,0

ο) — C

40

60

Frequency [kHz]

0

1

2

3

4

Amplitude.[ *1012 cm'3]

5

6

Figure 12. Comparison of spectral, correlation and stochastic characteristics (model and experiment); radius 30 cm

4. 4.1.

D E V E L O P M E N T O F 2D M O D E L S Problem

statement

1D model is unsuitable for calculations in the reflectometry, because the reflected probing beam contains the information not only from the point of reflection but also from any point belonging to its cone of propagation to the point of reflection. In order to modelling it, 2D calculations are required, and the coordinate χ becomes an equivalent of the poloidal direction in the tokamak: - D ^ V ) = % ( r ) . -10 has round section, the simplest method

159

Analysis of turbulence

3 cd

Figure 13. Comparison of spectral, correlation and stochastic characteristics (model and experiment); radius 29 cm

used to model it is the passage to the polar system of coordinates r, φ, where the origin coincides with the centre of the plasma, and the zero angle, with the optical axis of the central antenna. 4.2. Single fluctuation in polar coordinates The function of single fluctuation does not differ much from that in ID models. Instead of x, we find the dGß^

= fQ

φ

ixu)dF(u),

t ^

oo,

with the mixing distribution F{u)

= Ρ { κ ^ α [ 5 ( α / 2 ) Γ 1 / 2 < «J = f ^

GßA((u^)a/ß)

dGa/2 0,

dGßj(z),

= 0 for u < 0 . This fact plays an important role when simulating the r.v.'s with

fractionally stable distributions. Theorem 1 gives the approximate ( f o r large t) distribution of the particle coordinate in a continuous-time random walk: at the ith step it instantly moves to a new random point which is at the distance \Xj | f r o m the preceding one and then waits some time Γ ( + 1 before it undertakes the next step. It is easily seen that the situation under consideration is in some sense symmetric and can be turned over to investigate the asymptotic behaviour of the first passage time of the particle through a remote boundary. Namely, in addition to the above assumptions, w e restrict the conditions on the sequence X \ , X2, • • • and assume that these r.v.'s are nonnegative and £(X\)

e DNA(G(a,

1)) with a < 1.

Let M ( x ) , χ > 0, be the renewal process satisfying the relation M(x)

M(x)+1

Σ , χ , < χ < 1=1

Σ 1=1

x·•

for any x > 0. Then the r.v. M(x) Θ(χ)

=

Σ i= l

T'

can be interpreted as the 'first passage time' through the boundary χ by a continuous-time random walk. Just in the same way as that one used to prove Theorem 1, w e can obtain the following statement.

Fractionally stable

179

distributions

THEOREM 2. Under the above assumptions concerning the r.v.'s T\ and Χι, there exists a finite positive constant CQ = co(ct, β) such that p( { c0xa/P

< 4 =» )

f Jo

Q(t;ß,a,l)=

G(tua/ß;

β, 1) dG(u;a,

1),

χ - > oo.

Thus, we shall name distribution Q(x;a,ß, 1) and Q(t\ß,a, 1) f o r « , / ? e (0, 1] and θ = 1 coupled relative to space-time. Going over from finite dimensional distributions of process Σ (?) to the consideration of weak convergence of this process in Skorokhod space D([0, oo), M), we note that in [23] it has been proved that under conditions (6) and (7) the continuous-time random walk process weakly converge to the superposition Z{t) = A(B(t)) with corresponding normalisation. Here A (t) is the self-similar random process with stationary independent increments (Levy) process with Hurst exponent 1 /a, B(t) is the self-similar random process with Hurst exponent 1/β. Moreover, processes A(t) and Β(t) are independent and DB(t)~t2ß,

Ε

B(t)~tß.

It follows that increments of B(t) for β < 1 is non stationary. So B(t) is not a process with stationary increments. The process Z(t) is a self-similar one, but not a process with stationary increments. 4.

FSD DENSITIES

For FSD densities, we introduce the notation qix-,α,β.θ)

-

= OX

rg(Xye/°-,a,0)g(y:ß,l)yl»°dy. Jo

(10)

Using the properties of one-dimensional stable laws (see [1, 3]), one can derive some properties of FSD densities (10). (1) If 0 < a < 1 and θ = 1, the density q(x;a,ß, 1) differs from zero only on the positive semi-axis. In all other cases, the density is strictly positive on the whole real axis. (2) The reflection property qi-χ-,α,β,θ)

= q(x;

α,β.-θ)

holds. If θ = 0, then q{-x;a,ß,

0) = q(x\ct,ß,

0),

i.e., the FSD density with θ = 0 is symmetric about the origin χ = 0. (3) According to the reflection property, it suffices to consider FSDs for .y > 0 and to describe FSDs by means of their Mellin transforms (—1 < < a) /•OO

xsq{x\a.ß.6)dx.

q(s: α,β,θ)= Jo

(11)

180

V. Ε. Bening et al.

Applying it to (10) and using the relation (see [1]) !(,;«.*)3 ρ =

=

Jo

Γ(1 + / λ ί ) Γ ( 1 - ps)

( 1 + θ ) / 2

(12)

(13)

we obtain (—1 < Sfti < a ) qis-,α,β,θ)

= ρ - —

—

Γ(1 + /λ?)Γ(1 - ρ ί ) Γ ( 1 +

(14)

ßs/a)

(4) The densities q(x; α, β, θ) have moments of all orders if and only if a = 2. Moreover (note that if a = 2, then necessarily 0 = 0 , that is, in this case G(2,0) is the normal d.f. with zero mean and variance 2), for S > 0 we have oo

/ .oo |jc|

ι V * ; ß > 0) d*

=

ρ oo

- τ - J-oo / 2^/71

W

5

exp

ί

2\

\I

4 J\

/.oo dx

J/ο

x ^

n

d G { x \ β ,

1).

In particular, for η = 0, 1, 2 , . . . we have oo

Tin 4- 1 / 2 W

/ =

m2n+l(2,ß,0)

x2"q(x-,2,ß,0)dx 0. -OO

=

ν π Γ ( « Ρ + !)

If 1 < α < 2, then the second moment does not exist; if a < 1, then the first moment does not exist. (5) At the origin, an FSD density is equal to ,n ο Γ(1 + 1/«)Γ(1 — 1/α) q(0; α, β, mθ) = — — cos(0jr/2), ττΓ(1 - β / α ) whereas for the FSD function we have G( 0-,α,β,θ)

=

( \ - θ ) / 2 .

Note that q(x; α, β, θ) has an integrable singularity at the origin when a < 1, β < 1 and is zero when a < β. (6) If β = 1, then q(x;a,

1,0) = g ( x ; a , 0 ) ξ

dG(x;a,

0)/dx,

i.e., an FSD becomes a stable distribution. (7) If β < 1 and θ > 0, then q ( x ; a , ß , e ) ~ g ( x ; a , 0 ) / r ( l + ß ) ,

χ

oo,

i.e., the FSD densities have the same power asymptotics as stable distribution densities.

Fractionally stable

181

distributions

(8) If α = 2, then θ = 0 so that the FSD is expressed through the one-sided stable density:

This relation can be easily verified with the use of (11) - (14). In general, the FSD densities are not expressed in terms of elementary functions and for its representation the special functions are widely used (see Appendix A). Densities of the distributions are shown in Fig. 24 (see Appendix C) 5.

CHARACTERISTIC FUNCTIONS OF FSRVs

P. Levy, who defined and described the class of stable laws, introduced the term

second

characteristics i/(k;

α, Θ) = In g(k;

α. Θ)

so that the characteristic functions are expressed through the second characteristics via the relation g(k\a,

Θ) =

exp{i/f(A:;o!, (9)}.

(15)

The most important in what follows property of the second characteristics is that in the strictly stable case they are homogeneous in Euler's sense, i.e., f(ak\a,9) = a

a

f{k\a,e)

(16)

for any positive a. Turning back to (10), multiplying it by e x p ( - k x ) d x , integrating over the whole real axes, changing the order of integrations and using the second characteristics one can represent the FSCF in the form /»OO

q(k;a,ß,0)=

exp{-rlr(krß/a:a,e)}g(t;ß,

/ Jo

\)dt.

Taking into account property (16) /•OO

q{k\α,

β, θ) =

exp{-rß\lr(k;a,e)}g(t;ß,\)dt

/ Jo

and expanding the exponential function in the power series, we obtain:

n\

n=0

r

in

n-nß p

g(t;ß,\)dt.

The integrals present moments of subordinator of negative orders. It is known that Ε [S(ß)}~v

=

Γ Jo

g(r,

β. 1 ) r

v

d t

=

'(1 + v/jfi Γ(1 + υ)

(Π)

182

V. Ε. Bening et al.

hence -π

00

a α\

V^

' - i r ( k · α, θ)]η

Γ ( 1 + ηripβ / ρ) β) 1^11-

^

n\

Γ(1 + η β )

^

n=o

θ)]η

^ [-ψ(k\α,

Γ(1 + ηβ)

'

Comparing this expression with the series representation of the Mittag-Leffler function 00

w

xn

= n=0 Σ

v

we obtain the final result in the form qik-,α,β,θ)

= E

ß

{-f(k;a,9)).

(18)

According to (15), it reduces into SCF if β = 1: q(k;a,l,e) =

g(k-,a,9).

Formula (18) can be derived also from the following theorems proved in [1], T H E O R E M 3 . For any

0 < β < 1 and any

complex-valued

X,

poo β Ε β ( - λ ) =

If 1 / 2 < ß
0 . Then for

any complex-valued

λ

fOO Ε

6.

β σ

Ea{ku-ß°)g(u-ß,\)du.

{ λ ) = \

Jo

INVERSE POWER SERIES REPRESENTATION

It is known that stable densities can be represented in the form of power series asymptotic for a < 1 and convergent for a > 1: , m ^ g(*:«.g) = E

(-Ι)"'1 Γ(η/«+1) „_, „, Γ(«ρ)Γ(1 — n p ) X '

or in the form of inverse power series converging for a < 1 and asymptotic for a > 1 •I)""1 n\

where ρ = (1 +

θ)/2.

Tina

+

1)

T ( n p ) T ( \ - n p )

(19)

Fractionally

stable

183

distributions

Upon substituting the second series into (10) we obtain 00

ί\n—1

^

n\

T ( n p ) T ( \ - n p ) J0

^ n=\

n\

T(np)T{\-np)

n= 1

Γ(„«+1)

Γ

J0

r

nß

{

ß A ) d t

Using here (17) we obtain the corresponding series for FSD densities

n=1

For the fractionally stable distribution function we see that

n— 1

Similarly for (19) we find that , « m φ ; α , β , θ )

=

^ Σ

(-!)""'

T { n / a + \ ) T { \ - n / a ) Γ ( η ρ ) Γ { 1

-ηρ)Γ(1-ηβ/α)χ

•

n= 1

Expansions (21), (21) can be used to calculate FSD densities together with the representation for the Γ function Γ(1-χ)Γ(χ) = sinjrx In particular cases, the special representations below are more convenient. Namely, for a < 1, a = β 1 °°

q ( . x ; α . α , θ ) = π— t—1 sm{n α ρ ή ) \ χ \ ~ η α ~ λ , n= 1 q{x\a,a,9)=

1

°°

— — Vsin{nap(\ π χ r + 1 n= ι

\x\ > 1, -n))(-|x|a)",

|*| < 1.

(23) (24)

1 1

Equation (23) follows immediately from (21). Equation (22) has been obtained by analytic continuation: 00

f(x) =

ι

y>«(-*)" =

z-T

„=o

2 π ι

φ

rc+ioo

Γ(-Ζ)Γ(Ζ +

Jc-ioo

In order to find an expansion of / ( x ) in powers of χ arc of infinitely large radius to the left ψ Γ(-ζ)Γ(ζ +

\)a(:)xzd:.

, close the contour of integration by

\)a{z)xzd: = - f]a(-n) n=l

( - i )

.

184

V. Ε. Betting et al.

X

Figure 1. The graph of densities q(x; 1 / 2 , 1 / 2 , 1). The digits mean different number of series terms (23) and (24): curve 1, 10 items; 2, 30; 3, 100; 4, 1000

Results of calculation are presented in Fig. 1 for function q(x, 1 / 2 , 1 / 2 , 1 ) for number of terms of infinite series 10, 30, 100, 1000. The behaviour of relative error is presented in Fig. 2 for q(x, \/2, 1/2, 1) in comparison with the exact formula 1

q(x, 1 / 2 , 1 / 2 , 1 )

π*/χ(\ + χ)

We have strictly stable distribution for case a < 1, β = 1 and 0 = 0 , ( * . « . 1,0) =

π

-

'—'

m=ι

mi

V 2

/

The results of computation are presented in Figs. 3, 4, 23 (see Appendix C) and in Tables 4-18 (see Appendix D). 7.

INTEGRAL REPRESENTATION

Both convergent and asymptotic expansions are convenient tools of numerical analysis in the cases where the number of terms required to guarantee a reasonable accuracy is not very large. Otherwise one should prefer the integral representation of the distributions. From the

Figure 2. The graph of relative error q (x; 1 / 2, 1 / 2, 1). The digits mean different number of series terms (23) and (24): curve 1, 10 items; 2, 30: 3, 100; 4, 1000

Figure 3. The graphs of FSD densities q(x\ 3 / 2 , 1, 1/3). Block curve (formula from Appendix A) in comparison with series representation (light circle)

U Ε. Bening et al.

186

F i g u r e 4. Relative error of density computation q(x\

3 / 2 , 1, 1 / 3 )

computational viewpoint, the definite integral can be treated as the limit of a sequence of integral sums, i.e., as a series too; the existence of various schemes of numerical integration offers considerable scope for further improvements. It is clear that the presence of oscillating integrand plagues the computation. To avoid this trouble, the inversion formula has to be transformed into an integral of non-oscillating function. Let

g(x;a,0) = g(-x;a,~0) =

pOO ikx / e g{k\a, -Θ) dk. Jo

Omitting the special case a = 1, we assume that χ > 0 if α φ 1 and begin with the formula

f(k; α,-θ) = -\k\aexp{iae(n/2)signk}. The function g (k; α, —Θ) = exp{i/r (k\α, —Θ) admits the analytic continuation g + (z;a, Θ) from the positive semi-axis to the complex plane with the cut along the ray arg ζ = —3π/4. Considering the integral

J = f eixzg+(z;a,-6)dz = [ e~w{z'x) dz jl JL along the contour L which starts from zero and goes to infinity so that the W(z, x) takes only real values, after some manipulations we obtain for the stable density |x|l/(a-l)

/-W2

g(x\a,&) = -^ / βχρ[-\χ\α/(·α-ι)υ(φ;α,θ)}υ(φ;α,θ)άφ, JTl-a J-θπΙΙ t/

χ

Fractionally stable

187

distributions

where υ(φ\α,θ)

sin (αφ +

=

α/Ο-«*) c o s ( ( α -

αθπ/2)

1 )φ +

αθπ/2)

cos φ

cos φ

T h e similar expression takes place for the stable distribution function: /•OO Gix-,α,θ)

=

1 - J

g{x'\a,9)dx'

=

(1 -θ)/2

(25)

rn/2 + π~

/ exp J-θπΖ 2

ι

{-ν(χ,φ;α,θ)}άφ,

where ν{χ,φ\α,θ)

=

x"/(a_l)C/(0;a,Ö).

It is easy to derive the integral representation o f the F S D function. B y the definition, Q(x;

α, β,θ)=

f

q{x'\α,

J—CO

=

/

-oo /•O O

dx'

/· oo

X

/

β, θ)

dx'

dtg(t\ß,\)g(x'tß/a;a,9)tß/a

J0 G(xtß/a-a,e)g(f,ß,

l)

Jo Carrying out the evident manipulations

dt.

w/2

f

G(xtß/a]a,9)

=

(1-θ)/2

+ π~ι

/

exp{-ίβ/;ο·,6>)]"

1

η\

Γ(1 + Λ/(Ι Γ ( 1 + ηβ/(

-α)) 1 -α))

άφ.

χ >

0.

188

V. Ε. Bening et al.

This formula can be rewritten in terms of angular moments fjr/2 (υ"(α'θ»

= (1 „ +Λ θ)π/2η ΓJ-θχΐ2 '2 Ι-θπ/2

υ"(φ;α,θ)άφ

as follows: oo β

- ^ ^

+

, Σ ^ ^ η! η=ο

7 ' ΓΓ(1

+ " / " Γ - • α)) » ( Ο . + ηβ/(1

In the general case the integral representation can be written in terms of the Fox function (see Appendix).

8.

FOX FUNCTION REPRESENTATION

Fox function representation of FSDs is based on the Mellin transformation q{s\a, β,θ)

/»OO = I q(x;a, Jo

q(x;a,ß,9)

= — 2πι

ß,9)xsdx,

[ q(s;a, ß, 9)x~s~lds, Je

χ > 0.

Among the integral transformations, the Mellin transformation is the most convenient tool for investigation of FSD because any FSD has the form of the Mellin convolution (10). Its Mellin transform is nothing but the product of Mellin transforms of stable distributions under convolution. Indeed, multiplying both sides of (10) by xs dx and integrating over the positive semi-axis yields /»OO /»OO = I xsdx g(xyß/a;a,9)g(y, ß, \)yß/a dy Jo Jo /»OO /»OO = dyg(y;ß,\)yVa g{xyß'a-a,9)xs dx Jo Jo /»oo /»oo = / dyg(y,ß,\)y-sß/a g(z;a,9)zsdz Jo Jo

q(s;a,ß,9)

= g{s-a,9)g(-sß/a-ß,

1).

It is known that the Mellin transforms of stable densities are of the form [1] = S{S a

''

'

IW(l-c*V) r(ps)r(l

— ps)

=

Γ(1 + 5)Γ(1 — a's) Ρ

Γ(1+ρ*)Γ(1-ρ*)'

where a ' = 1 / a . Since ν,-Λπ

I i i z l l R Γ(1 — s) '

we obtain for (26) q(s;a,ß,9)

= ρ

Γ(1 + j ) f ( l - α ' ί ) Γ ( 1 +ce's) Γ(1 + /θί)Γ(1 - /θί)Γ(1 + a ' ß s )

(26)

Fractionally

stable

189

distributions

If we substitute —s — 1 for s, then a' q(x\α,β,θ)

f / h(s\a,ß,e)xsds, Je

=

2πι

χ >

0,

where his·,α,

β, θ) = o t q i - s -

l; α, β,

θ)

/οΓ(-ί)Γ(1 + a' + α'ί)Γ(1 - a' - a's) ~ α'Γ(1 - ρ - ρ ί ) Γ ( 1 + /> + /λϊ)Γ(1 - a ' ß - a ' _ Γ ( - 5 ) Γ ( α ' + or'j)r(l - a ' - a's) ~

The case tion:

a

r i l - p -

Represent

> β.

his\α,

ps)rip

β, θ)

+ ps)Til

(27)

ß s )

- a ' ß - a ' ß s ) '

in the standard form connected with Fox func-

his\α,β,Θ)

= — — — , C(s)Dis)

(28)

where Α(ί) = Γ(6ι -

ßis)T(b2

C(i) = Γ(1 -b3

- ß2s),

B(s)

+ ß3s),

=

Γ ( 1 - ax

+

«υ),

Dis) = Γ(α 2 - α 2 ί ) Γ ( α 3 -

a3s).

Thus, m — 2, η = 1, ρ = 3, and q = 3. Comparing (27) and (28) yields αι

=

1— α',

bi

= 0,

αϊ

— a';

a2

βχ

=

62 =

1;

=

1 — ρ,

a2

— p;

a3

= 1 — α'β,

a3

1 - α ' ,

β ι = α ' \

b3

=

β3

I -

ρ,

- α' β\ =

p.

Hence we find that ι

μ = Σ

ρ ßj

- Σ

7=1

a

J

=

1

+ +

-

ρ

α

' ~

ρ

-

α

'Ρ

=

1

-

a

'ß·

7=1

and if α > β, the integral (46) (see Appendix B) along the contour C separating the poles of A (s) and of Bis) can be expressed in terms of the Fox function as follows: ττ2\ 33

(

Ύ

V

(fli.«ι)

ia2,a2)

ia3,a3)

\

Q i ' ß l )

i^li

ib3.ß3)

)

ßl)

with parameters {a, a, b, β} given above. As the result we obtain the following formula for FSD density (x > 0 and a > β): r qix\a,

« m Ή 2 ΐ ( β, θ) = a Hit \x 33

V

0

"«'·«')

(0,1)

(1 - Ρ · Ρ ) , ,

Ο - α ' β . α ' β )

(1-α',α')

(1 - p , p )

\ 1.

)

190

V. Ε. Bening et al.

The case α < β. In order to consider the opposite case a < β, we replace χ by 1/x and s by —s — 2 in (27). After some typical manipulations with gamma functions, we find that J

2πί

Γ(-ρ-ρί)Γ(1

Je

+ p + ps)r(l

+a'ß

+

a'ßs)

Using again the standard form of the Mellin transform of the Fox function we find in this case Λ 0 ) = Γ (—a' - a ' j ) ,

B(s)

C(s) = Γ ( 1 + ρ + / μ ) Γ ( 1 + α'β + a'ßs),

D(s) = Γ ( - ρ - ps).

Now m =

= Γ (2 + ί)Γ(1 +

a' +

a's),

η = 2, ρ = q — 3. Comparison with (46) yields

a\ = —1,

«1 =

by = 0,

βι

1;

— 1;

«2 = —οι',

= a';

b2 = l - a ' ,

ß2=ct'\

a3 = —p,

«3 =

p\

b3 =

βι

p.

1 - ρ,

=

The condition μ = ßot' — 1 > 1 is valid if a < β and for this case and χ > 0 we obtain q(x;ct,ß,

9.

χ-

θ) =

2„,ήΜ(1 33

{ χ

(-1.1)

(-«'.«')

(-V,«')

(—ρ, ρ)

(-Ρ,Ρ)

\

(-α'β,α'β)

)•

SIMULATING STABLE RANDOM VARIABLES

There exists a large number of algorithms to simulate random variables of different kinds [24, 25]. Many of them use the inverse fimction method called also direct method based on the following theorem. THEOREM 5. F(x)

Let ξ be a random

a monotonically

lim x ^. a single

F(x)

inverse

distributed

=

increasing

0, linvc-^

fimction

F~l

on the interval

variable fimction

F{x)

=

1

(a —>

Since the function

—00

(ζ), ζ 6 ( 0 , 1 ) exists, (a, b) with

F(x)

uniformly and b

on the interval a derivative

00

and the random

are allowed). variable

(0,1)

and the £ =

and limits

Then F~l

the

(ξ)

is

density Pt(x)

Proof.

distributed

on (a, b) possessing

= F'(x).

(29)

is strictly increasing and

F f (*) = P{£ < x} = P{F-*(t) < χ } =

Ρ{ζ

F${x)

< F(x)}

= x, we see that = FK(F(x))

=

F(x),

and we arrive at (29). As an application of the theorem, we consider the following simple cases. The Cauchy variable Fß(l, 0).

Its probability density function is 2 pr(y) =

2

(n

+

(4y2)Y

According to the theorem on inverse function, Υ = (π/2) tan Ξ, where Ξ = π(ζ — 1 /2) is uniform on (—π/2, π/2).

(30)

191

Fractionally stable distributions

The Gaussian variable Yb(2,0).

Its probability density function is 1

( , y - a\2 )2\

J

(31)

Let Y ' be a random variable independent of Y and Y ' be distributed by the same law as Y. Let Q be the point on the plane (x, y) with the coordinates (Υ, Y'). Now we pass to the polar coordinates χ — a — r cos φ,

y — a = r sin φ

and see that Q has the distribution PQ(r,f)

= (1πσ2)~ι

= pR{r)p 0 which are with the the densities -

= (2π)

Ρφ(Φ)

By virtue of Theorem 5, ( 2 f Q

pR(r)

σ~2

Ίφ = ξχ,

J *

=

A

2

"

2

exp

r exp j - ^

j dr =

fc,

so we obtain Φ = 2πξι, Turning back to the coordinates (x,y), of Gaussian random variables

R =

we obtain two independent formulas for simulation

Y = yJ—2o 2 In £2 cos(2tt

=

„

π

'

2

·

Thus G ( x ; ß A ) =

[

G(x

Jo

I ϋ;β,

l)p($)d$,

where #

)

=

l/π,

0 < ϋ < π ,

and G ( x I ϋ \ β , 1) =

P{S(ß)

< χ I ϋ}

= εχρ{-χ-

β /

(

β

-»υ

β

(ϋ)}.

The right-hand side of this equation is the probability Ρ {Ε

> χ-

β Κ β

~

ι )

υ

β

( ϋ ) } = Ρ {Ε~ι

ß K l

< x

~

ß )

/ϋ

β

{ϋ)} =

Ρ { [ υ

β

( ϋ ) / E f ~

ß ) / ß

< χ),

E~X,

where £ is a random variable distributed with the density Ρ E(X) = x > 0. Thus, the random variables S ( β ) and [U β ( Φ / 2 ) / Ε ] ( ι ' β ) ' β have the identical distributions S(ß) = [υ

β

{Φ/2)/Εγ-

β )

'

β

.

(33)

This result was obtained by Kanter [26] Chambers' algorithm. Kanter's formula was generalised by Chambers et al. [27] to the whole family of stable variables: sinK