Analysis of the quiet-time statistics of edge electrostatic fluxes measured in tokamaks and stellarators B. Ph. Van Milligen 1, R. Sánchez 2, D. E. Newman 3 and B. A. Carreras 4 1 University of Alaska-Fairbanks, USA 2 Universidad Carlos III de Madrid, SPAIN 3 Asociacion Euratom-CIEMAT, Madrid, SPAIN 4 Oak Ridge National Laboratory, U.S.A. Data provided by: R. Balbin, C. Hidalgo, M.A. Pedrosa (CIEMAT), B. LaBombard (CMOD), J. Bleuel (w7AS) and JET, TJ-II and W7AS Teams

Introduction ¢ Self-organized-criticality (SOC) has found wide application in physical and earth sciences ¢ SOC describes the steady-state dynamics of driven systems with two disparate temporal scales: the drive and the local relaxation. ¢ CRITICAL: the SOC state is self-similar and statistically non-stationary. ¢ SELF-ORGANIZED: the SOC state is reached with no tuning. ¢ Transport takes place through avalanches.

Sandpile Model ¢ Critical slope: enforces time scale separation. ¢ Avalanches of all sizes and durations balance incoming flux in steady state --- POWER LAWS

SOC and MAGNETIC FUSION ê Turbulent transport appears when instability thresholds locally overcome. ê Excitation can propagate to neighbouring eddies, causing avalanching plasma transport. ê Consistent with: canonical profiles, transition from Bohm to gyro-Bohm scalings, perturbative ballistic transport phenomena.... : Diamond et al, Phys. Plasmas, 2, 3640 (1995) : Newman et al, Phys. Plasmas, 3, 1858 (1996)

Waiting-Times: an experimental test for SOC? í Avalanches can be thought as events that are excited at some time, t j. A waiting-time is defined as w j =t j -t j-1. í If avalanche durations were zero, the waiting-time PDF would be determined completely by the external drive. For uncorrelated t j, it must follow a Poisson distribution. í Some authors proposed that the randomness of the drive is essential for SOC dynamics (Boffetta et al, PRL 83, 4662 (1999)). Therefore, might looking for exponential waiting-time PDF be a necessity test for SOC ?

SOC model for SOLAR FLARES í Wheatland et al, Astroph. Journal 509, 448 (1998) Boffetta el al, Phys. Rev. Lett. 83, 4662 (1999) í Waiting-time PDF for solar data from the National Geophysical Database Centre at USA exhibit extended power-laws.

SOC model for MAGN. FUSION ê Spada et al, Phys. Rev. Lett. 86, 3032 (2001); Antoni et al, Phys. Rev. Lett. 87, 450 (2001). ê Waiting-time PDF of edge turbulent electrostatic fluctuations from RFX reversed- field-pinch (RFP) exhibit extended power-laws.

POISSON WAITING TIMES Why do uncorrelated triggerings follow an exponential? Because an exponential is the only function that verifies: Therefore, for some k>0:

Waiting Times: Problems? í Avalanches DO have duration d j >0. Therefore, not EVERY triggering can be detected due to avalanche overlapping. Detectable waiting-times are thus DISTORTED. í Detectable waiting-times are correlated. They are equal to w j =d j + q j, being q j the lapse of quiescent time between avalanches: quiet time. But SOC dynamics DO correlate d j 's (R. Sanchez et al, Phys.Rev.Lett. 88 (2002) )

Waiting Times vs Quiet times: Sandpile example

Waiting Times: Problems? í Therefore, even if the system drive is random, it might not be detectable from waiting-time statistics if overlapping is meaningful. í Luckily, it can be proved that quiet-times DO depend only on the drive, even in the presence of avalanche overlapping (R. Sanchez et al, Phys.Rev.E. (2002) in press). SO.. Use quiet-times instead?  Regretfully, that is not enough. It can also be shown (R. Sanchez et al, Phys.Rev.Lett. 88 (2002) ) that, even in the presence of non-random correlated drives, systems can exhibit SOC dynamics.

Random vs Coloured drive: Sandpile example

Quiet-Times: A solution? ê Quiet times-PDF can be used to probe the character of the external drive: ê Exponential quiet-time PDF implies random drive ê Power-law quiet-time PDFs implies self-similar non- random drive ê What about the dynamics? ê Conditional sampling of avalanches can be used to make quiet-times give information about internal dynamics. ê Duration-thresholding above beginning of self-similar range makes power-laws appear in quiet-time PDF

Duration thresholding: Sandpile example

Quiet-Times: Application to tokamaks and stellarators í In tokamak and stellarators, quiet-time analysis can be done for turbulent fluxes measured at the edge by means of Langmuir probes. í Since the probe only detects local flux, the signal will reflect the crossing of an avalanche by the probe location. Thus, quiet-times must be interpreted as the time between successive avalanche-crossings. í Another issue is that local fluctuations (say, within a single eddy) are intertwined with avalanches. The maximum scale related to them can be identified by multifractal analysis, and eliminated via m-point signal averaging.

Data smoothing and thresholding: W7AS

Quiet-Times: Application to tokamaks and stellarators ê The m-averaging process make local fluctuatin merge onto a quasi-continuous band that can then be eliminated by vertical-thresholding above the signal mean (we use Spada's prescription to improve this). ê Now, we apply quiet-time statistics, but assuming as events only those exceeding a prescribed duration threshold, D. For D = 0 (more precisely, the minimum found after treating the signal), quiet-time PDFs close to exponentials are found. As D increases, a power-law appears to develop as soon as the self-similar range of avalanche durations is crossed!

Quiet-Times: Application to tokamaks and stellarators ê Lack of statistics makes us use Rank-functions instead of PDFs, since the former require no-binning. ê Rank-functions for some quantity s are constructed by ordering the values in increasing order, and asigning to each distinct value s k a number: r k = r (k-1) + n k, being n k the number of times s k appears in the series. ê Any power-law or exponential behaviour in R(s) indicates power-law or exponential behaviour in P(s).

Quiet-Times: Application to tokamaks and stellarators ê Power-laws are looked for by fitting rank-functions to: forcing s2 << s1. In this way, for scales between them R(s)~s -k. ê The size of the power-law region is then defined as: D=log[s 1 /s 2 ] ê D >1 is requested for claiming power-law behaviour; also that exponential fits do not do better!

Quiet-Times: W7-AS data (m=32)

Quiet-Times: JET data (m=16)

Conclusions ê Quiet-time PDF can be used to look for SOC dynamics, but they only provide a consistency test! ê The edge of stellarator and tokamaks seems to be randomly-driven, exhibiting dynamics that appear to be consistent with SOC. The same behaviour has also been observed in the TJ-II heliac and the C-MOD tokamak. ê What happens with RFP?... may be the edge drive coming from within the reversal surface is not random, but self- similar. The dynamics might thus still be SOC.