Analysis of the quiet-time statistics of edge electrostatic

fluxes measured in tokamaks and stellarators

B. Ph. Van Milligen1, R. Sánchez2, D. E. Newman3 and B. A. Carreras4

1 University of Alaska-Fairbanks, USA

2 Universidad Carlos III de Madrid, SPAIN

3 Asociacion Euratom-CIEMAT, Madrid, SPAIN

4Oak 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

Turbulent transport appears when instability thresholds locally overcome.

Excitation can propagate to neighbouring eddies, causing avalanching plasma transport.

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?

Avalanches can be thought as events that are excited at some time, tj. A waiting-time is defined as wj=tj -tj-

1.

If avalanche durations were zero, the waiting-time PDF would be determined completely by the external drive. For uncorrelated tj, it must follow a Poisson distribution.

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

Wheatland et al, Astroph. Journal 509, 448 (1998) Boffetta el al, Phys. Rev. Lett. 83, 4662 (1999)

Waiting-time PDF for solar data from

the National Geophysical

Database Centre at USA exhibit extended power-laws.

SOC model for MAGN. FUSION

Spada et al, Phys. Rev. Lett. 86, 3032 (2001); Antoni et al, Phys. Rev. Lett. 87, 450 (2001).

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:

P(w>t+s) =P(w>s)P(w>t) ⇔ P(w>t) =e−kt

P(w>t+s)P(w>t)

=P(w>s)

Waiting Times: Problems?

Avalanches DO have duration dj>0. Therefore, not EVERY triggering can be detected due to avalanche overlapping. Detectable waiting-times are thus DISTORTED.

Detectable waiting-times are correlated. They are equal to wj=dj + qj , being qj the lapse of quiescent time between avalanches: quiet time. But SOC dynamics DO correlate dj's (R. Sanchez et al, Phys.Rev.Lett. 88 (2002) 068302)

Waiting Times vs Quiet times: Sandpile example

Waiting Times: Problems?

Therefore, even if the system drive is random, it might not be detectable from waiting-time statistics if overlapping is meaningful.

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) 068302) 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?

Quiet times-PDF can be used to probe the character of the external drive:

Exponential quiet-time PDF implies random drive Power-law quiet-time PDFs implies self-similar non-

random drive

What about the dynamics?

Conditional sampling of avalanches can be used to make quiet-times give information about internal dynamics.

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

In tokamak and stellarators, quiet-time analysis can be done for turbulent fluxes measured at the edge by means of Langmuir probes.

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.

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

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).

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

Lack of statistics makes us use Rank-functions instead of PDFs, since the former require no-binning.

Rank-functions for some quantity s are constructed by ordering the values in increasing order, and asigning to each distinct value sk a number: rk = r(k-1) + nk, being nk the number of times sk appears in the series.

Any power-law or exponential behaviour in R(s) indicates power-law or exponential behaviour in P(s).

P(s) =−1

maxR(s)[ ]dRds

Quiet-Times: Application to tokamaks and stellarators

Power-laws are looked for by fitting rank-functions to:

forcing s2 << s1. In this way, for scales between them R(s)~s-k.

The size of the power-law region is then defined as:

D=log[s1/s2]

D >1 is requested for claiming power-law behaviour; also that exponential fits do not do better!

R(s) =e−s/s1

1 + ss2( )k

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

Quiet-Times: JET data (m=16)

Conclusions

Quiet-time PDF can be used to look for SOC dynamics, but they only provide a consistency test!

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.

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.