Transcript TTF 2008 - University of Warwick

Statistical properties of edge turbulence in MAST spherical tokamak
and LHD stellarator
J.M. Dewhurst1, B. Hnat1, N. Ohno2,3, R.O. Dendy4,1, S.Masuzaki3, T. Morisaki3, A. Komori3, B.D.
Dudson5, G.F. Counsell4, A. Kirk4 and the MAST team4
1 Centre
for Fusion, Space and Astrophysics, University of Warwick, Coventry U.K.;
Science Institute, Nagoya University, Nagoya 464-8603, Japan;
3 National Institute for Fusion Science, Toki 509-5292, Japan;
4 Euratom/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, U.K.;
5 Physics Department, University of York, YO10 5DD, U.K
2 EcoTopica
1. Introduction
4. Autocorrelation function and skewness
Turbulence in the scrape-off layer (SOL) and divertor region of magnetically confined
plasmas is bursty and intermittent. Intermittent events, associated with density blobs
(filaments), have been linked with increased cross-field transport and are therefore a
subject of much study. Recent experimental evidence suggests that this edge turbulence
has generic statistical properties which emerge in the functional forms of the probability
density functions (PDFs) and the scaling of their higher moments [1-3]. The extent of
this universality across a range of confinement systems and operational regimes is an
important but unresolved issue. Here, we focus on the statistical properties of
measurements of the ion saturation current jsat from the edge region of the Mega-Amp
Spherical Tokamak (MAST) and the Large Helical Device (LHD) stellarator [4-6]. We
utilise modern statistical techniques which provide constraints on models and theory.
• Autocorrelation is related to power spectrum via Fourier transform
• Skewness measures the asymmetry of statistics, S=0 for the gaussian process
• Skewness is often thought of as a measure of nonlinear interactions and in
turbulent system can be related to the energy transfer rate
5. Absolute moment analysis [10]
Device type
Major radius, R
Minor radius, a
Spherical tokamak
0.85m
0.65m
Heliotron-type stellarator
3.9m
0.65m
Typical magnetic field strength
Probe type
Probe location
Sampling frequency
Typical length of time series
Shot numbers
0.5T
Reciprocating probe
Outboard midplane
500kHz
50ms / 25,000 samples
14219, 14222
2.5T
Reciprocating probe
Midplane
1000kHz
1s / 250,000 samples
76566, 77659
2. Data sets





Ion saturation current, jsat, collected by Langmuir probe in the midplane of the device
LHD data are stationary for much longer then MAST data
Power spectrum was used to select data with clean high frequency
regions
No low pass filter during data collection, so some aliasing is possible
Ion saturation current is defined as: jsat  ene cs , cs  sound speed
• All discharges show two distinct scaling regions
• We define following temporal scales:
• τac where the autocorrelation function falls below the threshold of ~0.1
• τm which separates two distinct scaling regions of the absolute moments
• For MAST discharges: τac≈τm≈30-50 μs; for LHD these are shorter
τac≈τm≈10-20 μs
• Region I with scaling of ~1 is consistent with scaling of coherent signal
• Temporal scale , τm , is similar to the observed lifetime of MAST [11]
6. Probability distribution Function
jsat signals from MAST Discharge 14222 and LHD discharge 76566
MAST
τ ≈ 4 μs
MAST
τ ≈ 64 μs
LHD
τ ≈ 64 μs
• PDF of aggregates, Isat(τ), is non-Gaussian on all temporal scales, but
evolves toward more symmetric and Gaussian-like form for large values of τ
• Extreme value distributions appear to provide a generic model for the PDFs,
while other functions (log-normal and gamma) fit only particular temporal
scales
3. Statistical methods [7,8]
Fluctuations on time scale τ given by: I sat 
LHD
τ ≈ 4 μs
t 
( j
t ' t
Scaling of absolute moments: Sm ( I sat , )  I sat
m
sat
(t ')  jsat ) / 

 ( m)
7. Averaged peak shape
Green-MAST
Blue-LHD
Probability density function (PDF) on time scale τ: P( I sat , )
PDFs are fitted with three model distributions [9]:
• Log-Normal: PLN ( I sat |  ,  ,  )  ( 2  I sat  exp
  ln( I sat )   
a
1
a 1
• Gamma: PG ( I sat |  , a, b)  (b (a)) I sat exp(
2
I sat
)
b
2 2
• Generalised Extreme Value:
1
1
 

 I     k  1  k ( I sat   )  1 k
1
 
PGEV ( I sat |  , k ,  ,  )  exp  1  k sat








 
 


• Observed statistical features can be related to the
average peak shape for MAST and LHD datasets.
• MAST peaks (blobs, filaments) are broader and
more asymmetric as compared to these from LHD
8. Conclusions
• Statistically, MAST and LHD Isat aggregates are different, but some generic
features are present
• PDFs are non-Gaussian and can be modelled by extreme value distributions
• Differences in statistics are related to the size/shape of coherent structures
References:
[1] B Ph van Milligen, R Sánchez, B A Carreras et al., Phys Plasmas 12, 052507 (2005)
[2] G Y Antar, G Counsell, Y Yu, B Labombard and P Devynck, Phys Plasmas 10, 419 (2003)
[3] R O Dendy and S C Chapman, Plasma Phys Control Fusion 48, B313 (2006)
[4] N Ohno, S Masuzaki, H Miyoshi, S Takamura, V P Budaev, T Morisaki, N Ohyabu and A Komori, Contrib Plasma Phys 46, 692 (2006)
For k>0 PGEV represents Fréchet distribution of maxima selected from the set
of realizations of the process with diverging second moment.
For k=0 PGEV represents Gumbel distribution of maxima selected from the set
of realizations of the process with converging second moment
[5] N Ohno, et al., 21st IAEA Fusion Energy Conference, Chengdu, China, Oct 16-21, 2006, EX/P4-20
[6] S Masuzaki, T Morisaki, N Ohyabu, A Komori et al., Nucl Fusion 42, 750 (2002)
[7] J M Dewhurst, B Hnat, N Ohno, R O Dendy, S Masuzaki, T Morisaki and A Komori, Plasma Phys. Control. Fusion 50 No 9, 095013 (2008)
[8] B D Dudson, R O Dendy, A Kirk, H Meyer and G F Counsell, Plasma Phys Control Fusion 47, 885 (2005)
[9] D. Sornette, Critical Phenomena in Natural Sciences; Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, Springer-Verlag, 2000!.
[10] B Hnat, B D Dudson, R O Dendy, G F Counsell, A Kirk and the MAST team, Nucl Fusion 48, 085009 (2008)
[11] A Kirk, N Ben Ayed, G Counsell, B Dudson et al., Plasma Phys Control Fusion 48, B433 (2006)