A Kirk

B Dudson

G F Counsell

M V Umansky

R O Dendy

T D Rognlien

X Q Xu

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Statistical analysis of plasma edge fluctuations in MAST andcomparison
with BOUT simulation results
B. Dudson1, A. Kirk2, R.O. Dendy2, G.F. Counsell2, M.V. Umansky3, T.D. Rognlien3,
X.Q. Xu3
1 Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, UK
ben.dudson@ukaea.org.uk
2 EURATOM/UKAEA Fusion Association, Culham Science Centre, Oxfordshire OX14 3DB
3 LLNL, Livermore, CA 94551
Introduction
In recent years many statistical techniques have been applied to edge turbulence measure-
ments from Langmuir probes and reflectometers in order to examine properties such as long-
time correlations which differentiate between models of turbulence. Fluid codes have developed
to the point where these statistical methods can be applied totheir output and compared with
experiment. The aim of the current work is to compare characte istics of edge turbulence from
the BOUT code, written by X.Xu at LLNL [1], with measurements taken from the Mega-Amp
Spherical Tokamak (MAST) [2] using some of these methods.
BOUT is a two-fluid Braginskii code which has a realistic geometry and includes closed and
open field lines as well as the magnetic x-point [1]. In the radial-poloidal plane, the grid is
obtained from UEDGE, based on an EFIT reconstruction. The code has sources of many ideal
and non-ideal instabilities which can lead to turbulence including high-n ballooning, dissipative
drift, shear Alfvén and peeling modes. Due to several specific features of spherical tokamaks,
the time-step is smaller than for a conventional tokamak. This makes observing saturated tur-
bulence over a long time in an ST quite challenging. See [3] for details of the BOUT simulation
of MAST.
Results
BOUT simulations were run for a MAST single-null L-mode plasma using temperature and
density profiles from Thomson scattering. A simulation was also run with double the density
and half the temperature. The results are compared with data from MAST ohmic L-mode single-
null plasmas with plasma currents of 700kA. Table 1 shows temperature and density values at
the separatrix for these plasmas and simulations.
Fig. 1a shows an ion saturation current (ISAT ) signal from MAST taken at the outboard mid-
plane with a reciprocating Langmuir probe. This shows intermittent bursts of varying sizes
32nd EPS Conference on Plasma Phys. Tarragona, 27 June - 1 July 2005 ECA Vol.29C, O-4.019  (2005)
Table 1: Plasma parameters
TSEP (eV) NSEP (1018m−3)
BOUT Low density case 20 6
BOUT High density case 10 12
12679 25±10 5±1
12850 25±10 7±1
12851 25±10 9±1
Figure 1: Isat signals
typical of this type of measurement. Fig. 1b shows an enlargedview of this signal over a period
of 170µs, the same time-length as anISAT measurement from the low density BOUT simula-
tion shown in Fig. 1c. The BOUTISAT signal shows similar features but lacks the small-scale
fluctuations present in the MAST data.
Frequency (kHz)
P
ow
er
 (
A
R
B
)
Frequency (kHz)
a
MAST plasmas BOUT simulations
P
ow
er
 (
A
R
B
)
1 10 100 1000
10
-15
10
-10
10
-5
10
0
1 10 100 100010
-15
10
-10
10
-5
10
0
b
low density
high density
12850
12851
12679
Figure 2: Fourier Power spectrum
The most widely used
method of analysing turbu-
lent signals is the Fourier
power spectrum. The spec-
tra from three MAST plas-
mas are shown in Fig. 2a,
whilst spectra from the
two BOUT simulations are
shown in Fig. 2b. For both
MAST plasmas and BOUT simulations, the power spectrum follows a power law for large fre-
quencies, but the gradient of this fall-off is much larger for the BOUT data than for the MAST
data. This indicates that small-scale structures present in xperimental turbulence are suppressed
in the simulation. This could reflect the absence of kinetic effects in BOUT which may give rise
to the small-scale structure seen in experimental data. Theeffect of numerical dissipation will
also be to damp high-frequency components. It is also possible that after only∼ 200µs the
turbulence is not yet fully developed and so there is less energy in small scale structures than
32nd EPS 2005; B.Dudson et al. : Statistical analysis of plasma edge fluctuations in MAST and comparison wit... 2 of 4
would be the case in a longer run.
In order to analyse the behaviour of turbulent signals over different temporal and spatial
scales, methods based on Hurst exponents have been applied toturbulence measurements from
tokamaks and other complex systems [4, 5, 6]. Differencing and rescaling is one such method
and involves calculating a set of probability distributionfu ctions (PDFs)P(δx,τ) of a quantity
δx(t,τ) = ∑t+τt̃=t
(
ISAT (t̃)− ISAT
)
whereISAT is the averageISAT signal. This is a measure of the
total density flux passing the probe during a time periodτ. The PDF of this then gives the
probability that during a time period of lengthτ a given a given flux passes the probe. If the
signal is statistically the same on all time-scales (self-similar), then these PDFs can be rescaled
by a simple power-law transformation onto a single underlying PDF - see [4] for further details.
MAST plasma 12850
10-4
10-3
10-2
10-1
(   s)
BOUT simulation
P
(0
, 
  
 )
1 10 100 1000
1.05±0.05
0.94±0.01
0.66±0.02
0.50±0.10
P
(0
, 
  
 )
100
10-4
10-3
10-2
10-1
100
(   s)
1 10 100 1000
Figure 3: Differencing and rescaling
The peakP(δx = 0) of
the PDFs as a function of
τ is shown in Fig. 3. Time-
scales over which the sig-
nal is self-similar are indi-
cated by power-law scaling
of the form P(δx = 0) ∝
τ−H , whereH is the Hurst
exponent. A typical result
from MAST has a Hurst exponentH ≃ 1 for τ less than 10− 50µs and a value ofH ≃ 0.7
for times longer than this [4]. The high density case does show a change of gradient at around
20µs, but the value of the Hurst exponent obtained is approximately 0.5, usually indicating no
correlations. A longer BOUT simulation would be needed to confirm this scaling at long times.
0 20 40 60 80 100
0
10
20
30
40
50
60
K
u
rt
o
s
is
-5
0
5
10
15
20
K
u
rt
o
s
is
0 20 40 60 80 100
Time difference (   s)Time difference (   s)
Bout simulationsMAST plasmas
low density
high density
12850
12851
12679
Figure 4: kurtosis
A simpler way of looking
for self-similarity is by look-
ing at how the ISAT signal
changes in time by examin-
ing the skewness and kurtosis
of δ I = ISAT (t + τ) − ISAT (t).
Fig. 4 shows the kurtosis of
δ I as a function of τ: κ =
〈
(δ I)4
〉
/
〈
(δ I)2
〉2
− 3. This gives a measure of intermittency at different scales- low val-
ues imply that the signal is constantly fluctuating by small amounts, whereas large values imply
that the signal consists of quiet periods and large fluctuations. For self-similar signals,κ should
32nd EPS 2005; B.Dudson et al. : Statistical analysis of plasma edge fluctuations in MAST and comparison wit... 3 of 4
be independent ofτ. This does not appear to be the case for either the BOUT data or the mea-
suredISAT data for small values ofτ ≤ 20µs. The value of the kurtosis at small values ofτ does
not appear to depend in a simple way on the density since intermit nt signals may be caused by
many different effects. For long times, both signals do apper to have approximately constant
kurtosis. Note that the values of the kurtosis from BOUT are smaller than for MAST plasmas
and in fact in BOUT are approximately zero for long times, corresponding to the kurtosis of a
Gaussian. This could be due to missing physics in BOUT which give rise to a more intermittent
signal seen in MAST plasmas, or because as with the power spectrum, the simulation has not
been run long enough for the rare large events which contribute to the kurtosis to be observed.
Conclusion
Quantitative methods have been used to characterise and compare turbulent signals from
MAST edge plasmas and from BOUT simulations. Significant similarities have been observed,
but also several differences. The kurtosis results are encouraging, since intermittency is one of
the main characteristics of edge turbulence, and its dependence on timescale appears to be re-
produced by BOUT simulations. However the differencing and rescaling results suggest that the
process which creates the long-term correlations seen in experimental data may not be present
in BOUT. There could be several reasons for the suppression of high-frequency components in
the Fourier power spectrum, which could be resolved by furthersimulations using a finer mesh.
Longer simulation runs will assist in determining whether theobserved differences are due to
an incomplete description of the physics in the BOUT model, or arise from the short duration
of the simulations so far.
This work was funded jointly by the United Kingdom Engineeringand Physical Sciences
Research Council and by the European Communities under the contra t of Association between
EURATOM and UKAEA. The views and opinions expressed herein do not necessarily reflect
those of the European Commission.
References
[1] BOUT documentation and publications:http://www.mfescience.org/users/xu
[2] A. Sykes et. al. Nucl. Fusion41 1423 (2001)
[3] M.V. Umansky, et. al. Sherwood conference 2005.
[4] B.D. Dudson, R.O. Dendy, A. Kirk, H. Meyer and G.F. Counsell Plasma Phys. Control.
Fusion47 (2005) 885 - 901
[5] B.A. Carreras, B.Ph. van Milligen, M.A. Pedrosa, et. al. 1998Phys. Plasmas 5(10) 3632
[6] Greenhough J, Chapman S C, Dendy R O, Ward D J 2003PPCF Vol. 45 p.747
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Statistical analysis of plasma edge fluctuations in MAST and comparison with BOUT simulation results

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Statistical analysis of plasma edge fluctuations in MAST and comparison with BOUT simulation results

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