Edge turbulence in the Mega-Amp Spherical Tokamak

228 Pages·2008·8.89 MB·English

Edge turbulence in the Mega-Amp
Spherical Tokamak
(cid:0)
Benjamin D. Dudson
Trinity College
University of Oxford
A thesis submitted for the degree of
Doctor of Philosophy
Trinity 2007
Abstract
In a fusion power plant the transport at the plasma edge must be carefully
controlled so that power is deposited onto material surfaces designed to
handle it. There has been growing interest in the role played in this
transport by field-aligned filamentary structures, particularly with the
advent of fast high-resolution cameras able to resolve their fast dynamics.
This thesis presents experimental analysis of filamentary structures at the
plasma edge in the Mega Amp Spherical Tokamak (MAST), and compari-
sonwiththeBOUT2-fluid3Dsimulationcode. Datafromareciprocating
Langmuir probe close to the plasma edge has been analysed using sta-
tistical techniques, demonstrating the robustness of the differencing and
rescaling method. Combined with autocorrelation function and Fourier
spectra, this has been used to demonstrate the presence of correlations
between filaments over long times, and the destruction of this correlation
inditheringH-modeplasmas. Cameraimagesrecordedatarateof100kHz
of L-mode plasmas have been compared to data recorded simultaneously
by the reciprocating probe. This shows that the statistical properties of
data from the reciprocating probe are separated into those due to the
structure of individual filaments and those due to correlations between
filaments.
Camera images have been systematically analysed using a automated and
semi-automated codes to determine the properties of filaments at the
plasma edge, finding mode numbers n ∼ 30, typical filament widths of
∼ 9cm, and a mean toroidal velocity of 3.55±0.05km/s in the co-current
direction. Radial velocities in the range of 0.5−1.5km/s are observed.
The BOUndary Turbulence (BOUT) code has been modified and used to
simulate MAST L-mode plasmas, showing agreement with many aspects
of the experimental data, including similar toroidal widths (∼ 5cm), ra-
dial velocities up to ∼ 2km/s and time-scales. The BOUT simulations
show the presence of an electrostatic drift-type wave 1 − 2cm inside the
3
separatrix rotating toroidally in the electron diamagnetic drift direction
(counter-current) with a toroidal mode-number n ∼ 15 − 35 and a fre-
quency of ∼ 50kHz. This wave is broken up by turbulent dynamics close
to the separatrix, forming many smaller plasma “blobs” (n ∼ 60 − 90)
whilst accelerating in the toroidal direction from ∼ 0.7km/s to ∼ 1.9km/s
counter-current. Some of these blobs are then accelerated radially out-
wards into the SOL.
Detailed analysis of the relative magnitudes of drive terms in the reduced
Braginskii equations solved by BOUT has been performed. This indicates
a change in the dynamics across the separatrix, from a regime where 3D
effects are important to essentially 2D dynamics in the SOL.
Acknowledgements
There are many people without whom this thesis would not have been
completed. In particular I thank my supervisor at UKAEA Fusion, Dr
Andrew Kirk, for his enthusiasm, support and good advice over the past
three and a half years. In addition to his constant encouragement, andrew
worked with N. ben Ayed to calibrate the Photron camera. This allowed
field-linestobemappedontoimages, andisusedinchapter3. Ialsothank
Prof. Richard Dendy for supervising the statistical analysis presented in
Chapter 4, Dr Chippy Thyagaraja for his help understanding the BOUT
equations, Prof. Justin Wark for acting as my supervisor at Oxford, and
Prof. Howard Wilson for his support and patience whilst this thesis was
being written.
I am also grateful to the magnetic fusion group at LLNL lead by Dr Tom
Rognlien for allowing me to use the BOUT simulation code and for their
welcomingnatureandhospitality, whichmademytimespentinLivermore
such a wonderful experience. In particular, I thank Dr Maxim Umansky
for introducing me to BOUT and patiently explaining the workings of this
complicated code and numerical methods in general.
I also thank the students at Culham for making these last three and a
half years there so enjoyable, and wish them all the best of luck for the
future.
Finally, i thank my parents for their love, support and encouragement
over my many years as a student.
Contents
1 Introduction 1
1.1 Tokamaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Diverted plasmas . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 The scrape-off layer . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 Spherical tokamaks . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.4 The Mega-Amp Spherical Tokamak . . . . . . . . . . . . . . . 12
1.2 Energy confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Fluid description of plasma . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Braginskii equations . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.2 Plasma drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Plasma instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 BOUT simulation code 32
2.1 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Previous work with BOUT . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 MAST simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.1 Extension to double-null plasmas . . . . . . . . . . . . . . . . 40
2.4.2 Code limitations and stability . . . . . . . . . . . . . . . . . . 44
2.5 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5.1 BOUT Shadow . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5.2 BOUT Camera . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Photron camera image analysis 59
3.1 Edge diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.1 Langmuir probes . . . . . . . . . . . . . . . . . . . . . . . . . 60
i
CONTENTS ii
3.1.2 Fast camera diagnostics . . . . . . . . . . . . . . . . . . . . . 62
3.1.3 Camera and Langmuir probe comparison . . . . . . . . . . . . 66
3.2 Average image intensity . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Filament analysis methods . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.1 Field-line matching . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Filament widths and mode numbers . . . . . . . . . . . . . . . 76
3.3.3 Tests with artificial data . . . . . . . . . . . . . . . . . . . . . 81
3.3.4 Toroidal velocity measurement . . . . . . . . . . . . . . . . . . 86
3.4 L-mode filament properties . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4.1 Filament velocities . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4.2 Dependence on plasma parameters . . . . . . . . . . . . . . . 91
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4 Statistical analysis techniques 100
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.1 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Previous and current work . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 Analysis methods applied to MAST data . . . . . . . . . . . . . . . . 106
4.3.1 Autocorrelation function . . . . . . . . . . . . . . . . . . . . . 106
4.3.2 Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3.3 Relating the ACF and power spectra . . . . . . . . . . . . . . 107
4.3.4 R/S Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.5 Growth of Range . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.6 First Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.7 Differencing and Rescaling . . . . . . . . . . . . . . . . . . . . 111
4.4 Artificial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5 MAST datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.6.1 MAST plasma 6861 (L-Mode) . . . . . . . . . . . . . . . . . . 117
4.6.2 Data shuffling . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.6.3 MAST plasma 9031 (Dithering H-Mode) . . . . . . . . . . . . 123
4.6.4 MAST plasma 5738 (H-Mode) . . . . . . . . . . . . . . . . . . 125
4.6.5 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . 128
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
CONTENTS iii
5 BOUT simulation results 131
5.1 Comparison with observations . . . . . . . . . . . . . . . . . . . . . . 131
5.1.1 Structure of BOUT results . . . . . . . . . . . . . . . . . . . . 134
5.1.2 Mid-plane fluctuations . . . . . . . . . . . . . . . . . . . . . . 145
5.1.3 Divertor leg fluctuations . . . . . . . . . . . . . . . . . . . . . 149
5.2 Mechanisms and drive terms . . . . . . . . . . . . . . . . . . . . . . . 155
5.2.1 Plasma quantities . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.2.2 Drive terms and BOUT shadow . . . . . . . . . . . . . . . . . 161
5.2.3 Summary of BOUT mechanisms . . . . . . . . . . . . . . . . . 166
6 Conclusions 171
6.1 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
A BOUT Code 186
A.1 Braginskii equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
A.1.1 Density equation . . . . . . . . . . . . . . . . . . . . . . . . . 188
A.2 Backward Differentiation Formula . . . . . . . . . . . . . . . . . . . . 191
A.3 Geometry and derivatives . . . . . . . . . . . . . . . . . . . . . . . . 192
B Photron image enhancement: Spice-Weasel version 1.0 195
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
B.2 Using Spice-Weasel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
B.3 Processing scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
B.3.1 Processing commands . . . . . . . . . . . . . . . . . . . . . . 201
B.3.2 Useful script components . . . . . . . . . . . . . . . . . . . . . 205
B.4 Technical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
B.4.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B.4.2 Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
B.4.3 Processing scripts . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.4.4 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
B.4.5 Synchronisation code . . . . . . . . . . . . . . . . . . . . . . . 216
List of Figures
1.1 Drawing of stellarator Wendelstein 7-X . . . . . . . . . . . . . . . . . 5
1.2 Schematic of a tokamak . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 X-point and divertor configuration . . . . . . . . . . . . . . . . . . . 8
1.4 Schematic of plasma flows in the SOL . . . . . . . . . . . . . . . . . . 10
1.5 Large and small aspect-ratio tokamaks . . . . . . . . . . . . . . . . . 11
1.6 Origin of the diamagnetic drift . . . . . . . . . . . . . . . . . . . . . . 21
1.7 Direction of the electron diamagnetic drift in a tokamak . . . . . . . 22
1.8 Good and bad curvature . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.9 Good and bad curvature in a tokamak . . . . . . . . . . . . . . . . . 24
1.10 Electron drift wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.11 Illustration of spinning eddies with fluid velocity u and vorticity ω . . 27
2.1 Block diagram of BOUT . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Example of poloidal mesh produced by UEDGE . . . . . . . . . . . . 35
2.3 Density and vorticity structure of a blob . . . . . . . . . . . . . . . . 37
2.4 BOUT simulation results for a MAST plasma . . . . . . . . . . . . . 38
2.5 BOUT MPI communications . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 BOUT communications for double null plasmas . . . . . . . . . . . . 43
2.7 Comparison of spatial differencing schemes . . . . . . . . . . . . . . . 45
2.8 Floor function used to prevent negatives . . . . . . . . . . . . . . . . 48
2.9 Cartoon showing the emergence of negative density/temperature . . . 49
2.10 Example outputs of BOUT camera . . . . . . . . . . . . . . . . . . . 56
2.11 3D visualisation of BOUT results . . . . . . . . . . . . . . . . . . . . 58
3.1 Example of I data from mid-plane reciprocating probe . . . . . . 61
SAT
3.2 Full view of MAST from photron fast camera . . . . . . . . . . . . . 63
3.3 Processing of fast camera images . . . . . . . . . . . . . . . . . . . . 64
3.4 Image of plasma edge for shot 15368 . . . . . . . . . . . . . . . . . . 65
3.5 Images showing radial expansion and splitting of filaments . . . . . . 66
iv
LIST OF FIGURES v
3.6 Images showing interaction between filament and Langmuir probe . . 66
3.7 Image intensity and I signals . . . . . . . . . . . . . . . . . . . . . 67
SAT
3.8 Diagram of MAST plasma with photron camera . . . . . . . . . . . . 68
3.9 Image intensity and Abel inversion . . . . . . . . . . . . . . . . . . . 69
3.10 Light emission due to transient events . . . . . . . . . . . . . . . . . . 70
3.11 Log-normal plots of image intensity . . . . . . . . . . . . . . . . . . . 71
3.12 I measurements as a function of radius . . . . . . . . . . . . . . . 72
SAT
3.13 Fitting of image projection parameters . . . . . . . . . . . . . . . . . 74
3.14 Field alignment with filamentary structures . . . . . . . . . . . . . . 76
3.15 Field-lines overlaid on mid-plane photron view A . . . . . . . . . . . 77
3.16 Diagram of filaments in mid-plane camera images . . . . . . . . . . . 78
3.17 Average image intensity as a function of toroidal angle . . . . . . . . 79
3.18 Measurement of filament width for overlapping peaks . . . . . . . . . 80
3.19 PDFs of quasi-mode number from image data . . . . . . . . . . . . . 81
3.20 Artificial data used to test analysis code . . . . . . . . . . . . . . . . 82
3.21 Results for artificial data N=30, FWHM = 5.88 . . . . . . . . . . . . 83
3.22 Results for artificial data . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.23 Semi-automatic analysis of filament toroidal velocities . . . . . . . . . 86
3.24 Plasma current, q and line-averaged density n for three CDN plasmas 88
95 e
3.25 Peak separation, mode numbers and peak widths . . . . . . . . . . . 90
3.26 Manual field-line fitting to filaments . . . . . . . . . . . . . . . . . . . 91
3.27 Toroidal velocity measured for shot 15236 . . . . . . . . . . . . . . . 92
3.28 Mode number and FWHM as a function of time . . . . . . . . . . . . 93
3.29 q and line-averaged density parameter space . . . . . . . . . . . . . 94
95
3.30 Mode number and FWHM as a function of q . . . . . . . . . . . . . 95
95
3.31 Mean mode number and FWHM against the fit . . . . . . . . . . . . 97
4.1 Power spectrum for Langmuir probe measurements . . . . . . . . . . 104
4.2 Graphs reproduced from Carreras et.al. . . . . . . . . . . . . . . . . . 105
4.3 Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.4 Hurst exponent artificial data . . . . . . . . . . . . . . . . . . . . . . 114
4.5 Paired D (left) and I (right) signals from three MAST plasmas . 115
α SAT
4.6 Analysis of I data from L-Mode plasma 6861 . . . . . . . . . . . . 117
SAT
4.7 Analysis of I data from L-Mode plasma 6861 . . . . . . . . . . . . 119
SAT
4.8 Randomly shuffled I data . . . . . . . . . . . . . . . . . . . . . . 121
SAT
4.9 Randomly shuffled I data . . . . . . . . . . . . . . . . . . . . . . 122
SAT
LIST OF FIGURES vi
4.10 Analysis of I data from dithering H-mode plasma 9031 . . . . . . 123
SAT
4.11 Analysis of I data from dithering H-mode plasma 9031 . . . . . . 124
SAT
4.12 Analysis of I data from H-mode plasma 5738 . . . . . . . . . . . . 125
SAT
4.13 Analysis of I data from H-mode plasma 5738 . . . . . . . . . . . . 126
SAT
4.14 Persistent magnetic signal during H-Mode MAST plasma 5738 . . . . 127
5.1 Poloidal profile of RMS density fluctuations . . . . . . . . . . . . . . 133
5.2 Density contours showing filaments detached from the plasma . . . . 135
5.3 Density contours showing distortions to the plasma edge . . . . . . . 136
5.4 2D Plot of density perturbation (n ) with electric potential (φ) . . . . 137
e
5.5 Density perturbations n˜ and electric potential φ . . . . . . . . . . . . 138
e
5.6 Correlation of density perturbations . . . . . . . . . . . . . . . . . . . 138
5.7 Mean toroidal mode number for low-density SND plasma . . . . . . . 140
5.8 Density perturbations at the outboard mid-plane . . . . . . . . . . . 141
5.9 Filament sizes in the direction of motion and trailing . . . . . . . . . 142
5.10 Radial and toroidal velocity distributions . . . . . . . . . . . . . . . . 144
5.11 I signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
SAT
5.12 Fourier Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.13 Differencing and rescaling . . . . . . . . . . . . . . . . . . . . . . . . 147
5.14 kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.15 Inner and Outer strike-point density fluctuations . . . . . . . . . . . . 150
5.16 Target probe layout and motion of strike-points . . . . . . . . . . . . 151
5.17 Example of signal from inner target probe . . . . . . . . . . . . . . . 152
5.18 I mean and fluctuation levels at strike-points . . . . . . . . . . . . 152
SAT
5.19 I mean and fluctuation levels for strike-points . . . . . . . . . . . 153
SAT
5.20 Fluctuation / mean I for inner and outer strike points . . . . . . . 154
SAT
5.21 δφ/T against δn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
e
5.22 Ion and electron temperature against density . . . . . . . . . . . . . . 157
5.23 Density-Temperature exponent T ∝ na . . . . . . . . . . . . . . . . . 158
i
5.24 Electron and ion temperature dependency . . . . . . . . . . . . . . . 159
5.25 Electron and ion temperature RMS fluctuation levels [eV] . . . . . . . 160
5.26 RMS percentage importance of density drive terms . . . . . . . . . . 164
5.27 RMS percentage importance of temperature drive terms . . . . . . . . 167
5.28 Dominant drive terms for density and temperature . . . . . . . . . . 168
5.29 Density and potential on flux surfaces . . . . . . . . . . . . . . . . . . 169
B.1 Gamma enhancement, factor > 1.0 . . . . . . . . . . . . . . . . . . . 202