Gas Puff Imaging Studies of Tokamak Edge Physics in the

National Spherical Torus Experiment

by

Yancey Sechrest

B.S., University of Arizona, 2007

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

2014

This thesis entitled:Gas Puff Imaging Studies of Tokamak Edge Physics in the National Spherical Torus Experiment

written by Yancey Sechresthas been approved for the Department of Physics

Prof. Tobin Munsat

Reader #2:

Reader #3:

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above

mentioned discipline.

iii

Sechrest, Yancey (Ph.D., Plasma Physics)

Gas Puff Imaging Studies of Tokamak Edge Physics in the National Spherical Torus Experiment

Thesis directed by Prof. Tobin Munsat

In order to be viable, Next-step fusion devices must overcome two pressing problems: they

must be able to achieve high levels of confinement while also handling potentially damaging heat

loads on material surfaces. The study of plasma edge physics promises solutions to both problems

because the plasma edge, being the boundary between confined and unconfined regions, plays a

key role in determining the global confinement and the plasma interaction with material surfaces

(e.g. edge transport barriers, pedestal evolution, and edge localized modes). However, the steep

gradients in density and temperature in the plasma edge that drive strong fluctuations in plasma

parameters require measurements of fluctuations with high spatial and temporal resolution. By

measuring drift scale (kyρs < 2) fluctuations for frequencies less than ∼ 200 kHz, Gas Puff Imaging

(GPI) meets these requirements while providing two-dimensional coverage at a large number of

measurement locations. This dissertation presents GPI studies of transitions from low to high

confinement regimes (L-H transitions) and Edge Localized Modes (ELMs). In 2010, a study of

L-H transitions with the GPI diagnostic revealed quasi-periodic reductions in the scrape-off-layer

turbulence levels during the 30 ms preceding the transition. The two-dimensional flow fields for

these “quiet-periods”, estimated from the GPI data by a pattern-matching velocimetry technique,

exhibit intriguing similarity with the Drift Wave - Zonal Flow paradigm, a leading candidate in

explaining L-H transitions. Following this study, a survey of GPI data from RF heated H-mode

plasmas near the L-H power threshold identified short-lived, coherent oscillations in edge emission

preceding the ELM crash. These observations provide detailed two-dimensional dynamics of the

growth, filamentation, and crash of the ELM event, which could improve our understanding through

comparison with nonlinear simulation. Cross diagnostic comparisons of GPI and Beam Emission

Spectroscopy measurements of edge fluctuations are also presented.

Dedication

To my lifeline, my family.

v

Acknowledgements

Two people deserve special recognition for their guidance and support. Chief among these

two is Tobin who was always patient and supportive. He encouraged me to explore and learn even

if it meant a slip in my productivity. Stewart Zweben also deserves special praise. His insight and

critical feedback have greatly influenced my critical thinking and writing, and I am very grateful

for his guidance.

During my work I have had the great opportunity to collaborate with many good scientists

who I admire. Among them are Ricky Maqueda, Devon Battaglia, and Dave Smith. Also, Jim

Myra, Dan D’ippolito, and Dave Russel of the Lodestar Research group. Our work would not have

been possible without the support of the whole NSTX team.

Grad school would have been nigh unbearable without the support of my close friends.

Whether watching movies at Steve and Carrie’s, playing Tetris Attack with Kevin, Climbing with

Travis and Andy, losing at Softball with Eric and the other Zeroes, or geeking out about all things

nerd with J.R., I can honestly say I wouldn’t have finished without all of you. Sami, Adam and

Carl also provided helpful discussion, or much needed stress-relief via lab basketball.

As noted in the dedication, my family has supplied me with a never-ending source of love

and support. Mom and Dad, you’ve always been willing to go to bat for me, and I’m incredibly

thankful for the opportunities you’ve provided for me. Tuck, thanks for always being there to chat,

whether about games or quarter-life crises. I’ll be ready to do this all again for your defense.

Finally, Tory. You’ve done so much to give me perspective, and keep me sane and grounded.

Thanks for putting up with me and my idiosyncracies.

Contents

Chapter

1 Introduction 1

1.1 Global Energy Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Nuclear Fusion and its Potential as an Energy Source . . . . . . . . . . . . . . . . . 2

1.3 Requirements for a Burning Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Magnetic Confinement and Force Balanced Equilibria . . . . . . . . . . . . . . . . . 6

1.5 The Tokamak Reactor Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 The Spherical Torus Concept and NSTX . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 Overview of Presented Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 The Gas Puff Imaging Diagnostic 12

2.1 The NSTX GPI System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Atomic Physics Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Direct Comparison of GPI and BES Measurements of Edge Fluctuations in NSTX 20

3.1 The Beam Emission Spectroscopy Diagnostic . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Plasma Conditions and NSTX Operation . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Relative Diagnostic Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Fluctuation Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Fluctuation Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6 Cross-Diagnostic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vii

3.7 Characteristic Time and Length Estimates . . . . . . . . . . . . . . . . . . . . . . . . 35

3.8 Gas Puff Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.9 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Velocimetry 44

4.1 Time Delay Estimation for Motion Estimation . . . . . . . . . . . . . . . . . . . . . 44

4.2 The Optical Flow velocity estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Pattern Matching velocity estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 The Hybrid Optical Flow and Pattern Matching Velocimetry Algorithm (HOP-V) . 51

4.5 A Note On Post-Processing Techniques: a Navier-Stokes Inspired Smoothing Algorithm 52

5 Measurement of 2D flows in the edge and SOL preceding L-H transitions 56

5.1 The L-H transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 The Drift-Wave Zonal Flow Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 The Predator-Prey model of the L-H transition . . . . . . . . . . . . . . . . . . . . . 67

5.4 GPI Observations of Flows preceding L-H transitions . . . . . . . . . . . . . . . . . . 69

5.5 Time averaged-flow profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.6 turbulence “Quiet-Periods” and quasi-periodic velocity fluctuations . . . . . . . . . . 72

5.7 Spatial Structure of 3 kHz mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.8 Flow shear and Reynolds stress calculations . . . . . . . . . . . . . . . . . . . . . . . 81

5.9 Quiet-periods as Limit Cycle Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Precursor Fluctuations During Small ELMs in NSTX 90

6.1 MHD stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2 An Overview of ELMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3 NSTX observations of ELM precursors . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.4 Operational Parameters and Plasma Conditions . . . . . . . . . . . . . . . . . . . . . 97

6.5 Precursor Oscillations in GPI Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . 99

viii

6.6 Wavenumber and Frequency Characterization of the Precursor Mode . . . . . . . . . 103

6.7 Quantification of Edge Deformation During Precursor Evolution . . . . . . . . . . . 105

6.8 Magnetic Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.9 Pedestal Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.10 Concluding Remarks on the Nature of the Precursor Mode . . . . . . . . . . . . . . . 111

7 Directions for Future Work 112

Bibliography 114

Tables

Table

3.1 Shot list for this study containing the shot number, toroidal field at the magnetic

axis, plasma current, neutral beam heating power, average density, and GPI puff

timing. Values are taken at the time of the gas puff. . . . . . . . . . . . . . . . . . . 23

3.2 Comparison of poloidal correlation lengths, decorrelation times, and velocities esti-

mated from GPI and BES correlation functions. . . . . . . . . . . . . . . . . . . . . . 38

3.3 Poloidal correlation length estimates from 60-0 ms before, and 30-90 ms after the

gas puff trigger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.1 Shot database for this study including the shot number, timeframe of interest,

toroidal magnetic field, plasma current, and RF heating power . . . . . . . . . . . . 98

Figures

Figure

1.1 Left: Diagram of NSTX design showing toroidal field (TF) coils, poloidal field (PF)

coils, center stack containing ohmic heating solenoid, and carbon wall tiles. Right:

Equilibrium reconstruction showing contours of poloidal flux. . . . . . . . . . . . . . 11

2.1 (a) Schematic of GPI system with view from machine center stack looking out.

(b) positioning of GPI view in R and Z with flux contours for typical operational

parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Diagram of GPI optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Cross-correlation map for each pixel with reference pixel at [17,40] for L-mode tur-

bulence. Green trace indicates field line trajectory through GPI gas cloud. . . . . . . 15

2.4 CR calculations of exponent for ne, α, and exponent for Te, β, used in estimation of

GPI Dα signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Comparison of Thomson scattering profiles of ne and Te with GPI emission profile

(blue curve) and R=140 cm BES radial array channels (black and red dashed lines).

The red dashed lines indicate the position of the inner and outer BES poloidal arrays. 23

xi

3.2 (a) R = 140 cm BES channel positions (red, diamonds) and GPI viewing area (blue

polygon) plotted over contours of poloidal flux for NSTX shot 141254. Flux surfaces

are labeled by their midplane r/a value, and the separatrix is indicated by the solid

black line. BES channel positions and the corners of the GPI view are plotted using

cylindrical coordinates. (b) BES inner poloidal array (red diamonds) plotted in

toroidal coordinates with GPI points (blue) for similar flux value. Magnetic field

line traces are plotted as dashed lines, and the dot-dashed line traces the generalized

poloidal direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Traces of (a) GPI raw (gray) and 〈I〉 (blue), and (b) δI. Plots (c) and (d) are BES

traces. All traces taken at the 0.85 ΨN location. . . . . . . . . . . . . . . . . . . . . 26

3.4 Traces of δIRMS/〈I〉 for BES (red) and GPI (blue) for ΨN = 0.85. . . . . . . . . . . 27

3.5 PDFs of (a) GPI and (b) BES intensity from 10 ms period. Dashed Lines indicate

Gaussian PDFs with similar mean and variance. . . . . . . . . . . . . . . . . . . . . 28

3.6 Scatterplot of skewness and kurtosis values for BES (filled triangles) and GPI (filled

squares) at r − rsep ≈ 2.9 cm, and BES (open triangles) and GPI (open squares) at

r − rsep ≈ 0.7 cm. The dashed line represents a second degree polynomial fit to the

GPI data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Contour plot of time-lagged cross-correlation between GPI and BES signals versus

time. Peak correlation value persists in time, but constant linear drift is present. . . 32

3.8 Traces of (a) time-lagged cross-correlation between GPI and BES, and (b) time traces

of GPI and BES intensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.9 Plots (a) and (b) are traces of cross-correlation vs. radius for GPI pixels with a BES

reference channel. Plots (c) and (d) are contour plots of the cross-correlation vs.

radius and time-lag. The black dashed line indicates the separatrix location, and the

dashed red line indicates the radial location of the BES reference channel. . . . . . . 33

3.10 Traces of (a) cross-coherence, (b) cross-spectral density, (c) cross-phase, and (d)

phase uncertainty for a BES coord and the closes GPI pixel in the R-Z plane. . . . . 34

xii

3.11 Plots of: the Cross-Coherence between poloidaly separated channels for (a) GPI and

(c) BES, and Autopower spectra for GPI (b) and BES (d). . . . . . . . . . . . . . . 36

3.12 plots of: (a) and (e) Time-lagged cross-correlations (solid) with envelope functions

(dashed), (b) and (f) zero-lag envelope peak correlation versus poloidal separation,

(c) and (g) envelope peak correlation versus time-lag to peak, and (d) and (h) time-

lag to envelope peak correlation versus poloidal separation. Values for poloidal cor-

relation length, and decorrelation time represent the 1/e length for a Gaussian fit to

the corresponding plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.13 Scatterplots comparing (a) poloidal correlation lengths estimates, (b) decorrelation

time estimates, and (c) TDE velocity estimates. . . . . . . . . . . . . . . . . . . . . 37

3.14 Time traces of low-pass filtered BES intensity for varying (a) Xsep = r − rsep and

(b) Z. Times are relative to the GPI gas puff timing. . . . . . . . . . . . . . . . . . . 39

3.15 Continous Wavelet Transforms of BES fluctuations normalized to 100 Hz low-pass

filter for shots 138845 (a) and 141249 (b). Time traces of the average GPI intensity

is plotted above the CWTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.16 (a) and (c): Block-averaged autopower spectra of fluctuations normalized to mean.

(b) and (d): coherence for BES channels separted by 4.8 cm. Black traces are

spectra for 60-0 ms before the gas puff trigger, and red traces are spectra for 30-90

ms following the trigger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1 Pattern Matching vector fields for (a) unsmoothed and (b) smoothed. Profiles of vx

and vy for unsmoothed, (c) and (d), and smoothed, (e) and (f). Red traces are the

imposed velocity field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Traces of (a) plasma current and injected neutral beam power, (b) diverter Dα light,

(c) Energy confinement times, and (d) plasma stored energy. Black traces are for

an ohmic L-mode shot, and red traces are for a neutral beam driven H-mode. L-H

transition timing indicated by dashed red line. . . . . . . . . . . . . . . . . . . . . . 58

xiii

5.2 Thomson scattering profiles of electron temperature (top) and electron density (bot-

tom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Traces of (a) fraction of GPI light in SOL, (b) GPI raw signal near separatrix, and

(c) normalized GPI fluctuation level. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4 Conceptual picture of the drift wave zonal flow paradigm. Gradients drive insta-

bilities to turbulent state. Turbulence drive anomalous transport via fluctuations.

Turbulence also self-generates zonal flow via Reynolds stresses. Zonal flows regulate

turbulence via shear suppression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.5 Illustration of the drift wave mechanism. The background density gradient is in the

−x direction, and the magnetic field is in z. E × B circulation around potential

perturbation (blue contours) pushes high density from −x and low density from +x

producing a propagation of the perturbation in +y. . . . . . . . . . . . . . . . . . . 63

5.6 Conceptual picture of limit cycle oscillation process. System evolves between a high

turbulence, low flow state and a high flow, low turbulence state. If the input power

is sufficient to steepen the density gradient during a period of suppressed turbulence,

then a mean flow shear develops and a transition to H-mode is observed. . . . . . . . 69

5.7 (a) Time averaged velocity field superimposed on time averaged GPI intensity con-

tours (gray contours). The separatrix is indicated by the dot-dashed line, and the

cropped field of view is indicated by the dashed box. The maximum time-averaged

velocity magnitude for the cropped region is 1.5 km/s, while the maximum instan-

taneous velocity magnitude is 7.4 km/s. The +x direction is radially outward. (b)

radial profiles of time averaged poloidal flow several shots for 10.5 µs preceding the

L-H transition. RMS values of the fluctuating velocity are shown for shot 135042. . . 71

5.8 Normalized GPI intensity (top), poloidal velocity (middle), and radial velocity (bot-

tom) ∼1 cm inside the separatrix. All traces represent poloidally averaged quantities

from shot 135042. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xiv

5.9 Traces from shot 135042 of Fsol filtered (a), and poloidally averaged poloidal velocity

inside (b), at (c), and outside (d) the separatrix. Fsol filtered has had the low

frequency components above zero frequency and below 1 kHz removed. Gray bars

indicate time periods where Fsol filtered is below 0.16. . . . . . . . . . . . . . . . . 74

5.10 Spectrogram of Fsol for the L-mode portion (t ≈ 0.215 − 0.245) of shot 135042

plotted with a linear color scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.11 Map of phase differences of ∼2.4 kHz fluctuations for: (a) GPI signal with reference

GPI signal, (b) vpol with reference vpol, (c) GPI signal with vpol. Reference signals

are at x ≈ 0cm, y ≈ 11 cm. Plots (a) and (c) have had the phase discontinuity

remapped so that the contour plots appear smooth. Plots cover the time range

t ≈ 0.241− 0.243 ms of shot 135042. The dashed-line indicates the separatrix. . . . . 77

5.12 Phase of ∼3 kHz mode plotted vs spatial cordinates for GPI signal and poloidal

velocity for periods of shots 135042-135045 with “rotating mode” visible in band-

pass filtered. Time periods are 512 frames (1.8 msec) and begin at: 135042(black)

t=0.241, 135043(gold) t=0.238, 135044(green) t=0.230, 135045(red) t=0.236 s. . . . 78

5.13 Coherence and phase plots versus frequency for poloidal velocity signals separated

by ∼12 cm poloidally at ∼1 cm inside the separatrix. . . . . . . . . . . . . . . . . . 80

5.14 Plot shows coherence of poloidal velocity vs. y separation for 2.8 kHz mode (�)

and background turbulence between 20-30 kHz (4) for poloidal velocity of shot

135042. Velocities are measured at ∼1 cm inside the separatrix. Solid lines are fits

to exponential decays with correlation lengths of 56 cm for the 2.8 kHz mode and 4

cm for the background turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.15 Autopower spectra of poloidal velocity at 3 radial positions for t ≈ 0.240 − 0.243 s

of shot 135042. The feature at 2.5 kHz does not appear to shift in frequency. . . . . 81

xv

5.16 Traces from shot 135042 of Fsol, poloidal velocity (km/s), shearing rate (MHz), and

the Reynolds stress (km2/s2). Poloidal velocity and Reynolds stress are taken at ∼1

cm inside the separatrix, and the shearing rate is taken at the separatrix. Traces

have been smoothed with a 3 point boxcar average. . . . . . . . . . . . . . . . . . . . 83

5.17 Poloidal velocity profiles for ∼30 ms of shot 135042 are plotted in (a). The averages

are over times where Fsol is greater than its mean value (dashed - bursty) and less

than its mean value (solid - quiet). “Error bars” indicate ±1 σ(standard deviation)

about the average value for the low Fsol case, and indicate the level of fluctuation.

Standard deviations are shown in (b), and skewness is shown in (c). . . . . . . . . . 84

5.18 Traces of Fsol, poloidal velocity (km/s), shearing rate (ξ), and the Reynolds shear

stress. Traces are from a SOLT turbulence simulation illustrating the bursty regime.

The frequency of bursts is 3.6 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.19 Radial profiles of the Reynolds shear stress averaged over the poloidal direction and

10 ms directly preceding the L-H transition for 4 shots. Error bars indicate estimated

level of uncertainty assuming uncertainties in the velocity measurments of 0.5 km/s. 86

5.20 Radial profiles of the average poloidal velocity, and the amplitude of the 3 kHz flow

feature during the quiet period oscillations. . . . . . . . . . . . . . . . . . . . . . . . 88

6.1 Time traces of RF heating power PRF , plasma current IP , Dα light, and line-

integrated density N for typical shot from this study. Neutral beams are also used

early in the shot for plasma conditioning. The shaded region indicates the time

period for traces plotted in Figure 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2 Multiframe image stills of an ELM event with precursor intensity fluctuations from

shot 141918. The time between frames is ∼7.5 µs. Distinct mode structure can be

seen in precursor oscillations leading to the ejection of the filament in the last two

frames. The approximate location of the separatrix is indicated by the dashed line. . 100

xvi

6.3 Time traces of (a) scrape-off layer fraction FSOL and (b) integrated edge intensity

Iedge corresponding to shaded timeperiod in Fig. 6.1. Traces show edge intensity

fluctuations preceding an ELM at 0.2425s, and an ELM-induced back transition at

0.245s. Low level fluctuations can also be seen near 0.244s. . . . . . . . . . . . . . . 101

6.4 Two-dimensional slices through GPI data with one cut at y=15.5 cm (a), and one

at x=10.5 cm (b). Precursor fluctuations are very distinct brightness pulses which

appear to move upward, or in the electron diamagnetic direction indicated by the tilt

of structures in (b). Structures are also seen to drift radially outward as indicated

by +x motion in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.5 Traces of (a) Average perpendicular velocities, and (b) FSOL for many ELM events

from the shots database. Velocities are measured ∼2 cm inside the separatrix. Tim-

ings are relative to the peak FSOL for each event. . . . . . . . . . . . . . . . . . . . . 102

6.6 Wavelet scalogram of integrated edge intensity, Iedge accompanied by the time trace

of the scrape-off layer fraction for shot 141919. The power spectrum shows significant

power at the 20 kHz scale during the ELM precursor fluctuations. The shaded region

in the wavelet scalogram indicates where edge effects become important, and the

white contour indicates the 95% significance level. . . . . . . . . . . . . . . . . . . . 104

6.7 Scatter plot of frequency of precursor fluctuations against the perpendicular wavenum-

ber derived from GPI intensity fluctuations. Red triangles are events that lead to

an ELM or back-transition, while blue squares are edge intensity fluctuations that

do not lead to an ELM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.8 Example (a) image frame from shot 141917 and (b) xedge function overlayed on

intensity contours. Maximum radial excursion relative to EFIT separatrix location

is plotted in (c), and the edge curvature, κ corresponding to this point is plotted in

(d). The Dashed line indicates the time point of image (a). . . . . . . . . . . . . . . 105

xvii

6.9 Scatterplot of minimum edge curvature, κmin and maximum radial excursion rela-

tive to the EFIT separatrix location for several ELM precursor events (diamond).

Intensity fluctuations not leading to an ELM are also included (square). . . . . . . . 107

6.10 Time traces of (a) Iedge and (b) low-pass filtered magnetic signals from shot 141917.

The black trace has been lowpass filtered at 200 kHz, and the red trace has been

bandpass filtered around 20 kHz. Magnetics traces are strongly correlated with

fluctuations in edge intensity as shown in the lagged correlation plot (c). Absolute

values of the correlation coefficient reach 0.8 for periods during the precursor activity.109

6.11 Pedestal parameters for height hped and width ∆ped are extracted from a modified

tanh fit to the electron pressure profile (inset) and a comparison of several shots is

presented. Squares are 0.6-1.0 MW RF shots, triangles are Ohmic H-mode shots,

stars are ∼ 1.0 MW NBI shots, and the diamond is a 4 MW NBI heated shot. . . . . 110

Chapter 1

Introduction

1.1 Global Energy Challenges

As populations increase and industrialization spreads, the energy needs of the world will

continue to rise. Providing for the world’s energy needs over the next century poses a number

of serious challenges. Since the industrial revolution, much of the world’s energy production has

come from the burning of fossil fuels (e.g. oil, coal, and natural gas) [21], and the burning of these

fuels has led to the release of massive quantities of greenhouse gases (GHG). The increase in GHG

emission has been correlated with the rise of global mean temperatures, and the current scientific

consensus is that anthropogenic GHG is driving significant changes in the global climate [66]. These

changes in climate have wide ranging effects including impacts on water supply and crop yield, and

harsher impacts of natural dissaters on natural and human systems [67].

In addition to the environmental impacts, it must be recognized that fossil fuels are a finite

resource, and many concerns surround the remaining reservoirs of these resources, their accessibility,

and their ability to meet future energy needs [41]. While predictions of the depletion time of fossil

fuel reservoirs are uncertain, one prediction is that oil and gas resources will be depleted in 40

years, and coal resources will be depleted on the order of 100 years [125]. As fossil fuels become

more scarce and difficult to extract, other energy generation methods will need to be tapped. These

include nuclear fission, biofuels, and renewable sources such as hydroelectric, wind and solar [21].

Scientific innovation will be required to meet the energy needs of the future in a sustainable way

with minimal environmental impact.

2

1.2 Nuclear Fusion and its Potential as an Energy Source

Fusion is a nuclear reaction in which two atomic nuclei collide at high energy and fuse to

create a single, heavier nucleus in addition to other energetic products. The cross-section for a

fusion reaction depends on the energy, or temperature, of the reactants. The fusion reactions

with cross-sections that peak at the lowest energies, and consequently are easiest to achieve in lab

experiments, are:

D + T = 4He(3.52MeV ) + n(14.06MeV ), (1.1)

D +D = 3He(0.82MeV ) + n(2.45MeV ), (1.2)

D +D = T (1.01MeV ) + p(3.03MeV ), (1.3)

D +3 He = 4He(3.67MeV ) + p(14.67MeV ). (1.4)

The cross-section for Deuterium-Tritium (D − T ) fusion is much greater than the other processes

at experimentally viable energies in the range 10-100 keV, and peaks around 100 keV or 1.16

trillion Kelvin. For this reason, first generation fusion reactors are expected to operate using D−T

fusion, and so it is currently the most important reaction. Tritium is radioactive and a controlled

substance, however, so research experiments often use D −D fuel (i.e. pure deuterium).

Fusion energy production promises a number of benefits over other energy solutions. For

comparison, a fission reaction involving U −235 yields 0.86 MeV per nucleon, and a D−T reaction

produces 3.52 MeV per nucleon. In terms of specific energies, Coal has a specific energy of ∼30

MJ/kg, fission with U−235 has a specific energy of 8.3×107 MJ/kg, and D−T fusion has a specific

energy of 3.4× 108 MJ/kg. In addition to its high specific energy, the deuterium in D− T fuel is a

light element and naturally abundant. Tritium is radioactive, and more difficult to obtain though

options do exist. The most appealing option is for the fusion reactor to breed the required tritium

fuel in a lithium blanket surrounding the device as lithium is relatively abundant. Other sources of

tritium include heavy water fission reactors. Compared to other fuels used for energy production,

D − T fuel has a high energy density, and is relatively abundant source of energy, assuming the

3

issues surrounding tritium supply can be resolved.

While a fusion reactor will become radioactive due to neutron bombardment of construction

materials, the disposal of the low level radioactive waste from a fusion reactor is expected to be

simpler and significantly cheaper than disposal of high level radioactive waste from a fission reactor

[124]. The bulk of the radioactive waste from the fusion reactor can be disposed of by shallow

land burial, and radioactivity of all materials decays to below safety limits on the time scale of 100

years or less. Still, the cost of waste disposal must be included in assessments of the economics of

a possible fusion reactor.

The idealized picture of fusion energy is incredibly appealing, especially in light of the envi-

ronmental and resource related challenges currently on the global energy horizon. Fusion promises

energy free of GHG emissions from a relatively abundant, high energy density fuel source. The only

drawback being the production of easily managed, short-lived radioactive waste. For this reason,

the study of fusion has continued to captivate scientists for the better part of a century, though at

times a working fusion reactor seems to resemble Ahab’s white whale.

1.3 Requirements for a Burning Plasma

To fuse, two particles must collide with sufficient energy to overcome the Coulomb repulsion

barrier. As discussed in the previous section, this requires a great deal of energy. To drive an

appreciable number of fusion reactions, the population of plasma ions must be heated to very high

temperatures (T ≥ 10 keV). Intuitively, we can expect that the yield of fusion energy will depend

on the density of plasma ions and the plasma temperature. The actual rate of fusion reactions is

given by [158]

R =

∫ ∫σ(v′)v′f1(v2)f2(v2)d

3v1d3v2, (1.5)

where v′ is the relative velocity, σ(v) is the cross-section for the fusion reaction, f is the distribu-

tion function, and subscript 1, 2 refers to ion species. Using Maxwellian distributions for fi and

4

performing some simplifications this becomes

R = 4πn1n2( µ

2πT

)3/2 ∫σ(v′)v′3 exp

(− µv′2

2T

)dv′. (1.6)

This has the formR = n1n2〈σv〉, so the thermonuclear power per unit volume can then be expressed

as

pTn = n1n2〈σv〉E (1.7)

where E is the energy released per reaction. For a given n = n1 + n2, the power is maximized for

n1 = n2 = 12n, so the power can be written as

pTn =1

4n2〈σv〉E . (1.8)

In D − T fusion, the resulting neutron carries 4/5 of the released energy, and the helium

nucleus, or α particle, carries the remaining 1/5. The neutrons are neutral particles, and are not

affected by electric and magnetic forces. Therefore they easily escape the plasma, and, ideally,

deposit their energy into a coolant used for heat exchange and energy production. The alpha

particle, though, is a charged product, and so will be confined with the plasma where it transfers

its energy to the plasma ions through collisions. Thus the alpha particle will lead to plasma heating

per unit volume given by

pα =1

4n2〈σv〉Eα, (1.9)

and the total α particle heating is given by

Pα =

∫pαd

3x (1.10)

In steady-state operation, the power loss PL is balanced by external heating PH and Pα

giving

PL = PH + Pα. (1.11)

The power loss is defined as

PL = −dWdt

=W

τE(1.12)

5

where

W =

∫3nTd3x (1.13)

is the plasma stored energy and τE is the energy confinement time. If the α heating is large enough

to completely offset the power loss PL then the external heating systems can be turned off, and

the fusion reaction will be self-sustaining. This is called ignition, and the plasma is said to be a

burning plasma. A rough estimate of the requirements for ignition may be obtained from the above

definitions by assuming a constant density and temperature. The result is

nτE >12

〈σv〉T

Eα. (1.14)

In the temperature range 10-20 keV this can be approximated as

nTτE > 3× 1021m−3keV s. (1.15)

The triple product nTτE is revealed as an important metric of plasma performance. This makes

sense intuitively because this relation states that a burning plasma requires a sufficient energy

density nT be maintained for an adequate time τE so that many fusion reactions can occur.

Another important quantity is the plasma Q factor, the ratio of thermonuclear power pro-

duced to the externally supplied heating power

Q =14n

2〈σv〉EVPH

. (1.16)

Important values of Q are:

• Q = 1: heating power is equal to thermonuclear power generated, break even point ignoring

electrical efficiencies.

• Q = 5: Alpha particle heating of D − T fusion equals supplied heating power.

• Q ∼ 5: D−T Break even assuming electricity generation efficiency of 33%, heating efficiency

of 75%.

• Q ∼ 20: Reactor relevant Q values. Assumes above efficiencies and 25% power recirculation.

6

• Q→∞: Burning Plasma. No External heating required.

The Japanese tokamak JT-60 currently holds the world record for achieving Q=1.25, but this

number is extrapolated to D − T fusion yield based on an experimental result using D − D fuel

[48]. The International Thermonuclear Experimental Reactor (ITER) currently under construction

in Cadarache, France is expected to achieve Q > 5 [104].

1.4 Magnetic Confinement and Force Balanced Equilibria

As we saw in the last section, a viable fusion reactor must be able to confine a dense plasma

long enough to heat the plasma to sufficient temperatures for fusion reaction to take place. From

basic electromagnetism, we know that charged particles moving in a magnetic field experience a

force F = qv×B. The direction of motion and the resulting force are always perpendicular, so the

particles will undergo circular motion in a plane perpendicular to B described by the equation

ma⊥ = mv2⊥rL

= qv⊥B. (1.17)

The radius of the orbit is then rL = mv⊥/(qB). The particles motion in the parallel direction is

unaffected. The conclusion from this is that charged particles in a magnetic field will be confined

to the magnetic field lines. This simple concept is the basis for the field of magnetic confinement

fusion.

The simplest description of a plasma is given by the magnetohydrodynamic (MHD) model

that treats the plasma as a single conducting fluid. The plasma is assumed to be neutral, so the net

charge is zero. Furthermore, fluid motions are taken to be slow compared to characteristic times

scales of the plasma so that the displacement current is neglected. The MHD set of equations are:

∂ρm∂t

+∇ · (ρmU) = 0 (1.18)

ρm∂U

∂t+ ρmU · ∇U = J×B−∇P (1.19)

J = σ(E + U×B) (1.20)

d

dt(Pρ−γm ) = 0. (1.21)

7

When combined with Maxwell’s equations this yields a complete set of equations describing the

plasma.

The condition for static plasma equilibrium is given by Eqn. 1.19. For an equilibrium with

zero flow this gives the equation for a force-balanced equilibrium

J×B = ∇P. (1.22)

From this it can be seen that both

B · ∇P = 0 (1.23)

and

J · ∇P = 0, (1.24)

so the current and magnetic field lie in surfaces of constant pressure. Hopf’s Theorem from topology

tells us that the geometric form of the closed constant pressure surface that satisfies the above con-

straints must be a torus. Thus plasma equilibria and confinement devices with closed surfaces are

inherently toroidal objects; from self-organized systems like the spheromak and field-reversed con-

figuration, to externally imposed equilibria such as the tokamak, and reversed field pinch. Toroidal

equilibria may be described using cylindrical coordinates with the center axis of the torus chosen

as the z-axis, and φ then corresponds to the long dimension around the torus or toroidal direction.

The short dimension around the torus is called the poloidal dimension. The radius from the z-axis

to the torus center line is the major radius, and the radius from the torus centerline to the plasma

boundary is the minor radius.

For axisymmetric equilibria derivatives with respect to φ are zero, so B = ∇×A gives

Br = −∂Aφ∂z

(1.25)

Bz =1

r

∂

∂r(rAφ). (1.26)

Thus Br and Bz can be expressed in terms of a single function ψ = rAφ. The function ψ is related

to the poloidal magnetic flux in the torus. The requirement that ∇ ·B then gives

Br∂ψ

∂r+Bz

∂ψ

∂z= B · ∇ψ = 0, (1.27)

8

and B is found to lie in surfaces of constant ψ. It follows then that the plasma pressure may be

expressed as a function of ψ. Similarly, the φ component of the force balance equation gives

B · ∇F = 0, (1.28)

where the function F = rBφ is related to the poloidal current flux.

The above relations can be used to derive a single equation that, upon solution, gives the form

of the axisymmetric magnetostatic equilibrium. This equation is derived from the radial component

of the force balance equation

JφBz − JzBφ =∂P

∂r. (1.29)

Using Ampere’s law and the definitions of ψ and F this can be expressed as

r∂

∂r

1

r

∂ψ

∂r+∂2ψ

∂z2= −µ0r2

d

dψP (ψ)− 1

2

d

dψF (ψ)2. (1.30)

This is the Grad-Shafranov equation, and its solution gives the equilibrium flux function ψ in

terms of the two free functions P (ψ) and F (ψ). Solutions are usually computed numerically by

first specifying the two free functions. An important class of solutions are those with constant ψ

surfaces forming nested toroidal surfaces as these solutions describe a plasma equilibrium that may

be magnetically confined and isolated from material surfaces. Example equilibrium flux surfaces

for the low aspect ratio spherical torus NSTX is shown in Fig. 1.1.

1.5 The Tokamak Reactor Concept

In the previous section we outlined some of the requirements for an axisymmetric plasma

equilibrium. Now we can put these pieces together to conceptualize a device that could in theory

confine a plasma in stable equilibrium. First, remember that Hopf’s theorem tells us that the

equilibrium should be toroidal, so we will begin by imposing a toroidal field using a set of external

magnet coils. For magnetic coils with a straight leg on the inside of the torus, the toroidal field

can be obtained from the integral form of Ampere’s law, and is given simply by Bφ = µ0NI/2πr

where N is the number of coils and I is the current per coil. Conveniently, this simplifies the

9

Grad-Shafranov equation since

F = rBφ = constant. (1.31)

Using the toroidal component of Ampere’s law and the Grad-Shafranov equation with ∂F/∂ψ = 0,

we see that a toroidal current

Jφ = rdP

dψ(1.32)

is required to provide equilibrium. One solution is to drive a toroidal current in the plasma in-

ductively by driving a time dependent magnetic flux with a solenoid located on the z-axis. This

confinement scheme, characterized by a strong toroidal field imposed by external magnet coils and

a toroidal current carried by the plasma, is the Tokamak. In practice, several sets of poloidal field

coils are employed in addition to the toroidal field coils for equilibrium shaping, and a vertical field

coil is used to counterbalance the hoop force.

The tokamak equilibrium can be understood conceptually by examining the single particle

drifts. For a toroidal device, the imposed toroidal field generally falls off as Bφ ∝ 1/r and the

radius of curvature RC points in −r. The gradient drift,

vg =w⊥qB2

b×∇B, (1.33)

and the curvature drift,

vC =2w‖

qBRCb× RC (1.34)

are both in z. Here w⊥ and w‖ are the perpendicular and parallel kinetic energy. Driving a toroidal

current in the plasma produces a poloidal field, so the magnetic field lines become helical and lie

in nested toroidal surfaces. As the particle travels around the tokamak it will spend some of the

time drifting inward, toward the center line of the torus, and some of the time drifting outward,

away from the center line. The net result is that particles are confined about the starting surface.

1.6 The Spherical Torus Concept and NSTX

The low aspect ratio (ratio of major radius to minor radius), spherical torus concept was orig-

inally explored by Peng and Strickler [113]. The compact design offers a number of compelling

10

advantages including reduced construction costs and equilibria characterized by high natural elon-

gation, large plasma current, and strong paramagnetism [113]. The low aspect ratio design requires

the removal of neutron shielding protecting the toroidal field coils at the center stack, however, and

as a consequence normal conducting copper magnets become necessary. Additionally, engineering

considerations lead to a reduced toroidal field relative to the conventional tokamak design. To be

feasible then, the spherical torus must operate at high β ≡ 2µ0〈p〉/B2v where Bv is the vacuum

toroidal magnetic field at the plasma geometric center and 〈p〉 is the volume averaged pressure.

The plasma β is loosely related to the ratio of thermonuclear power output to energy expended

establishing the magnetic field [158]. Fortunately, equilibrium and stability studies have indicated

that the low aspect ratio design yields improved confinement, and optimized, high-β equilibria are

found with low aspect ratio, high plasma-generated bootstrap current fraction, and high elongation

[100, 73].

The National Spherical Torus Experiment (NSTX) [109, 76] is a low aspect ratio R/a = 1.27

tokamak experiment. The major radius of the device is R = 0.85m and the minor radius is a =

0.67m. The machine can operate at toroidal magnetic fields up BT = 0.55 T with plasma currents

of Ip = 1.4 MA. An engineering schematic of the device and a typical equilibrium reconstruction

is included in Fig. 1.1. Up to 7 MW of neutral beam injection (NBI) heating power and 6 MW

of high-harmonic fast wave (HHFW) heating at 30 MHz is available for auxiliary heating. Central

temperatures of ∼ 1 keV, average densities of ∼ 6.0e19 m−3, and pulse lengths of up to ∼ 1 s

have been achieved. Most critically, experiments with active suppression of resistive wall mode

instabilities have demonstrated high-β operation achieving β = 39% [118].

1.7 Overview of Presented Work

This dissertation details studies of tokamak edge physics in NSTX using the Gas Puff Imaging

diagnostic. The GPI diagnostic is described in Chapter 2, and Chapter 3 presents the characteriza-

tion of edge turbulence and fluctuations with GPI and their comparison with Beam Emission Spec-

troscopy measurements. Chapter 4 details several imaging-based velocity extraction techniques,

11

Figure 1.1: Left: Diagram of NSTX design showing toroidal field (TF) coils, poloidal field (PF)coils, center stack containing ohmic heating solenoid, and carbon wall tiles. Right: Equilibriumreconstruction showing contours of poloidal flux.

and Chapter 5 examines turbulent flows in the edge preceding the transition to high-confinement

operation using these techniques. Finally, Chapter 6 presents GPI observations of Edge Localized

Mode dynamics including the characterization of a coherent precursor in GPI intensity.

Chapter 2

The Gas Puff Imaging Diagnostic

Gas Puff Imaging (GPI) is a diagnostic capable of imaging edge turbulence in fusion devices.

During operation of the diagnostic, a neutral gas puff is discharged into the edge of the plasma and

the visible light emission from collisional excitation of the neutral gas with the plasma is imaged

on timescales faster than the turbulence autocorrelation time (typically . 10µs). This allows for

the 2D field of turbulent fluctuations in the edge of the device to be imaged, and the motion of

the turbulent structures to be tracked. GPI has been used on a number of fusion experiments

including NSTX, Alcator C-mod, TEXTOR, RFX-mod and EAST [97, 169, 127, 1, 90]. GPI has

been used to study edge turbulence and blobs [2, 27, 97, 169, 87, 106], turbulent velocity fields

[147, 105, 167, 123, 26], ELMs [96, 122], and L-H transitions [167, 161]. This chapter details the

design and operation of the NSTX GPI diagnostic and the atomic physics basis for GPI.

2.1 The NSTX GPI System

The Gas Puff Imaging system on NSTX [97, 170] uses a fast-framing camera to capture

images of line-filtered emission from the collisional excitation of a neutral gas by the background

plasma. A schematic of the diagnostic can be seen in Fig. 2.1(a), and the position of the view in

machine R and Z is shown in Fig. 2.1(b). The gas puffing is delivered by a manifold mounted on the

outer wall behind the RF limiter shadow 20 cm (∼28 ◦ poloidally) above the outboard midplane.

Gas is injected through thirty 1 mm diameter holes spaced evenly along the 29 cm length of the gas

manifold. The gas injection system is tilted with respect to the horizontal to produce a gas cloud

13

that is elongated along the direction of the local magnetic field at the edge (∼35-50◦ pitch angle).

Deuterium, Helium, and Argon have been used for the neutral gas species. Deuterium emission

competes with background thermal emission of the plasma, so Helium and Argon typically have

higher signal to background ratios. During an observation period, gas flow rates of 6×1021 atoms/s

are maintained for ∼ 50 ms.

Figure 2.1: (a) Schematic of GPI system with view from machine center stack looking out. (b)positioning of GPI view in R and Z with flux contours for typical operational parameters.

A diagram of the optical system can be seen in Fig. 2.2. The gas cloud is viewed through a

re-entrant window approximately 70 cm away. Light from the gas cloud is imaged onto a 400 pixel

× 400 pixel coherent fiber optic bundle by a 1 in. F/1.3 lens with a 25mm focal length. The image

from the fiber bundle is then imaged onto a fast-framing camera using a 50mm focal length, 1 in.

F/1.3 lens on the fiber bundle and a 8 mm, 0.5 in. F/1 lens on the camera. An interference filter

is used to select spectral line emission for the chosen neutral species. Typically Deuterium is used

for the gas species, and the Balmer-α (Dα : n = 3→ 2) line at 656 nm is imaged.

Many cameras have been used over the life of the GPI diagnostic system, but the current

iteration, which recorded most of the data discussed in this thesis, uses a Vision Research Phan-

tom v710 fast-framing camera. The camera uses a CMOS sensor capable of capturing images at

14

Figure 2.2: Diagram of GPI optical system

1280×800 resolution with framerate that depends on pixel count. For the GPI system, Images are

captured at a framerate 400 kHz with a 64×90 resolution and 2.2 µs exposure time. The sensor has

a readout noise of 29 electrons per pixel at 30.2◦C, and a 23200 electron full well capacity. Typical

turbulence autocorrelation times in the edge of NSTX are on the order of ∼10 µs, so turbulence

time scales are well resolved by the time per frame of 2.5 µs of the camera. For data discussed in

Chp. 5, images from two Phantom v7.3 cameras were interleaved to achieve a 285 kHz framerate

at 64×64 resolution and 3 µs exposure times.

Empirical studies have demonstrated that edge turbulence correlations are significantly longer

along the magnetic field than across[97, 170]. Thus turbulent structures are filamentary in nature

with 3-5 cm cross-sections in the cross-field directions and several 10’s of centimeters along the

field. To maximize the cross-field resolution, the viewing direction is tilted by 40◦ with respect to

horizontal so that the view is approximately aligned with the local direction of the magnetic field at

the intersection with the gas puff during standard operation. The x and y coordinates of the camera

image are approximately perpendicular and tangent to surfaces of constant flux, respectively. Thus,

the image x coordinate is approximately the radial coordinate, and the y coordinate is then the

generalized poloidal coordinate.

The pixel resolution of the optical system at the location of the gas puff is ∼4 mm. The gas

cloud does extend ∼24 cm along the line of sight [97], however, so the resolution is degraded by

15

the coupling of the elongated structures with the magnetic field line curvature and misalignment

of the optical system with the magnetic field. To illustrate the smearing of pixels, the cross-

correlation map of GPI data for typical L-mode turbulence is shown in Fig. 2.3 with the projection

of a magnetic field line trajectory through the gas cloud. Field line curvature reduces the radial

resolution to ∼ 3 cm. Misalignment of the line of sight with the magnetic field pitch angle can

reduce the poloidal resolution by up to ∼2 cm, but here the poloidal resolution is ∼ 0.5 cm. For

typical NSTX edge parameters, the GPI light emission is localized to the region with Te, Ti . 0.3

keV, and ion sound gyroradius ρs . 1 cm. Thus, fluctuations that satisfy 〈kpol〉ρs . 2.0, including

typical drift turbulence scales [151] of 〈kpol〉ρs ≈ 0.1− 1, are well resolved by the GPI diagnostic.

Figure 2.3: Cross-correlation map for each pixel with reference pixel at [17,40] for L-mode turbu-lence. Green trace indicates field line trajectory through GPI gas cloud.

2.2 Atomic Physics Basis

The system under consideration is a neutral deuterium gas immersed in a plasma character-

ized by electron density ne, ion density ni, and electron temperature Te. Our interest is to describe

the dependence of the neutral Dα emission on the plasma parameters. The present description will

16

loosely follow the presentation in the DEGAS neutral transport code user manual [139]. The Dα

emission rate will be given simply by S = N3A23, where N3 is the population of the n = 3 excited

state of deuterium and A23 is the Einstein coefficient for spontaneous decay from level n = 3 to

n = 2. Neutral atoms in a plasma undergo a wide array of collisional and radiative transitions that

distribute its population over a great many excited states, so the above equation for the deuterium

emission is deceptively simple. The time evolution of a given atomic state under the influence of

these various interactions is given by the Collisional Radiative (CR) model:

dNm

dt= −{

∑n<m

Kd,nmne +∑n>m

Ke,nmne +Ki,mne +∑n<m

Anm}Nm

+∑n>m

Kd,mnNnne +∑n<m

Ke,mnNnne +∑n>m

AmnNn

+{βrad,m + βdia,m +Kr,mne}neni. (2.1)

The interactions included in the model and the meaning of each term is as follows:

• Collisional dexcitation from level m to n by electron impact at a rate Kd,nm

• Collisional excitation from level m to n by electron impact at a rate Ke,nm

• Ionization from excited state m by electron impact Ki,m

• Spontaneous radiative decay from state m to n at a rate Anm

• Radiative recombination to state m at a rate βrad,m

• Diaelectronic recombination to state m at a rate βdia,m

• Three-body recombination to state m at a rate Kr,m

In words, the model accounts for the collisional and radiative transitions out of state m given by the

negative terms proportional to Nm, the collisional and radiative transitions into state m from all

other states given by the terms proportional to Nn, and the recombination of plasma ions into state

m given by the final 3 terms. For the temperatures and densities relevant to the Tokamak edge, the

electron collision processes dominate, so all collisional terms in the model are mediated by electrons.

17

This model was originally developed for hydrogen by Bates, Kingston and McWhirter [6], and

many calculations have been perform since [5, 99, 34, 35, 46, 74, 15]. Alternatively, this system of

equations can be written compactly in matrix form as

n = Mn + Γ, (2.2)

where the transitions due to collisions, radiation, and ionization are combined into the matrix M,

and the recombination terms have been gathered into the vector Γ.

Equation 2.1 defines a large (technically infinite) system of coupled linear differential equa-

tions. A number of assumptions are leveraged to make the problem tractable. First, it’s typically

assumed that the population for states above some sufficiently highly excited atomic state is neg-

ligible so that the summation can be truncated. Second, a quasi-steady state approximation is

invoked for states above the n = 1 state which allows for their time derivative to be set to zero.

The reasoning behind this is that the timescales for the evolution of the excited states are much

faster than the evolution of the ground state, so that these states can be assumed to reach equi-

librium instantaneously. These assumptions reduce the system to a countable number of coupled

linear algebraic equations, and one differential equation for the time evolution of the ground state.

To solve the system of equations, we truncate the system at Q+ 1 states, and then separate

the n = 1 state from the Q excited states and treat it explicitly. For the excited states, eqn. 2.2 is

rewritten to yield,

0 = MQnQ + M1n1 + ΓQ. (2.3)

This system of equations can then be solved (e.g. by inversion of MQ) to give the population of

the excited states in terms of the ground state population n1 and the plasma parameters. This is

usually written in the form,

nq = C1,qn1 + Ci,qni, (2.4)

where C1,q is the coupling to the ground state and Ci,q is the coupling to the “continuum”.

18

The solutions for the Q excited states may then be substituted into the time evolution

equation for the n = 1 state to yield,

dn1dt

= −n1Seff + neniReff , (2.5)

with the effective ionization rate Seff and effective recombination rate Reff defined as,

Seff =∑m≥1

C1,mKi,m, (2.6)

Reff = −∑m≥2

Ci,mKi,m +∑m≥1

(βrad,m + βdia,m +Kr,mne). (2.7)

Transport terms, source terms, and etc. can be added in to eqn. 2.5 depending on simulation

needs.

The procedure for estimating the Dα emission is first to collect atomic data for the rates

of collisional and radiative transitions, ionization, and recombination as function of ne and Te.

Calculations of these terms are discussed in Burgess and Summers 1976 [15], and data has

been tabulated by Janev and Smith 1993 [72]. Then, solving the excited state equations with

n1 = 1 and ni = 0 gives C1,q, and solving the equations with n1 = 0 and ni = 1 gives Ci,q.

With these quantities, we can obtain the excited state population as a function of n1 and ni, and

multiplying by the rate of spontaneous emission A23 gives the Dα emission rate in terms of these

quantities. Recombination is typically negligible at temperatures relevant to the tokamak edge, so

ignoring the ni terms and substituting the approximate ne and Te dependence of the C1,q term

yields the GPI Dα emission rate,

S = n1nαe T

βe , (2.8)

where α and β are obtained from the CR model calculations and depend on the local values of ne

and Te. These values have been tabulated [20] for densities and temperatures relevant for the edge

of NSTX, and plots of their dependence on plasma parameters are presented in Fig. 2.4. Typical

plasma parameters near the peak of the GPI emission profile are ne ≈ 1013 cm−3 and Te ≈ 30 eV,

thus GPI Dα emission has the approximate form S ≈ n1n0.7e T 0.3e .

19

Figure 2.4: CR calculations of exponent for ne, α, and exponent for Te, β, used in estimation ofGPI Dα signal.

The simple CR model discussed above can be extended to account for molecular interactions,

or the presence of meta-stable states. Details on the inclusion of molecular interactions and their

rates can be found in Stotler 1996 [140]. CR models for helium including the treatment of the 21S

and 23S meta-stable states is discussed in included references [45, 50]. In addition, a more formal

description of CR models and a procedure for determining their validity is described in Greenland

1998 [51].

Experimental observations with Gas Puff Imaging have been compared to the Monte Carlo

neutral transport code DEGAS 2 which also computes a CR model for the given gas puff species

[142, 141, 138, 20]. Good agreement is found between time average profiles of GPI emission and the

simulation results. Also, absolute light emission values have been compared for deuterium puffs,

and again good agreement is found [20].

Chapter 3

Direct Comparison of GPI and BES Measurements of Edge Fluctuations in

NSTX

Understanding the physics of the edge and pedestal regions of tokamak plasmas is essential to

the operation of high performance fusion devices. Turbulence in the edge enhances transport above

neoclassical levels which severely limits the achievable confinement. Experiments have observed that

during the transition to high confinement (H-mode) an edge transport barrier (ETB) is formed,

and a suppression of the transport is achieved [149, 103, 152]. The reduced transport at the edge

leads to a buildup of pressure with steep gradients in temperature and density. This yields a

“pedestal” in the density and temperature profiles that the core profiles sit atop, thus the height

and width of the pedestal significantly impacts the achievable performance of the device. The steep

gradients of the pedestal provide a source of free-energy for MHD instabilities localized to the edge,

accordingly called Edge Localized Modes (ELMs), that can exhaust large amounts of stored energy

and particles. These ELMs can damage plasma facing components, therefore controlling ELMs

is a major concern for current and next-step fusion devices [42]. The nature of microturbulence

that drives transport in the edge, the development of the ETB during the L-H transition and

subsequent suppression of turbulence, the evolution of the pedestal, and the dynamics of ELMs

all underscore the importance of understanding this region of the plasma, and the need for highly

resolved diagnostic information.

In this chapter, we examine two diagnostics that meet this need, Gas Puff Imaging (GPI) and

Beam Emission Spectroscopy (BES). Measurements of edge turbulence made with BES and GPI

21

are directly compared in MHD quiescent H-mode operation that is free of large ELMs. This study

tests our understanding of the operation of, and interpretation of data from these two diagnostics.

The goal of this comparison is to provide experimental verification that the measurements agree

where modeling of diagnostic response suggests they should, and that observations are consistent

between the two diagnostics. In addition to increased confidence in the measurements that this

provides, this study lays the groundwork to use the diagnostics more collaboratively in the future.

Furthermore, the relatively close proximity of the BES viewing volume to the GPI gas manifold

allows the local effects of the GPI neutral gas puff on the plasma to be quantified.

3.1 The Beam Emission Spectroscopy Diagnostic

Similar to the GPI diagnostic, the Beam Emission Spectroscopy diagnostic measures light

emission from the collisional excitation of neutral atoms. Where the GPI diagnostic uses a gas puff

to localize the observation, the BES diagnostic images Dα line emission that is localized to the

intersection of optical sight lines with the neutral heating beam. The velocity of beam neutrals in

combination with the viewing geometry produces a Doppler shift which isolates the beam Dα line

from thermal emission from the bulk plasma. The Dα emission is collected by a high-throughput

optical assembly and imaged onto a set of optical fiber bundles. The fiber bundles transmit the light

to a set of collimating lenses and transmission filters that selectively pass the doppler shifted Dα

emission. Finally, PIN photodiodes measure the Dα light intensity. The dependence of measured

emission on plasma parameters has been studied (see Hutchinson,et al. [64] and references

therein for review), and for typical parameters in the NSTX pedestal the light emission can be

approximated by δIBES/〈IBES〉 ≈ 0.5δne/〈ne〉 for modest fluctuation levels, δne/〈ne〉 ≤ 10%

[64, 44]. The sensitivity of light emission to other parameters, including temperature, is found to

be much weaker than the density dependence.

The NSTX BES system, discussed in detail in references [128, 129], consists of two separate

arrays of viewing channels with one view centered at R = 130 cm (r/a ≈ 0.45) and another view at

R = 140 cm (r/a ≈ 0.85). For the current study we make use of only the outer 28 channel array,

22

and an illustration of the channel positions is presented in Fig. 3.2. The radial array provides

coverage from r/a ≈ 0.4 to well into the scrape-off layer, and two poloidal arrays are positioned at

r/a ≈ 0.85 and in the scrape-off layer. The optical view is aligned with the local magnetic field

at the position of the heating beam, an angle of 37◦ with respect to horizontal, to optimize cross-

field resolution. Spot-sizes at the neutral beam cross-section are 2-3 cm, and detailed point-spread

function calculations indicate minor image distortions from field line misalignment and atomic state

lifetimes. Thus, turbulent fluctuations with k⊥ρs ≤ 1.5 are well resolved. Data is acquired at a

2 MHz sampling rate, and frequency-compensating, wideband preamplifiers provide photon-noise

limited measurements at frequencies up to 400 kHz.

3.2 Plasma Conditions and NSTX Operation

The shots selected for this study are from a subset of the NSTX 2010 run campaign in which

both the GPI and BES diagnostics were operational, and each shot is chosen to be MHD quiescent

and free of large ELM events during the GPI observing period. Table 3.1 details the list of chosen

shots and shot parameters. The shots are selected from 2 different experiments carried out on

different run days. Typical ne and Te profiles for these shots are plotted in Fig. 3.1 along with

the GPI emission profiles and the R=140 cm BES radial array channel locations. The GPI light

emission is localized around the last closed flux surface (LCFS), and the light emission decays to

its half maximum value near the normalized flux value, ΨN of 0.8. This position corresponds to

densities of 2 − 5 × 1013 cm−3 and temperatures of 100 − 200 eV. The R=140 cm BES array has

radial views located at normalized flux values between ∼ 0.3 − 1.2, spanning the SOL, gradient

region, and pedestal top. This study focuses on the near-SOL and gradient region to compare with

the GPI measurements.

3.3 Relative Diagnostic Positioning

A schematic representation of the BES and GPI views is illustrated in Fig. 3.2(a) and (b).

Fig. 3.2(a) shows the relative position of the GPI view (blue polygon) and the BES channels (red

23

Table 3.1: Shot list for this study containing the shot number, toroidal field at the magnetic axis,plasma current, neutral beam heating power, average density, and GPI puff timing. Values aretaken at the time of the gas puff.

Shot BT [T] Ip [MA] PNBI [MW] 〈n〉 [cm−3] tGPI [s]

138845 0.34 0.8 3.8 5.9× 1013 0.58138846 0.34 0.8 3.8 5.9× 1013 0.58138847 0.33 0.8 3.8 6.0× 1013 0.58141249 0.36 0.7 2.9 3.3× 1013 0.48141254 0.37 0.8 2.9 3.4× 1013 0.48

Figure 3.1: Comparison of Thomson scattering profiles of ne and Te with GPI emission profile (bluecurve) and R=140 cm BES radial array channels (black and red dashed lines). The red dashedlines indicate the position of the inner and outer BES poloidal arrays.

diamonds) in the (R, Z) plane. BES channels and corners of the GPI view are plotted using their

cylindrical R and Z coordinates, ignoring the different toroidal angles. Flux surfaces are indicated

by dashed contours and labeled with their midplane r/a value, and the separatrix is indicated by

the solid black contour. In Fig. 3.2(b), The BES inner poloidal array channels (red diamonds)

are plotted in toroidal and machine-poloidal coordinates along with GPI pixels (blue points) with

similar values of poloidal flux. Magnetic field line traces in these coordinates are indicated by

dashed lines, and a trace of the generalized poloidal coordinate passing through the GPI points is

24

indicated by the dot-dashed line.

Figure 3.2: (a) R = 140 cm BES channel positions (red, diamonds) and GPI viewing area (bluepolygon) plotted over contours of poloidal flux for NSTX shot 141254. Flux surfaces are labeledby their midplane r/a value, and the separatrix is indicated by the solid black line. BES channelpositions and the corners of the GPI view are plotted using cylindrical coordinates. (b) BES innerpoloidal array (red diamonds) plotted in toroidal coordinates with GPI points (blue) for similarflux value. Magnetic field line traces are plotted as dashed lines, and the dot-dashed line traces thegeneralized poloidal direction.

The GPI view is centered 28◦ above the outboard midplane, and the radial extent of the view

typically captures the edge region, separatrix, and SOL. The R=140 cm BES array is positioned

at 12◦ above the outboard midplane and provides similar radial coverage. The separation between

the lower-left corner of the GPI view and the intersection of the BES radial and inner poloidal

array, as seen in Fig. 3.2(b), is 16◦ in the toroidal direction and 3◦ in the poloidal direction.

The physical distance between these two points is 35 cm, and the distance along the generalized

poloidal direction from the GPI corner to the intersection with the field line passing through this

BES channel is ∼ 20 cm.

The comparisons presented in Sections 3.4, 3.6, and 3.5 use the BES channel at the intersec-

tion of the radial and inner poloidal array as reference for shots 141249 and 141254. In R and Z,

this channel is located just inside of the inner edge of the GPI view ∼ 2 cm above the bottom of the

GPI view. For shots 138845-47, the BES channel one channel inside from the intersection of the

radial array with the outer poloidal array is used for comparison. This channel lies ∼ 4 cm out from

25

the inner edge and ∼ 1 cm above the bottom edge of the GPI view. GPI pixels used in comparisons

are chosen to overlap in R and Z with the BES reference point to within 1 cm. This ensures that

the comparison is done for similar flux surfaces, but field lines passing through the measurement

locations are still separated by ∼ 20 cm in the direction perpendicular to the magnetic field due to

the toroidal separation. For the calculation of poloidal correlation lengths presented in Section 3.7

shots 141249 and 141254 use the BES inner poloidal array, and shots 138845-47 use the outer BES

poloidal array. GPI correlation lengths are calculated for poloidal separations about the center

of the GPI view at the radial position of closest flux surface overlap with the given BES poloidal

array.

3.4 Fluctuation Levels

Based on the collisional radiative model, GPI emission is given by IGPI ∝ nαe Tβe , and α and

β are tabulated for a given values of ne and Te [142]. Near the 0.85 normalized flux position, the

plasma conditions are ne = 2 − 5 × 1013 cm−3 and Te = 100 − 200 eV. Under these conditions,

GPI exponents are α = 0.7 − 0.6 and β = 0.12 − 0.03, thus for small fluctuations the density

dependence is at least 5 times stronger than the temperature dependence. Similarly, BES emission

varies with plasma density to the 0.6− 0.5 power, and temperature dependence of the emission is

typically negligible. Therefore, it is expected that both diagnostics predominantly measure density

fluctuations, and, for small fluctuation amplitude, the ratio of GPI to BES RMS fluctuation levels

normalized to mean is expected to be ∼ 1.2.

The BES light emission is subject to low-frequency, < 4 kHz, fluctuations in the neutral

heating beams. In the analysis presented in this section, we account for this slow oscillation by

applying a Gaussian convolution filter (GCF) with a 4 kHz e−1 frequency cutoff to time series of

both diagnostics. The slowly varying “mean” intensity, 〈I〉 is then defined as the GCF of the raw

trace, and the fluctuating signal is defined as δI = I−〈I〉. By extension then, the RMS fluctuation

26

level is given by,

δIrms/〈I〉 =〈(I − 〈I〉)2〉1/2

〈I〉, (3.1)

where 〈·〉 represents the GCF with 4 kHz frequency cutoff. Time traces of raw (I), mean (〈I〉).

The GCF is strictly positive-valued in the time domain, which ensures that the resulting RMS

fluctuations are real-valued at all times. Fluctuating (δI) intensity for both GPI and BES are

plotted in Fig. 3.3. GPI RMS fluctuation levels are plotted in Fig. 3.4. Generally for shots in this

database, GPI RMS fluctuation levels are ∼ 18% of the mean which is ∼ 9 times greater than the

BES values of ∼ 2% at this location. This large discrepancy in fluctuation levels is well above the

expected ratio of 1.2.

Figure 3.3: Traces of (a) GPI raw (gray) and 〈I〉 (blue), and (b) δI. Plots (c) and (d) are BEStraces. All traces taken at the 0.85 ΨN location.

The large discrepancy in RMS fluctuation levels suggests that either BES is underestimating,

or GPI is overestimating the fluctuations at this location. Temperature dependence is not expected

to be important, as discussed above, and both diagnostics have similar radial localization. The

poloidal resolutions are different, which could lead to a difference in measured fluctuation levels

due to an averaging over small scales, but applying a spatial filter to the GPI observations cannot

account for the observed discrepancy. Large plasma fluctuations can alter the neutral gas density of

27

the GPI gas cloud, and this may induce fluctuations in light emission at smaller radii. This effect is

commonly referred to as “shadowing”, and a simple discusion of the effect can be found in [142]. To

account for the observed discrepancy, the neutral density perturbation at the measurement location

would need to be of order 10%. Shadowing is a complex effect involving full 3D plus time dynamics,

and efforts to better quantify the effect are currently underway. For now, shadowing remains as a

possible explanation.

Figure 3.4: Traces of δIRMS/〈I〉 for BES (red) and GPI (blue) for ΨN = 0.85.

The NSTX fast scanning probe [11] is capable of making measurements of edge fluctuations

in NSTX H-mode plasmas for ψN > 0.9. Probe data from Type-III ELMing H-mode plasmas with

Bt = 0.45T , 1.3 MW of NBI power, and 0.8 MA plasma current have been presented in [12]. For

these shots, it was observed that normalized density fluctuations increased from ∼ .20 at ΨN = 0.9

to a peak of ∼ 1.50 at Ψn = 1.0. In the SOL, Normalized fluctuation levels were relatively flat

at a value near 1.0. These H-modes are significantly different than those presented in this paper.

Most notably these shots were at higher Bt, lower NBI power, and Type-III ELMs were present.

Therefore, a comparison with results presented here is not useful. However, future experiments

could be designed to compare normalized fluctuation levels between all three diagnostics, and this

may help resolve the observed discrepancy between the GPI and BES fluctuation levels.

28

3.5 Fluctuation Statistics

Fluctuation statistics (i.e. PDFs and their moments) provide a well defined and experimen-

tally accessible way of characterizing turbulence. The statistical description of plasma turbulence

seeks to explain these quantities and gain insight into the underlying physical mechanisms in the

process. Much effort in the study of turbulence has focused on the understanding and prediction

of the PDFs of turbulent quantities and the search for a universal distribution or universal features

of turbulent distributions (for a review of the statistical description of plasmas see Krommes[85]).

Furthermore, observations of non-Gaussian statistics and intermittency have led to a number of

realizations about the importance of coherent structures and the nature of turbulent transport in

the edge and SOL regions (e.g. [107, 13, 14]).

Figure 3.5: PDFs of (a) GPI and (b) BES intensity from 10 ms period. Dashed Lines indicateGaussian PDFs with similar mean and variance.

As mentioned previously, BES emission is subject to neutral beam oscillations below 4 kHz,

so PDFs are created from high-pass filtered, 10 ms time traces for both GPI intensity and BES

intensity. The PDFs and their normalized skewness (3rd moment) and normalized excess kurtosis

values (4th moment) are shown in Fig. 3.5(a) and (b). Moments are normalized to the relevant

power of the variance, and the normalized excess kurtosis is the difference of the normalized 4th

moment of the PDF from Gaussian statistics. The first two moments of the PDF, mean and

variance, are largely meaningless here due to the filtering of the low-frequency contributions and

differing intensity scales, but the desired information is captured by δIrms/〈I〉 as seen in Fig. 3.4.

29

Skewness is a measure of the asymmetry of a distribution, while kurtosis is a measure of the relative

weight of the tails. Deviations from a Gaussian PDF (represented by the dashed line in Fig. 3.5) are

indicated by deviations of the normalized skewness from S = 0 and the normalized excess kurtosis

from K = 0. For this shot, the GPI distribution is slightly skewed toward positive values, and a

small positive kurtosis indicates more weight is present in the tails relative to a normal distribution.

The BES distribution has negligible skew and small positive kurtosis.

The skewness and kurtosis values for a given PDF can be used as a reduced description

of the full PDF. This simplification enables easy comparison of PDFs for different shots, plasma

conditions, or radial locations, so that trends in the fluctuation PDFs may be identified (e.g. see

analysis in Labit et al. [88]). Figure 3.6 plots skewness and kurtosis values calculated from PDFs

of ∼ 10 ms time blocks of intensity fluctuations that have been high-pass filtered at 4 kHz. Five

successive time blocks from each shot are analyzed for t = 0− 50 ms after the GPI emission peak.

Skewness and kurtosis values from 2.9 cm (filled symbols) and 0.7 cm (open symbols) inside the

separatrix are both included in the figure.

Figure 3.6: Scatterplot of skewness and kurtosis values for BES (filled triangles) and GPI (filledsquares) at r − rsep ≈ 2.9 cm, and BES (open triangles) and GPI (open squares) at r − rsep ≈ 0.7cm. The dashed line represents a second degree polynomial fit to the GPI data.

Near r − rsep ≈ −2.9 cm, both GPI and BES intensity may be approximated by I ≈ Cnαe ,

30

where α ≈ 0.6 for GPI and α ≈ 0.5 for BES. Therefore, it is expected that the measured PDFs

would be similar at this location, and rough agreement is apparent in Fig. 3.6. Averaging values

for the 6 shots yields GPI skewness of S = 0.24 ± 0.25 and kurtosis of K = 0.38 ± 0.44, and BES

skewness of S = −0.01 ± 0.13 and kurtosis of K = 0.11 ± 0.34. Values are close to a normal

distribution within variances for both diagnostics, but GPI does show a shift in the average values

toward small positive skewness and kurtosis. Raising a random variable with a skewed distribution

to a fractional power will shift the skew to negative values though, so it is possible that small

differences could be explained by the difference in dependence on plasma parameters.

GPI skewness values at both radial locations vary by about ±0.5, and points appear to

follow a quadratic trend when kurtosis is plotted vs. skewness. Fitting the data to a second degree

polynomial yields the dashed line plotted in Fig. 3.6. Similar quadratic trends have been observed

in GPI data of edge plasma fluctuations across a number of machines [120] and in probe data from

TORPEX [88]. In addition, analytic models attempting to illuminate the physical mechanisms

determining this trend have been explored [86]. Realizability constraints, derived by invoking the

positive semidefiniteness of the mean of any nonnegative quantity, yield the relation [85] K ≥ S2−2,

so a quadratic trend is perhaps not unexpected. BES values, however, exhibit a high degree of

scatter in skewness and kurtosis for the outer radial location, and they do not clearly exhibit a

quadratic relationship between S and K.

The previous observations using GPI by Sattin et al. [120] measured skewness and kurtosis

values across a wide range of radial locations from 6 cm inside the separatrix to 6 cm into the SOL,

and a simple model was then put forward to describe the observed PDFs as the sum of two inde-

pendent basis functions. Both the sum of two gamma distributions and two beta distributions were

explored. The correspondence of GPI measurements and probe measurements was not addressed,

though, thus GPI measurements were effectively assumed to measure density fluctuations.

Caution should be taken, though, as interpreting the observed trends in GPI data is quite

difficult, for a number of reasons. The functional form of the emission determined by the atomic

physics necessarily implies a complicated dependence of the distribution of the intensity on the

31

distributions of underlying plasma parameters, both ne and Te. Furthermore, the functional form

of the GPI intensity depends on the plasma parameters which change significantly across the edge

and SOL regions, and order unity fluctuations are likely to complicate the interpretation of both

GPI and BES diagnostics near the separatrix. Finally, it is unlikely that ne and Te fluctuations can

be considered to be independent, thus understanding the trends of moments and their relation to

the underlying plasma fluctuations would likely require a focused effort to model the edge plasma

with synthetic GPI and BES diagnostics.

3.6 Cross-Diagnostic Analysis

As discussed in previous sections, both the GPI and BES diagnostics should predominantly

measure density fluctuations near the 0.85 normalized flux position. In this section, we employ cross-

correlation and cross-spectral analysis to estimate the similarity between time traces and spectral

content of the signals. Due to technical issues with the BES data acquisition for these shots, there

exists an unknown time offset between the GPI and BES time records, and, compounding this

difficulty, a constant drift is present in the BES clock time kept by one diagnostic relative to the

other. Fortunately, this drift is readily visible in the time-lagged cross-correlation function versus

time, shown in Fig. 3.7, and can therefore be easily corrected. Time traces have been high-pass

filtered to removed the < 4 kHz fluctuations which are problematic due to beam fluctuations.

Finding the maximum correlation value for each time point yields a 1D function for the lag-time

to peak correlation versus time. Then, a clock-correcting offset and dilation factor can be obtained

by performing a linear fit of this function, and we utilized this dilation factor to align the BES time

base with GPI in the analysis contained in this section. An unresolved absolute time offset between

the two signals still exists, however, and this offset limits the potential use of the correlation analysis

in some applications. Note that this peak correlation value is persistent in time, and that the peak

correlation value of R ≈ 0.5 is significanly above the background in this case.

With the correction applied to the BES time base, Fig. 3.8(a) shows the traces of time-

lagged cross-correlation, and time traces of intensity fluctuations for a GPI pixel and a BES channel

32

Figure 3.7: Contour plot of time-lagged cross-correlation between GPI and BES signals versus time.Peak correlation value persists in time, but constant linear drift is present.

normalized to standard deviations are compared in Fig. 3.8 (b). For shots listed in Table 3.1, peak

correlation values often exceed R = 0.6, and a high degree of similarity is visually apparent in

traces of the normalized fluctuations.

Figure 3.8: Traces of (a) time-lagged cross-correlation between GPI and BES, and (b) time tracesof GPI and BES intensity.

The cross-correlation vs. radius for GPI pixels correlated with a single BES channel is plotted

in Fig. 3.9(a) and (b), and contour plots of the time-lagged cross-correlation vs. radius are included

in Fig. 3.9(c) and (d). The x-axis for these plots is the difference of the radial location of the GPI

pixels mapped to the midplane and the midplane outer separatrix radius. The r− rsep location for

33

the BES reference channel is illustrated by the dashed red line. The radial array of pixels ∼ 2 cm

above the bottom of the GPI view is used for this comparison.

The correlation functions in Figures 3.9(a)-(d) exhibit a number of interesting features. First,

the cross-correlation features for all shots are significantly radially extended, and strong correlation

exists up to ∼4 cm into the SOL. Second, shots in the 141254 series generally exhibit a roughly

linear time shift in the peak correlation time vs. radius. This is due to the radially extended

wavefronts for this structure being tilted in the radial vs. poloidal plane. Shots in the 138845 series

exhibit distinctly different behavior. For these shots, the peak correlation time is roughly constant

inside of r − rsep ≈ −2 cm. At r − rsep = −1 cm the maximum correlation values are slightly

suppressed, and outside of this location a time shift or anti-correlation is observed. This may be

indicative of a strong sheared flow present at r − rsep = −1 cm. It should be noted, however, that

the GPI light falls off rapidly inside of r − rsep ≈ −4 cm.

Figure 3.9: Plots (a) and (b) are traces of cross-correlation vs. radius for GPI pixels with a BESreference channel. Plots (c) and (d) are contour plots of the cross-correlation vs. radius and time-lag. The black dashed line indicates the separatrix location, and the dashed red line indicates theradial location of the BES reference channel.

The cross-spectral density may be used to provide a measure of the shared frequency content

34

between the BES and GPI signals. Traces of (a) cross-coherence, (b) cross-spectral density, (c)

cross-phase, and (d) phase uncertainty comparing a BES channel and a nearby GPI pixel in the

R-Z plane are shown in Fig. 3.10. Similarities are seen for the low frequencies in the auto and cross

spectral densities plotted in Fig. 3.10(b), and a peak near 10 kHz is observed by both GPI and

BES. The BES auto spectral density shows a second feature near 80 kHz that is not seen in the

GPI spectrum. This 80 kHz feature is seen only in this shot, and it is likely dominated by noise

due to the higher noise floor in the GPI spectrum.

Cross-coherence measures the constancy of the relative phase of similar frequency components

between signals. The cross-coherence spectrum in Fig. 3.10(a) demonstrates that a well-defined

phase relationship exists between the two diagnostics for the 10 kHz frequency band, and similar

peaks in the cross-coherence spectrum are seen in all shots included in Table 3.1. GPI observations

show intensity fluctuations propagating downward (-y, ion diamagnetic drift direction) through the

view, therefore the strong cross-coherence suggest that fluctuations, which are likely extended along

field lines, propagate coherently between the GPI and BES views. This corresponds to a distance

of ∼ 20 cm in the generalized poloidal direction perpendicular to the magnetic field.

Figure 3.10: Traces of (a) cross-coherence, (b) cross-spectral density, (c) cross-phase, and (d) phaseuncertainty for a BES coord and the closes GPI pixel in the R-Z plane.

35

3.7 Characteristic Time and Length Estimates

Estimates of the poloidal correlation length and the decorrelation time provide a characteristic

length scale and a characteristic lifetime for the fluctuations, respectively. Aside from characterizing

the fluctuations, the correlation lengths, decorrelation times, and scalings of these quantities with

other parameters may offer insight into the underlying instability driving the turbulence (e.g see

references [130, 131]). We do not address the scaling here, but instead compare the estimates

produced by the BES and GPI diagnostics. Auto-spectral density functions and cross-coherence

functions between pixels separated by 2.4 cm, 4.8 cm, and 7.2 cm are plotted in Fig. 3.11. As

discussed previously, peaks near 10 kHz and 80 kHz are present in the BES coherence spectrum,

but only the 10 kHz peak is present in the GPI spectrum. This 80 kHz feature is only present

in shot 141254, but a strong coherence feature at low frequencies, between about 0-20 kHz, is

present in all of the shots. Therefore, for the following analysis a band-pass filter is used with a

low frequency cutoff of flc = 4 kHz and a high frequency cutoff of fhc = 50 kHz. The low frequency

cuttoff is chosen to eliminate contamination of beam fluctuations in the BES signal, and the high

frequency cutoff is placed at the point where the first coherence peak crosses the√N noise floor,

where N is the number of time blocks used for the coherence calculation. This method of estimating

decorrelation times is similar to the generalized cross-correlation method of time-delay estimation

[84], but an ad-hoc filter based on the coherence spectrum is used here.

After applying this filter, time-lagged cross-correlations are calculated using 4 ms long time

blocks for varying pixel separations, and envelope functions are calculated using the Hilbert trans-

form. Individual estimates of the correlation function are then averaged over 60 ms. Block-averaged

correlation functions for GPI and BES are shown in Fig. 3.12(a) and (e) respectively, and envelope

functions calculated with the Hilbert transform are overlaid. Poloidal correlation lengths can be

estimated by a Gaussian fit to the zero-lag envelope correlation versus pixel separation shown in

Fig. 3.12(b) and (f). Similarly, decorrelation times can be estimated by a Gaussian fit to the

envelope peak correlation versus time to peak correlation shown in Fig. 3.12(c) and (g). Finally, a

36

Figure 3.11: Plots of: the Cross-Coherence between poloidaly separated channels for (a) GPI and(c) BES, and Autopower spectra for GPI (b) and BES (d).

velocity for structures can be estimated by Time-Delay Estimation (TDE) from the plot of time to

peak correlation versus poloidal separation shown in Fig. 3.12(d) and (h). Structures are observed

to propagate downward (-y, ion diamagnetic drift direction) in the GPI view, and the magnitude

of the TDE velocity is estimated from Fig. 3.12(d) and (h).

Estimates of poloidal correlation lengths, decorrelation times, and TDE velocities for several

shots are compared in Fig. 3.13(a) and (b), respectively, and results are tabulated in Table 3.2.

Dashed lines in Fig. 3.13(a) and (b) indicate perfect agreement and where the absolute difference

equals 40% of the mean. All values show good agreement at the ±40% level, though BES velocity

estimates derived from the correlation functions are consistently lower than GPI velocities. All

velocities are in the ion diamagnetic drift direction (-y in the GPI view).

Poloidal correlation length estimates have also been performed using the full extent of the

GPI view, and this method is able to resolve the first anti-node of the spatial correlation function.

The results of the calculation using the full view strongly suggest that the 4-point estimate used

in Fig. 3.12 and 3.13 significantly underestimate the correlation length. During the 2010 run

37

Figure 3.12: plots of: (a) and (e) Time-lagged cross-correlations (solid) with envelope functions(dashed), (b) and (f) zero-lag envelope peak correlation versus poloidal separation, (c) and (g)envelope peak correlation versus time-lag to peak, and (d) and (h) time-lag to envelope peakcorrelation versus poloidal separation. Values for poloidal correlation length, and decorrelationtime represent the 1/e length for a Gaussian fit to the corresponding plot.

Figure 3.13: Scatterplots comparing (a) poloidal correlation lengths estimates, (b) decorrelationtime estimates, and (c) TDE velocity estimates.

campaign, it was discovered that BES poloidal arrays did not have the poloidal coverage to resolve

the first anti-node, and, due to data acquisition constraints, only 4-channels of the BES poloidal

array were typically available. Therefore, the 4-point estimate is used for comparison here, and

this estimate does appear to be consistent between the two diagnostics. The upgraded BES view

38

Table 3.2: Comparison of poloidal correlation lengths, decorrelation times, and velocities estimatedfrom GPI and BES correlation functions.

BES GPI BES GPI BES GPIShot LPOL [cm] LPOL [cm] τc [µs] τc [µs] vg [km/s] vg [km/s]

141249 11.1 ± 1.1 16.4 ± 1.0 22.6 ± 2.2 27.4 ± 0.1 4.3 ± 1.0 7.5 ± 0.9141254 13.3 ± 1.9 15.0 ± 0.7 29.4 ± 4.8 23.6 ± 1.8 5.1 ± 1.4 7.1 ± 0.8138845 18.1 ± 5.4 14.9 ± 0.4 42.0 ± 0.1 41.4 ± 3.1 4.0 ± 0.9 5.7 ± 0.4138846 19.0 ± 8.1 14.9 ± 0.7 31.9 ± 0.8 34.1 ± 2.7 5.3 ± 1.7 6.2 ± 0.5138847 14.4 ± 4.9 14.2 ± 0.6 30.3 ± 10.0 36.4 ± 2.4 4.2 ± 1.4 5.9 ± 0.6

for NSTX-U has been designed to address this issue, and should be able to resolve anti-nodes of

the correlation function.

3.8 Gas Puff Effects

During the GPI observation period ∼ 3 × 1020 neutral deuterium atoms are puffed into the

edge, and the total electron content of the plasma before the GPI pulse is ∼ 6× 1020 electrons [20].

It is estimated that ∼ 20% (6 × 1019) of the gas puff atoms become ionized inside the separatrix,

but due to losses by edge particle diffusion the global density increase is expected to be ≤ 3% [168].

Previous work [168] has explored the possible perturbing effects of the GPI gas puff on the edge

plasma and turbulence, and found that edge electron density and temperature changed by ≤ 10%

at the Thomson scattering location far from the GPI puff preceding and up to the peak in the GPI

emission. Furthermore, edge turbulence quantities as measured by the GPI diagnostic did not show

significant variation during the gas puff. Still, the effect of the gas puff on the local density and

temperarture in the gas cloud could not be measured. A cursory look at the BES measurements

was included in the previous study, and it was found that the mean BES Dα emission increased

and closely followed the GPI Dα emission during the GPI puff. Here we provide a more detailed

examination of the GPI puff effects on the BES fluctuation spectra and poloidal correlation length.

The close proximity (∼ 40 cm) of the BES sight-lines to the GPI gas cloud provides us with

an opportunity to examine the local effects of the gas cloud on the edge turbulence. The low-pass

39

filtered, mean BES intensity is seen to increase concurrently with the GPI gas puff, and the increase

at different radial and poloidal locations is captured by Fig. 3.14. The largest increase is seen at

Xsep = r − rsep = −0.8 cm and Z = 7.6 which corresponds to the bottom of the GPI view. Data

from BES channels above Z = 7.6, closer to the center of the gas puff, is not available in these

shots. The increase in signal decays at locations further from the gas cloud (decreasing R and Z),

and typically the effect of the gas puff on the mean BES signal is negligible inside of Xsep ≈ −12

cm.

Figure 3.14: Time traces of low-pass filtered BES intensity for varying (a) Xsep = r − rsep and (b)Z. Times are relative to the GPI gas puff timing.

The effect of the gas puff on the fluctuations is illustrated by Fig. 3.15(a) and (b) which show

Continous Wavelet Transforms (CWT) for δI/I, the BES fluctuations normalized to the 200 Hz

low-pass filtered trace. Time traces of frame-averaged GPI emission are plotted above the CWT.

The low-pass filter is used to capture only the slow variation due to the GPI puff. Beam fluctuations

manifest as a coherent feature near 900 Hz in both figures, and weaker, less coherent harmonics can

be observed up to ∼ 4 kHz in average spectra for long time blocks. In Fig. 3.15(a) the amplitude

of the 1-20 kHz fluctuation band increases with the GPI emission. Shots 138845-47 each share this

behavior, but in shots 141249 and 141254 any increase in amplitude in this band is less clear due

to a feature centered at 10 kHz with a ∼ 10 kHz width that appears to be insensitive to the GPI

puff.

Block-averaged BES frequency spectra for δI/I before and during the gas puff, plotted in

40

Figure 3.15: Continous Wavelet Transforms of BES fluctuations normalized to 100 Hz low-passfilter for shots 138845 (a) and 141249 (b). Time traces of the average GPI intensity is plottedabove the CWTs.

Fig. 3.16(a) and (c), illustrate this increase in fluctuations more clearly. Here, autospectral power

estimates are obtained by averaging spectra from 23 time blocks ∼2.6 ms in length. Time blocks

are taken at 60-0 ms before, and 30-90 ms following the gas puff trigger. Fluctuations in the 1-10

kHz band increase signifcantly during the gas puff for Shots 138845-47, and a smaller increase is

seen only below ∼4 kHz in Shots 141249 and 141254. Coherence spectra (using raw traces without

normalization) for BES channels separated by 4.8 cm in the poloidal direction are plotted in Fig.

3.16(b) and (d). Before the gas puff, a strong coherence peak is seen between 10-20 kHz for shots

138845-47, and during the gas puff the coherence values for this feature are reduced by ∼0.2. In

contrast, Shots 141249 and 141254 show only minor changes in the coherence spectrum with the

gas puffing, and the feature near 10 kHz appears to shift to slightly higher frequency.

Poloidal correlation lengths offer another tool to assess possible changes in the turbulence

due to the GPI neutral puff. Correlation lengths are estimated by the same method used in the

previous section for 60-0 ms preceding, and 30-90 ms following the gas puff trigger. The results are

tabulated in Fig. 3.3. Poloidal correlation lengths for all shots become shorter during the gas puff,

although only two shots are different from zero within uncertainties.

The large, ∼300% increase in BES mean intensity levels during the gas puffing is a striking

feature of the analysis in Section 3.8. To better understand the effects of the gas puff on the

BES measurements, DEGAS2 simulations [141] have been performed, and it is found that gas

41

Figure 3.16: (a) and (c): Block-averaged autopower spectra of fluctuations normalized to mean.(b) and (d): coherence for BES channels separted by 4.8 cm. Black traces are spectra for 60-0 msbefore the gas puff trigger, and red traces are spectra for 30-90 ms following the trigger.

Table 3.3: Poloidal correlation length estimates from 60-0 ms before, and 30-90 ms after the gaspuff trigger.

Shot LPOL pre-puff LPOL post-puff Diff.

138845 19.7 ± 3.3 [cm] 17.7 ± 4.6 [cm] -2.0 ± 5.7 [cm]138846 20.5 ± 2.6 17.3 ± 6.7 -3.1 ± 7.2138847 19.5 ± 2.6 13.3 ± 3.5 -6.2 ± 4.3141249 16.2 ± 2.6 10.5 ± 1.1 -5.6 ± 2.8141254 14.5 ± 1.0 13.2 ± 1.8 -1.4 ± 2.1

puffing increases the neutral deuterium content at the intersection of the BES view with the neutral

beams by ∼ 1e16/m3. This is 3 orders of magnitude less than the plasma density at this location,

therefore it is not expected that neutral-neutral collisions will play a significant role. Furthermore,

the increase in local plasma density is presumably of this order or less, so it is unlikely that the

increased emission is attributable to an increased plasma density.

The gas puff does lead to an increase in Dα signal on the BES channels, but the BES system is

designed to use the doppler shift and interference filters to reject most of this background emission.

42

Simulations estimate the thermal Dα signal on the BES channels to be . 6.6e19 photons/(m2 sr s).

The BES etendue is 2.3 mm2 sr, optical fiber losses are 42%, and attenuation by the interference

filter is 5e−4% at the unshifted Dα wavelength. Thus BES is expected to measure 3.2e6 photons/s,

or a 65 µV signal output. This is several orders of magnitude less than the ∼1 V signal levels

typically seen, and clearly does not account for the increase seen in the mean BES signal.

3.9 Summary of Results

Comparisons of BES and GPI measurements of edge fluctuations in NSTX have been pre-

sented. The BES and GPI diagnostic views share coverage over the range of normalized flux from

Ψn ≥ 0.8 into the SOL at similar poloidal angles, but are separated in toroidal angle by ∼ 16◦.

Near Ψn = 0.85 both diagnostics are expected to predominantly measure density fluctuations. The

measured fluctuations are therefore expected to have similar characteristics, and many similarities

are observed. Direct comparison of GPI and BES fluctuation measurements on field lines separated

by ∼ 14 cm in the direction perpendicular to B yields strong correlations, R > 0.6, and strong

cross-coherence between ∼ 5−15 kHz. Fluctuation PDFs and their moments show good agreement

at Ψn = 0.85, and both GPI and BES distributions are close to Gaussian at this location. Cor-

relation lengths, decorrelation times, and TDE velocity estimates all show good agreement within

±40%. It should be emphasized that, given the physical separation of the measurement locations,

the observed agreement is very strong.

There are some important differences, however. Measured GPI fluctuation levels normalized

to the mean are a factor of 5-6× greater than the measured BES fluctuation levels. This discrepancy

is unresolved, but could be explained by either an underestimate of the fluctuation level by BES at

this location or an overestimate by GPI. It is possible that large fluctuations in the plasma density

near the separatix may induce fluctuations in the neutral density, thus an increased fluctuation

level would be measured by GPI at smaller radii due to the modulated neutral density. In addition

to the dissimilarity of fluctuation levels, differences in the fluctuation PDFs are present at larger

radii, but this is not unexpected due to large-fluctuation amplitudes and increased temperature

43

dependence of the light emission. Also, comparing the 4-point estimate of the correlation length to a

calculation utilizing the full GPI spatial information revealed that the 4-point estimate consistently

underestimates the poloidal correlation length.

Finally, effects of the neutral deuterium puff used for GPI on the BES mean and fluctuations

are quantified. BES mean intensity levels are seen to increase by as much as 300% near the

separatrix position. Sensitivity of the BES mean to the neutral puff decreases with distance of the

measurement location from the center of the gas cloud. In addition, CWTs and frequency spectra

for fluctuations normalized to the 200 Hz low-pass filtered BES signal show an increase in relative

fluctuation levels in the 1-10 kHz band, but only one shot shows a significant change in the poloidal

correlation measured before and during the gas puff.

Chapter 4

Velocimetry

The estimation of motion is a very general problem arising in computer vision, image analysis,

and signal processing. A great many techniques, spanning a diverse range of fields, have been

developed to handle this problem. This chapter examines the common techniques of Time Delay

Estimation, Optical Flow, and Pattern Matching for application to velocity extraction from image

sequences. Each of these techniques has been employed extensively in analyzing Imaging data of

plasma systems.

4.1 Time Delay Estimation for Motion Estimation

The estimation of time delay of signal arrival between spatially separated measurements,

or simply time delay estimation (TDE), is a common problem encountered in signal processing.

It has played an important role in a diverse array of fields including radar and sonar detection,

oceanography, seismology, and speech localization and tracking. TDE techniques have also been

used extensively In the field of plasma physics to measure turbulent velocities [147, 71, 70, 164, 60,

56, 26, 28]. This section describes the basic TDE technique.

The simplest conception of TDE is a two-point, one dimensional velocity estimation tech-

nique. Consider two zero-mean time traces, X(t) and Y (t) measured at locations separated by a

distance ∆x. This could be two probes, two pixels from an image, or any spatially separated mea-

surements. The full time traces are made up of N time points Xi, Yi with a time between samples

dt. To estimate the velocity at time tn = ndt, the full arrays are broken into smaller time blocks

45

Nb in size starting at index n. From these subarrays, the cross-correlation function, RXY (τ = ldt)

is calculated by shifting one subarray relative to the other, and it is given by,

RXY (τ) =1

Nb

n+Nb−1∑i=n

Xi ∗ Yi+l. (4.1)

It’s assumed here that enough data points exist in the full arrays for an unbiased estimate of RXY

can be calculated at the maximum lag value. The correlation coefficient function is the normalized

correlation function given by,

ρXY (τ) =RXY (τ)√

RXX(0)RY Y (0), (4.2)

and is bounded between −1 < ρ < 1. For a given signal pulse propagating between the two

measurement locations, the peak of the correlation coefficient function occurs at a time lag of τpeak,

and the velocity estimate v(tn) = ∆x/τpeak. The cross-spectral density function is the Fourier

transform pair of the cross-correlation function, so correlation functions can be calculated in the

frequency domain as an alternative to the above description. Common extensions of this procedure

can be made to account for motion in 2D, to include 3 or more points in the estimate, and to

estimate sub-dt time delays.

A number of complications may arise for real applications. First, if the signal pulse expe-

riences significant dispersion during propagation then the peak of the cross-correlation coefficient

function will not coincide with the peak of the envelope function introducing an error in the veloc-

ity estimate. This issue is discussed in some detail in Bendat and Piersol [9]. Second, spatially

extended structures moving orthogonally to the separation direction can yield a non-zero velocity.

For example, a plane wave propagating in the direction orthogonal to the separation with a velocity

v and wave fronts aligned at an angle θ with respect to the propagation direction will be measured

moving at a velocity vo = v tan(θ) in the orthogonal direction. The impact of spatial structure on

TDE and possible solutions have been discussed in [43].

Simple TDE techniques have been implemented for use with NSTX GPI data, however their

use is limited by several issues. First, TDE techniques necessarily produce velocity measurements at

a fraction of the original time resolution, and the GPI time resolution of 2.5 µs is currently enough

46

to only marginally resolves the turbulence dynamics with associated ∼10 µs autocorrelation times.

Therefore, TDE velocimetry applied to the NSTX GPI data is only capable of extracting the slowly

evolving, mean velocity components. Second, correlation structures of edge turbulence are often

spatially extended and tilted or sheared with respect to the propagation direction, so significant

errors are introduced as discussed in the previous paragraph. The addition of a fast photomultiplier

array, as has been done in Alcator C-mod, could increase the time resolution to the point where

TDE techniques become more feasible, and the implementation of more advanced, multi-point TDE

algorithms could overcome the errors introduced by the spatial structure of the turbulence.

4.2 The Optical Flow velocity estimation

The optical flow approach to velocity estimation is derived from the assertion that brightness,

I(x, t) is conserved from frame to frame. The original implementation was described by Horn and

Schunk[61], but a great many extensions have been proposed since[4, 136]. In analogy to the

continuity equation in fluid mechanics, the constraint equation can be expressed as

∂tI +∇ · [vI] = 0. (4.3)

Typically, the more rigid constraint of constant brightness is imposed to give

∂tI + v · ∇I = 0, (4.4)

∂tI + u∂xI + v∂yI = 0. (4.5)

This is equivalent to the continuity equation for the case of divergence free flow. The intensity field

is discretized on the image pixel grid giving one such equation for each pixel, and also two unknown

velocity components for each pixel. Following the procedure of Srinivasan and Chellappa [136],

the velocity field is decomposed into a set of K basis functions φk so that

u =∑k

ukφk , v =∑k

vkφk. (4.6)

Substituting eq. 4.6 into eq. 4.5 gives

∂tIp +∑k

ukφk∂xIp +∑k

vkφk∂yIp = 0, (4.7)

47

where the pixel index p has been included here explicitly. If error is allowed for in the above

equation by introducing an error term εp to the right hand side then the velocity components may

be solved for by a least squares minimization. The procedure is to construct the sum over all pixels

of the squared errors and then set each of the partials of this function with respect to the individual

uk and vk to zero. This process yields 2K equations

0 =∑p

∂tIpφk′∂xIp +∑k

ukφkφk′∑p

∂xIp∂xIp +∑k

vkφkφk′∑p

∂yIp∂xIp

0 =∑p

∂tIpφk′∂yIp +∑k

ukφkφk′∑p

∂xIp∂yIp +∑k

vkφkφk′∑p

∂yIp∂yIp (4.8)

Solving the above equations yields the coefficients uk and vk, and then eq. 4.6 can be used to

recover the full velocity field.

Writing the above in matrix form can reduce the least squares solution to a relatively simple

matrix problem. Equation 4.7 can be expressed in matrix form by using the following definitions,

Apk =

φk∂xIp , k even

φk∂yIp , k odd

and,

xk =

uk , k even

vk , k odd

With these definitions, the equation can be rewritten as

Ax = b, (4.9)

where the time derivatives have been collected into b. It can be shown that the least squares

solution, eq. 4.8 can be rewritten as

ATAx = ATb. (4.10)

Then the solution is provided by inverting ATA,

x = (ATA)−1ATb. (4.11)

48

The procedure to solve for the velocity field is to first construct the arrays A, and b from

the partial derivatives of the intensity field and the chosen basis functions φk. Then, the velocity

coefficients are found by eq. 4.11, and the full velocity field is recovered by eq. 4.6. This process is

then repeated for each frame in the image sequence to give a velocity map v(x, y, t) for each image

at the original image resolution.

The implementation used to examine NSTX GPI data is discussed in Munsat, et al. [105].

The 2D basis functions are taken to be separable, i.e.

φk(x, y) = φi(x)φj(y), (4.12)

with k = Nx ∗ j + i where Nx is the image x dimension. The chosen set of 1D basis functions are

the complete set of discrete second-order coiflet functions. This is a complete set of orthogonal

functions on the discrete pixel grid that are defined as scaled and translated transformations of the

‘mother’ wavelet.

Optical flow estimation schemes experience two major limitations. First, the derived velocity

field cannot measure flow along intensity contours. This can be seen in the original definition of

the constraint equation, eq. 4.5 where the velocity is dotted into the intensity gradient so that only

the components in the direction of the intensity gradient are constrained. This is more commonly

referred to as the ”aperture problem”, and places severe constraints on the extracted velocity field.

One attempt at overcoming this problem is to use spatial smoothing or some other neighborhood

function that uses nearby information to correct this problem. This can help ameliorate the issue,

but the problem can still persists for scales larger than the chosen neighborhood. At a more

fundamental level, there is no information in the intensity field by itself that can be used to

disambiguate motion along contours of intensity. To resolve this kind of motion requires extra

information, further assumptions, or more constraints on the problem. Second, If the motion of

structures is not adequately time resolved the structures can become “lost” by the algorithm. This

is known as temporal aliasing, and effectively imposes an upper bound on the extracted velocity

above which the algorithm gives inaccurate results. Tests have shown that structures that move

49

less than half their spatial width are well tracked by the algorithm[105].

4.3 Pattern Matching velocity estimation

Pattern matching, block matching, or region based matching algorithms [4, 3] have been

employed extensively in the computer vision community for velocity estimation of image sequences,

and this method is frequently used to perform motion estimation for video compression. The

technique used for velocity estimation of NSTX GPI data is adapted from Wildes, et al. [159]

with some modification, and the full details of the method and tests of its performance may be

found in Munsat, et al. [105]. The general concept of the technique is to use a block-matching

algorithm to find the displacement vector that minimizes the error between a subsection of an

image and a subsection of the subsequent image for a specified error functional. In this manner, a

two-dimensional velocity field is obtained for each image in the sequence. The original algorithm

was developed by Munsat, et al. [105], but a number of upgrades and extensions have been

included as part of work performed for this thesis.

For a given image sequence I(x, y, t), the algorithm begins by segmenting the first image into

K subsections (tiles) positioned at (Xk, Yk). Then, for each tile in the image a two-dimensional

search of the subsequent image is performed that seeks to minimize the function

Ferr = FID + λFS , (4.13)

where FID is the absolute intensity difference between image subdivisions, FS represents a velocity

field smoothness constraint, and λ is a free parameter that sets the relative weighting between the

two functions. The intensity difference contribution for the kth tile is

FID =

∫tile|I(Xk, Yk, t)− I(Xk + ∆x, Yk + ∆y, t+ ∆t)|dxdy, (4.14)

where d = ∆xx+ ∆yy is the displacement vector, ∆t is the time between frames, and the integral

is over the tile area. The smoothness contribution is

FS = (1− exp(−τc2s)), (4.15)

50

where

cs =

√dvxdx

2

+dvxdy

2

+dvydx

2

+dvydy

2

, (4.16)

and the velocity derivatives are calculated at (Xk, Yk). The velocity derivative tolerance is set by

the free parameter τ . Typically, the values of λ and τ are chosen such that FS is comparable to

twice the standard deviation of I(x, y, t) multiplied by the tile area when c2s exceeds the tolerance

1/τ . The displacement vector d that minimizes the combined error yields a velocity estimate for

tile k of v(Xk, Yk, t) = d/∆t, and this minimization process is repeated for each tile and each image

to yield a velocity field v(x, y, t).

The choice of error functional described above carries with it a number of consequences.

First, the use of the absolute intensity difference effectively assumes that, to a good approximation,

brightness is conserved between frames. Second, by imposing a smoothness constraint on the

velocity field, it is assumed that the actual flow field does not have velocity gradients that exceed

∼ 1/√τ . In practice it is found that imposing a ?soft penalty? on the smoothness of the velocity

field allows for some abrupt variation, but the occurrence of unphysical or spurious vectors is

reduced. Finally, the smoothness constraint requires knowledge of the local velocity field in the

neighborhood of the kth tile. An initial guess for the velocity field may be provided in any number of

ways, but, fundamentally, accurate velocity information is not known prior to the error minimization

process. To resolve this issue, an iterative, multi-resolution technique is employed to estimate the

velocity field, where the first iteration carries out the minimization process using only the intensity

difference component. The workflow can be summarized as follows,

(1) Segment image into tiles.

(2) For each tile, find displacement that minimizes FID.

(3) Repeat step 2 for each frame.

(4) For each tile, find displacement that minimizes Ferr.

(5) Repeat step 4 for each frame

51

(6) Repeat steps 4-5 for N iterations

(7) Subdivide Tiles.

(8) Repeat steps 4-7 until final resolution is reached.

Upon completion of this procedure a two-dimensional velocity field is obtained for each frame in the

image sequence at a specified final resolution that is some fraction of the original image resolution.

4.4 The Hybrid Optical Flow and Pattern Matching Velocimetry Algorithm

(HOP-V)

The HOP-V algorithm described by Munsat, et al. [105] seeks to combine the optical

flow and pattern matching techniques to overcome some of the limitations of each. The pattern

matching algorithm uses the spatial structure of the intensity field in a neighborhood around the

chosen pixel in estimating the velocity. Often for real image data the intensity gradient is not one-

dimensional, so the pattern matching method is able to overcome the aperture problem. However,

the pattern matching technique needs an initial guess to be provided for the velocity field. The

optical flow solution can fill this need by providing a reasonable first guess to be used by the

pattern matching algorithm. The HOP-V algorithm proposes to apply the optical flow and pattern

matching techniques to image data in series to achieve the best results. Details of the performance

of this technique can be found in Munsat, et al. [105].

Chapter 5 will examine the application of the HOP-V algorithm to GPI data of an oscillatory

L-mode like state preceding the L-H transition in NSTX. In theory, one could use the optical flow

result, and some assumption on its validity, to reduce the extent and convergence time of the

pattern matching 2D error minimization. In practice, however, it is found that the application of

the optical flow and pattern matching techniques in series is extremely time consuming, and results

from the combined technique do not appear to out perform the pattern matching technique seeded

with a random velocity field. Still, a more sophisticated wedding of the two techniques could out

perform the individual application of either. Regardless, a random velocity seeding is performed

52

for the results discussed in chapter 5, and the optical flow technique is not used.

4.5 A Note On Post-Processing Techniques: a Navier-Stokes Inspired

Smoothing Algorithm

The velocity estimation routines discussed above generally assume that translation is the only

allowed transformation of the intensity. For example, the optical flow algorithm assumes constant

brightness, DI(x, t)/Dt = 0. This is a good assumption for the case of rigid body motion that

these routines were originally developed for, but this can be inappropriate for the velocity fields of

neutral fluids and plasmas. One method to make the extracted velocity fields more ”fluid-like” is to

apply a Navier-Stokes inspired iterative smoothing algorithm to the output of the above mentioned

velocity extraction routines. An adaptation of the original technique developed by Doshi and

Bors [32, 33] has been implemented, and a brief discussion is included here.

The Navier-Stokes equation for an incompressible Newtonian fluid is

∂tv + v · ∇v = −∇p+D∇2v + f (4.17)

where p is the pressure, D is the diffusion coefficient, and f is the “body forces” present in the

system. The smoothing routine attempts to approximate fluid like evolution of the velocity field in

a 3 step process iterative process. First, an anisotropic diffusion process is applied to the velocity

field. Second, velocity information is advected forward and backward in time. Finally, the original

velocity field is updated with the smoothed vector by a weighted averaging,

vi(k + 1) = (1− ε)vi(k) + εvSi (k) (4.18)

The diffusion step employs an anisotropic kernel that smooths the velocity field mainly along

the direction of the local flow, and attempts to preserve velocity gradients. This is done using

components of the local Hessian matrix of the flow,

S =

∂xxu ∂xyu

∂yxv ∂yyv

53

The diffusion is applied using a gaussian kernel with the covariance matrix S. The smoothed

velocity field is given by

vDc (k + 1) =

∑xi

vi(k) exp[−(xi − xc)TS−1c (xi − xc)]∑xi

exp[−(xi − xc)TS−1c (xi − xc)(4.19)

where vDc (k+1) is the diffused velocity at position xc for iteration k+1, Sc is the covariance matrix

computed at position xc, and vi(k) is the velocity vector at position xi in the neighborhood N(xc).

Following the diffusion, a median filter is applied to the velocity field to improve the performance

of the diffusion in the presence of outliers.

The advection process is approximated by a weighted averaging of the velocity v(x, t) and

v(x+vdt, t+dt). This is preformed both forward in time and backwards in time. For each velocity

vi(xn, ym, t) = {ui(xn, ym, t), vi(xn, ym, t)} in the vector field at time t and iteration i, the velocity

at time t+ 1 for the forward in time advection process is given by

ui+1(xn′ , ym′ , t+ 1) =wnm

2ui(xn, ym, t) + (1− wnm

2)ui(xn′ , ym′ , t)

vi+1(xn′ , ym′ , t+ 1) =wnm

2vi(xn, ym, t) + (1− wnm

2)vi(xn′ , ym′ , t) (4.20)

where the weights are given by

wnm = exp[−√

(xn′ − xc)2 + (ym′ − yc)2], (4.21)

and (xc, yc) = (xn, ym) +vi(xn, ym, t)dt. The averaging is preformed in a 2×2 grid point neighbor-

hood around (xc, yc). Admittedly, this is somewhat of a kludge, and an improvement would be to

implement a second-order accurate integration scheme for the numerical advection (e.g. See Rood

[114])

Results of the application of this Navier-Stokes post-processing to output of a pattern-

matching velocity extraction method for a synthetic sheared-flow test case can be seen in 4.1.

The velocity field smoothness constraint was removed for this test, and as a result some noticeable

velocity discontinuities exist in the resulting unsmoothed velocity field in Fig. 4.1(a). Also, signifi-

cant edge effects can be seen at the top and bottom of the frame where the structures leave the field

54

of view. After application of the Navier-Stokes smoothing the discontinuities have been suppressed,

and the edge effects have been lessened as seen in Fig. 4.1(b). Profiles for the unsmoothed vx and

vy and smoothed vx and vy can be seen in Fig. 4.1(c) and (d), and Fig. 4.1(e) and (f) respectively.

The affect of the smoothing is to decrease the overall variation in the flow, but smoothing also leads

to a slight underestimation of the peak velocity. Smoothing also pulls the average velocity up at

the edge of the view.

The results of the Navier-Stokes smoothing are quite promising, though some errors are

introduced near maxima and minima of the flow that are common to all smoothing operations. The

algorithm is an iterative adjustment of the original flow field requiring a number of calculations

to be done for each velocity vector, so application of this method to real data can be quite time

consuming. In practice it is found that the time costs of applying this post-processing to the velocity

fields is generally not worth the marginal gain in accuracy compared to the HOP-V routine with the

added smoothness constraint. Still, if a fast algorithm with an improved treatment of the advection

could be developed then this technique could be quite useful for denoising or correction of aperture

effects of velocity fields extracted by optical flow or other velocimetry techniques.

55

Figure 4.1: Pattern Matching vector fields for (a) unsmoothed and (b) smoothed. Profiles of vx andvy for unsmoothed, (c) and (d), and smoothed, (e) and (f). Red traces are the imposed velocityfield.

Chapter 5

Measurement of 2D flows in the edge and SOL preceding L-H transitions

Since its discovery more than 30 years ago [156], the high confinement mode, or H-mode,

and the associated transition from the lower confinement L-mode has been studied extensively as

evidenced by the great many experimental and theoretical reviews [149, 17, 52, 23, 63, 68, 137, 155,

49]. In this chapter, a GPI study of L-H transition dynamics on NSTX is presented. In this study,

flows and turbulence behavior in the edge of NSTX preceding the L-H transition is characterized,

and observations of oscillations between a low and high turbulence state preceding the transition

are presented. First, however, a brief overview of the L-H transition accented by observations of

L-H transitions in NSTX is presented, and a brief discussion of theories of the transition is included.

Following this, the drift wave zonal flow paradigm and associated predator-prey model of the L-

H transition are discussed in detail. Finally, the experimental observations are evaluated in the

context of the predator-prey model.

5.1 The L-H transition

With the application of auxiliary heating by neutral beams or RF waves, a transition is

often observed in magnetic confinement devices from a state of low confinement to a state of high

confinement where plasma fluctuations and transport are drastically reduced at the edge of the

device. Originally observed in a divertor tokamak [156], the improved confinement regime has been

realized on a wide array of magnetic confinement devices including divertor and limiter tokamaks,

stellarators, and even in a mirror machine with limiter biasing [119]. At the transition to H-mode

57

an abrupt drop (sub ∼100 µs) in the light emission from recycling neutrals generated by plasma

material interactions is observed, and plasma density begins to increase. These two observations

imply an increase in confinement, and provide an indicator for the transition. A transport barrier

is formed just inside the separatrix with a ∼1 cm width that reduces edge fluctuation levels and

particle, momentum, and energy transport. As a result edge pressure gradients steepen, energy

confinement times improve by a factor of ∼2, and subsequent increases in plasma β and stored

energy are seen.

Traces of operational and plasma parameters for 2 different NSTX shots are plotted in Figure

5.1. One shot, 141740, is an ohmic L-mode shot, and the other, 138692, is a neutral beam heated

H-mode. The L-H transition occurs at 0.166 s as indicated by the abrupt drop in the divertor Dα

emission. Following the transition to H-mode, the energy confinement time increases steadily, and

the density and temperature rise leading to an significant increase in stored energy. The energy

confinement time for the H-mode case is comparable to the Ohmic case, but still greater than

typical auxiliary heated L-mode energy confinement times. The difference in the L and H-mode

density and temperature profiles are illustrated by Fig. 5.2. Steep gradients in both density and

temperature are developed at the edge during H-mode indicating the presence of an edge transport

barrier. In NSTX, the H-mode density profile tends to become hollow with a peak near 85% of the

minor radius.

GPI measurements exhibit a significant drop in fluctuations at the L-H transition as seen

in Fig. 5.3. The fraction of GPI light contained in the SOL, FSOL is a clear indicator of the

transition. Average values of FSOL drop by ∼7% at the transition, and fluctuations in FSOL are

significantly reduced. This implies a significant drop in transport into the SOL and an improvement

in confinement. Raw GPI signals inside the separatrix increase in mean value at the transition,

and the normalized fluctuation level drops significantly.

A number of theories have been put forth to explain the creation of the edge transport

barrier and thus the transition to H-mode. One such class of theories argue for a change in plasma

conditions leading to the stabilization of certain instabilities at the edge as the mechanism for

58

Figure 5.1: Traces of (a) plasma current and injected neutral beam power, (b) diverter Dα light,(c) Energy confinement times, and (d) plasma stored energy. Black traces are for an ohmic L-modeshot, and red traces are for a neutral beam driven H-mode. L-H transition timing indicated bydashed red line.

producing an H-mode. Theories involving the stabilization of Ideal MHD ballooning and peeling

modes, drift resistive ballooning modes, collisional drift waves, or drift-Alven waves have been

developed, and each found varying success in predicting transitions on various machines [49, 23].

However, none of these theories were able to consistently explain experimental results across all

machines and all transition types (e.g. neutral beam, sawtooth, or pellet injection triggered).

Another class of theories that has found considerable success in describing the H-mode tran-

sition propose the suppression of turbulence and transport via E×B shearing [126, 10, 69, 148, 23].

Theoretical work showed that a shear in the E × B velocity can act to reduce the radial corre-

lation length of turbulence or de-phase the density and potential fluctuations, thus reducing the

59

Figure 5.2: Thomson scattering profiles of electron temperature (top) and electron density (bot-tom).

Figure 5.3: Traces of (a) fraction of GPI light in SOL, (b) GPI raw signal near separatrix, and (c)normalized GPI fluctuation level.

turbulence and transport. A sufficient requirement for the quenching of turbulence by flow shear

is expressed as ωE×B ≥ γMAX , where ωE×B is the shearing rate and γMAX is the maximum linear

60

growth rate [157, 16].

Experimental observations have confirmed the development of a layer of Er shear at the edge

of the plasma with a ∼1 cm width at the H-mode transition, and this shear layer is correlated

with the reduction in turbulent fluctuations, transport, and radial correlation lengths as expected

[49, 155, 17]. Furthermore, it has been shown that changes in Er precede changes in the density

and temperature profiles [53], and transitions have even been induced by biasing of the edge with

probes [145, 17]. Taken together, these observations point to the development of Er shear as the

driving mechanism for the transition.

Even still, with all the experimental and theoretical work that has been performed on the

H-mode transition, clear evidence which could determine the mechanism that creates the Er shear

layer and triggers the transition into H-mode has escaped us. A number of mechanisms have

been proposed to explain the creation of a sheared flow layer at the edge including ion orbit loss,

neoclassical poloidal flows, Stringer spinup of poloidal flow, and zonal flows driven by turbulent

Reynolds stress [49, 23]. Other theories include Bifurcation theories and phase-transition models

[23]. Very recently, many experimental studies have indicated the importance of the interaction

between Zonal flows (m=0,n=0 potential structures) and turbulence in the L-H transition [39, 40,

25, 98, 121, 153, 162, 95], and several studies suggest nonlinear transfer of energy from turbulence

to the Zonal flow may provide an H-mode trigger mechanism [153, 95, 26]. In addition, the NSTX

GPI study of flows preceding the L-H transition discussed in this chapter presents a number of

results with qualitative similarity to the limit cycle oscillation of Predator-Prey models of the L-H

transition [80, 81]. In light of these results, the following section will give a closer look at models

of edge transport barrier formation including zonal flows.

5.2 The Drift-Wave Zonal Flow Paradigm

The most complete theory of transport in magnetically confined plasma devices is the neo-

classical theory which describes the diffusion of particles, heat, and momentum due to coulomb

collisions and geometrical effects. In magnetic confinement devices, experimentally measured trans-

61

port often significantly exceeds the neoclassical predictions, thus this enhanced transport is termed

‘anomalous’ transport. Experiments have indicated that anomalous transport is due to turbulent

transport driven by plasma instabilities. The interactions between the plasma equilibrium, micro-

instabilities, and anomalous transport is encapsulated in the Drift Wave paradigm (blue boxes of

Fig. 5.4). More recent work in geophysical and planetary flows (see Busse 1994 [18]) and plasma

turbulence [30] has highlighted the importance of the interaction of turbulence with mesoscale

flows. These Zonal flows are self-generated by the turbulence, and can interact with, and even reg-

ulate, the turbulence via shearing. This motivated the shift to a Drift Wave Zonal Flow (DWZF)

paradigm that describes a self-regulating turbulence system. This section seeks to motivate each

of the interactions included in the DWZF paradigm which will be important for the Predator-Prey

model of the LH transition described in the following section.

Figure 5.4: Conceptual picture of the drift wave zonal flow paradigm. Gradients drive instabilitiesto turbulent state. Turbulence drive anomalous transport via fluctuations. Turbulence also self-generates zonal flow via Reynolds stresses. Zonal flows regulate turbulence via shear suppression.

The drift wave is a class of instabilities that tap into the free energy provided by gradients in

plasma pressure. As gradients are inherent to magnetically confined plasmas, the drift wave is one

of the most fundamental instabilities. Experimental evidence also points to these low frequency

modes as a major driver of transport in tokamaks and other magnetic confinement schemes. The

discussion of drift waves included here loosely follows that in Bellan [8]. The basic mechanism can

be illustrated by a simple example with a density gradient in the x direction, magnetic field in z,

and the plasma is taken to be collisionless. Ion temperature is assumed to be less than electron

62

temperature, so that ion pressure forces may be ignored. Electron thermal velocity is taken to be

much greater than parallel phase velocity of the perturbation (ω/kz � vTe) so that the electron

parallel momentum balance yields a Boltzmann response

ne1ne0

= −qeφ1Te

. (5.1)

Since ω � ωci and ions are assumed to be cold, the lowest-order perpendicular ion motion given

by the ion momentum balance equation is simply E×B drift,

Ui1 =−∇φ1 ×B

B2. (5.2)

Invoking quasi-neutrality gives ne1 = ni1. Then substituting Eqn. 5.1 and 5.2 into the linearized

ion continuity equation and Fourier transforming yields

ω = −kyTeeB

∂xni0ni0

= kyvde. (5.3)

Thus the presence of a pressure gradient in a magnetized plasma provides for the propagation of a

wave of coupled density and potential perturbations in the electron diamagnetic drift direction, vde.

This simplified drift wave mechanism is illustrated in Fig. 5.5. E×B circulation around potential

perturbations carries plasma from low density areas to high density areas and vice versa. The net

result is that the perturbation propagates in +y.

The inclusion of collisional drag between electrons and ions in the momentum balance equa-

tions results in a phase shift between the density and potential responses given by

ne1ne0

= −eφ1Te

−ikyvde + k2zTe/νeime

−iω + k2zTe/νeime. (5.4)

This effectively shifts the contours of the density and potential perturbations in Fig. 5.5 relative

to each other, so that the E × B motion about the potential structure reinforces the density

perturbation. Thus the wave is destabilized.

If a successive approximation of the ion perpendicular motion is used in the derivation of

collisionless drift waves sketched above then the next order is found to include the ion polarization

63

Figure 5.5: Illustration of the drift wave mechanism. The background density gradient is in the−x direction, and the magnetic field is in z. E×B circulation around potential perturbation (bluecontours) pushes high density from −x and low density from +x producing a propagation of theperturbation in +y.

drift

Ui1 =−∇φ1 ×B

B2− 1

ωciB

d

dt∇φ1 (5.5)

with d/dt being the convective derivative given by d/dt = ∂t − U · ∇. Retaining this term in

the derivation and keeping quasi-linear combinations of first order fluctuating terms yields the

Hasegawa-Mima equation [57], the simplest drift wave model that includes a nonlinear interaction.

This nonlinearity originates from the action of the convective derivative on the E×B motion. The

result is a single field equation for the nonlinear evolution of wave amplitudes

∂tφk + iωkφk =1

2

∑k=k1+k2

Nk,k1,k2φk1φk2 (5.6)

with

Nk,k1,k2 =1

1 + k2(k1 × k2 · z)(k22 − k21), (5.7)

and

ωk =ky

1 + k2. (5.8)

Here, quantities have been normalized in the following way, φ = Lneφ/ρsTe, x = x/ρs, t =

ρsωcit/Ln, and L−n 1 = ∂x lnn0. Multiplying Eqn. 5.6 by φ∗k gives an equation for the evolution of

64

wave energy

1

2∂t|φk|2 + iωk|φk|2 =

1

2

∑k=k1+k2

Nk,k1,k2φk1φk2φ∗k. (5.9)

The sum on the right describes the nonlinear three wave coupling process. This process is ubiquitous

in plasma turbulence due to the inherent nonlinearity of the convective derivative, and it describes

the spreading of turbulence in wavenumber space. Instabilities driven in some range of unstable

wavenumbers will transfer energy to other, linearly stable modes in a cascade mediated by three

wave interactions. Effectively, this process ensures that instabilities with sufficiently strong driving

will produce a turbulent spectrum of fluctuations. Many extensions to this simple picture exist

(e.g the two field Hasegawa Wakatani model [58]), but the simple models serve to illustrate the

fundamental characteristics. Taken together, the drift-wave mechanism and the three wave coupling

process describes the formation of a gradient driven turbulence state in magnetically confined

plasmas.

Next, we will look at the implications of the turbulent state on the fluxes. Neoclassical

theory considers plasma variables n, p, T , E, B, etc. to be stationary, however experiments observe

significant fluctuating components of these quantities. These fluctuations can drive transport which

is not captured by the neoclassical theory. As a demonstration of this for electrostatic fluctuations,

consider the fluid moment equation for ion momentum:

min∂tU +minU · ∇U = neE + neU×B−∇p+ R (5.10)

where R is the collisional friction term, and other quantities have their usual meanings. We take

all quantities save B to be fluctuating quantities with fluctuation A defined as A = 〈A〉+ A and 〈·〉

denotes the ensemble average. Upon dropping the kinetic terms on the left hand side, performing

an ensemble averaging of the equation, and rearranging, the quasi-linear perpendicular particle

flux, Γ⊥ = nU⊥ is found to be [115]

〈Γ⊥〉 = − 1

eB2∇〈p〉 ×B +

1

B2〈n〉〈E〉 ×B +

1

B2〈nE〉 ×B +

1

B2R×B (5.11)

The first two terms on the right hand side describe the within flux surface diamagnetic and equi-

librium E × B particle flux contributions. The third term gives the fluctuation driven E × B

65

particle flux, and the final term describes the particle flux driven by collision processes. This sim-

ple demonstration reveals that particle flux can be driven by electrostatic fluctuations if the density

and electric field perturbations are correlated. The particle flux can easily be calculated from the

perturbed density and potential found in the collisional drift wave model described above giving [8]

Γx =k2yρ

2s

1 + k2yρ2s

k2y|φ1|2

2B2

νeime

k2zTe

dn0dx

. (5.12)

This flux is proportional to the density gradient, so the net effect is an enhanced diffusion driven

by drift wave fluctuations.

The remaining interactions in the DWZF paradigm requiring motivation are the interactions

between the turbulence and the zonal flow. In the context of toroidal plasmas, the zonal flow (see

Fujisawa 2009 [47] for review) is a potential structure which is constant on a flux surface, so

it has a toroidal mode number n = 0 and poloidal mode number m = 0. It has a finite radial

wavenumber which lies between the turbulence scales, on the order of the ion gyroradius, and the

system scales described by the minor radius of the device. This scale range can be expressed as

λr ≈ 10 − 50ρi. The zonal potential structure gives rise to radial electric which drives a poloidal

flow in the plasma. Since the zonal flow is a poloidal flow that is constant on a flux surface, it will

drive a pressure perturbation due to the poloidal asymmetry of the toroidal geometry. Essentially

more plasma exists on the outside of the torus than the inside due to the geometry. This pressure

perturbation may be relieved by driving a toroidal flow, or driving an oscillating flow that couples

to a pressure perturbation. The former is the stationary or zero mean frequency zonal flow, and the

latter is the geodesic acoustic mode. In addition to the previously mentioned characteristics, the

geodesic acoustic mode is associated with an m=1 density perturbation and is predicted to oscillate

at frequency between 4-12 kHz in the NSTX edge [167]. Zonal flows have also been discovered in

neutral fluid systems, and are seen, for instance, in the atmospheres of the major planets in our

solar system (e.g. Jupiter’s stripes) [18].

Now, let’s examine the self-generation of the zonal flow by the turbulence. In a sense, this

process has already been described by the three wave coupling process described in the Hasegawa

66

Mima model (Eqn. 5.6). In 2D turbulence, the cascade of wave energy due to three wave processes

is often such that wave energy is transported to larger scales, and eventually this energy condenses

at the lowest wavenumbers. The zonal flow generation occurs by a modulational instability, a non-

local process in wavenumber space involving the coupling of two high-k drift waves into a low-k zonal

mode [30, 47]. Alternatively, the generation of large scale structured flows can be understood as

the result of work done by Reynolds stresses. Averaging the poloidal momentum balance equation

over flux surface yields the equation for evolution of the zonal flow [31, 148]

∂t〈Uθ〉 = ∂r〈uruθ〉 − µθ〈Uθ〉 (5.13)

where µθ is the poloidal flow damping rate, and 〈·〉 denotes a flux surface averaging. The first term

on the right hand side clearly represents a generation of zonal flow by the divergence of turbulent

driven momentum flux, the Reynolds stress.

The back reaction of the zonal flow on the turbulence via flow shearing can be demonstrated

by a simple dimensional argument [148]. Consider an eddy in a background sheared flow uy(x).

The shear flow will stretch the eddy by a distance ∆ys in a time τs given by

∆ys = τs∆xe∂xuy. (5.14)

The factor ∆xe∂xuy gives the difference in y velocity across the eddy. During this time, the eddy

will advect a fluid element a distance

∆xs = τsux = ∆xeτsτe

(5.15)

where ∆xe and τe are the eddy x dimension and eddy lifetime, respectively. If the shear is strong

(τs < τe) then ∆ys, the shear distortion, can exceed the turbulence coherence length, and the eddy

will lose coherence due to interaction other turbulent flows. The eddy lifetime is then set by τs,

and Eqn. 5.15 shows that the shear-wise eddy dimension is reduced. Thus the turbulence becomes

decorrelated, the correlation length across the flow is shortened, and subsequently the transport

is reduced. Furthermore, it has been shown [10] that this shear interaction can also reduce the

amplitude of the turbulent fluctuations.

67

The discussion in this section introduced the Drift Wave Zonal Flow paradigm, and provided

motivation for the various interactions illustrated in Fig. 5.4. In the next section we will examine

a theory for the L-H transition based on these interactions.

5.3 The Predator-Prey model of the L-H transition

Motivated by the possibility for Zonal Flows to regulate the turbulence amplitude and trans-

port, a simple 0D model of the L-H transition was proposed by Kim and Diamond [80, 81]. The

model system evolves equations for the Turbulence amplitude ε, the zonal flow shear V ′ZF , and

the gradient of the density profile N = −(Ln/n)∂rn. Since the zonal flow generation is directly

dependent on the turbulence amplitude, a sustained state of reduced turbulence (i.e. an H-mode)

requires the inclusion of a mean flow shear. The model incorporates this by considering a reduced

momentum balance equation given by

〈V 〉 = − 1

eBz

1

n

dpidr. (5.16)

This assumption neglects the affects of toroidal and zonal flows in the momentum balance, and

furthermore does not address the question of the generation of an Er during the H-mode. In

this regard, the model serves only to demonstrate how Zonal Flows may affect a transition. A

more sophisticated treatment of the momentum balance is required to fully understand the edge

transport barrier formation. It should be kept in mind that the model is only a 0D model, and

thus only a qualitative description of the L-H transition dynamics is offered.

The model equations are as follows:

∂tε = εN − a1ε2 − a2V′2ε− a3V

′2ZF ε, (5.17)

∂tV′ZF = b1

εV ′ZF1 + b2V

′2− b3V ′ZF , (5.18)

∂N = −c1εN − c2N +Q, (5.19)

V ′ = dN2. (5.20)

The evolution of the turbulence amplitude (Eqn. 5.17) is given by, from left to right, linear

68

growth driven by the density gradient, nonlinear saturation, suppression by mean flow shear, and

suppression by zonal flow shear. The evolution of the zonal flow shear (Eqn. 5.18) is given by

turbulent generation of zonal flow by Reynolds stress, and zonal flow damping. The factor 1/(1 +

b2V′2) represents the suppression of zonal flow drive by mean flow shear as described in Kim and

Diamond [80]. The density evolution (Eqn. 5.19) is affected by turbulent diffusion, neoclassical

diffusion, and auxiliary heating input. Finally, Eqn. 5.20 closes the system of equations by using

Eqn. 5.16 to express the mean flow shear in terms of the diamagnetic flow. In addition to the

simplified motivations provided in the previous section, this set of equations and the interactions

included in them can be justified by a wave kinetic treatment of the problem [94].

The evolution of the system toward an H-mode state can be studied by varying the input

power Q as a function of time. The evolution exhibits three main phases. First, at low input

power drift waves are driven unstable, and ε grows linearly. Second, the zonal flow begins to grow

when the turbulent drive overcomes the damping. As the flow grows it begins to interact with the

turbulence, and a limit cycle phase is observed. Finally, at high input power the density profile

grows, driving a mean flow shear, and the turbulence is suppressed as the system transitions into

an H-mode like state.

The limit cycle dynamics and the transition to H-mode can be described by conceptual picture

given by Fig. 5.6. The limit cycle evolves in the following way. Gradient driven instabilities drive

growth of turbulent fluctuation level. When the zonal flow damping is overcome by the turbulent

drive, zonal flow amplitude begins to grow. Once the zonal flow is strong enough it begins to

suppress the turbulence via shearing. As the turbulence decays, the zonal flow drive decreases

accordingly. Eventually the zonal flows are damped away, and the turbulence begins to grow again.

A transition to a quiescent, H-mode state is possible if the input power is strong enough to develop

a mean flow shear during the period of reduced turbulence.

Many experimental studies in recent history have observed qualitatively similar dynamics

to this limit cycle oscillation [39, 40, 25, 98, 121, 153, 162, 95]. Most notably, perhaps, is the

recent work by Schmitz which details the 1D plus time dynamics of the turbulence amplitude and

69

Figure 5.6: Conceptual picture of limit cycle oscillation process. System evolves between a highturbulence, low flow state and a high flow, low turbulence state. If the input power is sufficientto steepen the density gradient during a period of suppressed turbulence, then a mean flow sheardevelops and a transition to H-mode is observed.

radial electric field at the edge using Doppler Backscattering. The promising similarity between the

experimental results and the simple model has motivated the extension of the 0D model to a 1D

plus time dynamical model [101]. In the next section, we’ll examine a set of experiments on NSTX

that exhibit qualitative similarity to limit cycle oscillations described by the simple 0D model.

5.4 GPI Observations of Flows preceding L-H transitions

The interaction between drift wave turbulence and zonal flows make it clear that flow orga-

nization can play an important role in magnetic confinement devices. Recently, Zweben, et al.

[167] reported on the observation of ’quiet periods’ in the National Spherical Torus Experiment

(NSTX)[77] edge prior to the L-H transition. These quiet periods were observed with the GPI

diagnostic, and they found that the GPI Dα light emission during these periods resembled that of

H-mode. The quiet periods occurred at a frequency of ∼3 kHz, and were correlated with changes in

the direction of the poloidal flow. They also found that the dimensionless poloidal shearing values

70

were correlated with the quiet periods.

In this chapter, we analyze the same database of shots as Zweben, et al. [167]. However,

we use the hybrid optical flow and pattern matching velocimetry (HOP-V) technique [105] to derive

2D velocity fields using GPI data from the NSTX edge region. This technique yields time resolved

velocity fields with a spatial resolution of ∼1.5 cm, at 16×16 spatial points, and time resolution of

3.5 µs. Using the derived velocity fields and GPI data we report on three main results: (1) detailed

characterization of space and time evolution of zonal flow features, (2) calculation of turbulence

shear stress and its radial profiles, (3) relationship between zonal flow and turbulent bursts.

5.5 Time averaged-flow profiles

GPI data for each of the shots discussed in this chapter (#135021-23, #135041-46) have an

image size of 64×64 pixels, and the HOP-V algorithm calculates velocities for tiles of 4×4 pixels.

This produces a 16×16 point velocity field at the temporal resolution of the image sequence as

illustrated by Fig. 5.7a. In the Figure the time averaged velocity field for a period of approxi-

mately 10 milliseconds during L-mode operation is superimposed upon time averaged contours of

GPI intensity. The GPI intensity falls off near the top and bottom of the view due to optical

vignetting and the limited size of the GPI emission cloud. The velocity field has been cropped to

an approximately 10.4 cm × 14.8 cm (7×10 points) section in the center of the GPI field of view

to avoid the region of reduced signal, and to avoid edge effects introduced by structures that are

not tracked as they travel out of the GPI view. Also, due to the emission profile of the GPI gas

puff, the intensity fluctuations outside this boxed region are not high enough to consistently yield

strong correlations for the HOP-V algorithm, and as a result velocity data from these regions is

often not reliable. All analysis (poloidal averaging, etc.) in this work will be limited to the boxed

region of Fig. 5.7a. The approximate location of the separatrix (dot-dashed line) is determined

by the NSTX-standard model LDRFIT, and is uncertain up to ±2 cm. The exact location of the

separatrix is not, however, a critical aspect of the analysis. The radial position is displayed relative

to the separatrix position.

71

Figure 5.7: (a) Time averaged velocity field superimposed on time averaged GPI intensity contours(gray contours). The separatrix is indicated by the dot-dashed line, and the cropped field of viewis indicated by the dashed box. The maximum time-averaged velocity magnitude for the croppedregion is 1.5 km/s, while the maximum instantaneous velocity magnitude is 7.4 km/s. The +xdirection is radially outward. (b) radial profiles of time averaged poloidal flow several shots for 10.5µs preceding the L-H transition. RMS values of the fluctuating velocity are shown for shot 135042.

(a)

(b)

radial

poloidal

Figure 5.7 serves to illustrate some general characteristics of the observed flow. In movies of

the GPI signal (viewable in the multimedia section of Zweben, et al. [167]), intensity structures

(“blobs”) are typically seen to move with positive y-velocity (upward, in the electron diamagnetic

direction) in the inner region (x ≈ -5-0 cm), while slowly moving radially outward. As the structures

cross the separatrix and move into the scrape off layer (SOL), their poloidal velocity reverses

direction, and they move downward (in the ion diamagnetic direction). Figure 5.7b shows a similar

72

pattern with negative average velocities outside the separatrix, while flow reversals tend to average

to zero near the separatrix. Average velocities are on the order of 1.5 km/s poloidally and .5

km/s radially, and instantaneous velocities can be significantly higher, as discussed below. Several

centimeters inside the separatrix, the average poloidal velocities are positive. Thus a non-zero time-

averaged poloidal flow sheared is present at the separatrix. A similar pattern in the time-averaged

flow field is seen in shots 135041-135046, and similarities can be seen in Figure 5.7b. In addition

most all shots show a clear downward average flow outside the separatrix. ‘Error bars’ in Figure

5.7b indicate RMS levels of the fluctuating poloidal velocity for shot 135042, so it can be seen that

the velocity fluctuations are on the order of the mean flow values.

5.6 turbulence “Quiet-Periods” and quasi-periodic velocity fluctuations

Figure 5.8: Normalized GPI intensity (top), poloidal velocity (middle), and radial velocity (bottom)∼1 cm inside the separatrix. All traces represent poloidally averaged quantities from shot 135042.

To illustrate temporal behavior, time traces for normalized GPI intensity, poloidal velocity,

and radial velocity are shown in Fig. 5.8 (shot 135042, t ≈ 0.241-0.243 s). In the Figure, each

trace represents a poloidal average, localized radially approximately 1 cm inside the separatrix,

73

which is near the time-averaged GPI intensity maximum. Gray outlines of Vp and Vr indicate the

level of uncertainty in these values assuming random, statistically independent uncertainties in the

individual velocity measurements of 0.5 km/s (∼0.5 pix/frame). In the averages shown, the level

of uncertainty is ∼0.15 km/s.

The GPI intensity is normalized by subtracting from each pixel its mean value taken over the

full 17000-frame exposure and then dividing by the mean, so a value of 1.0 indicates a fluctuation

which is on the order of the mean value. Typically, poloidal velocities are seen to range between

±10 km/s. Radial velocities are generally on the order of a few kilometers per second, and can

approach the minimum threshold of measurable velocities of ∼0.5 km/s. As mentioned previously,

the poloidal velocity changes from being predominantly positive inside the separatrix to negative

in the SOL. Near the separatrix, the poloidal flow is seen to fluctuate around zero with periodic

reversals.

Time traces for the GPI intensity and poloidal flow quantities, like those in Fig. 5.8, of all

shots examined here exhibit distinct quasi-periodic oscillations/bursts at a dominant frequency of

3 kHz. To compare the poloidal flow oscillation with the behavior of the turbulence, we define The

quantity Fsol as the fraction of the GPI Dα intensity contained in the SOL to total image intensity,

and a time trace of Fsol is plotted in Fig. 5.9a along with poloidal velocity traces at three radial

locations (chosen inside, at, and outside the separatrix, shown in Fig. 5.9b,c,d respectively). A

spike in Fsol corresponds to intensity structures (blobs) passing into the SOL. In this way Fsol can

be used as a proxy for turbulent transport at the edge. Periods of low Fsol indicate little activity

in the SOL and no or few occurrences of blobs, similar to GPI observations of H-mode plasmas.

These periods are described as “Quiet-periods” [167], and are indicated in Fig. 5.9 by the vertical

gray bars. Periods of high Fsol indicate increased SOL turbulence activity and frequent creation

and ejection of blobs.

Low frequency components of Fsol between 0-1 kHz have been removed from the filtered

Fsol trace in Fig. 5.9a. These frequencies are dominated by the slow rise and fall of the total

GPI emission due to the dynamics of the gas puffing. The gray bars indicate where the filtered

74

Figure 5.9: Traces from shot 135042 of Fsol filtered (a), and poloidally averaged poloidal velocityinside (b), at (c), and outside (d) the separatrix. Fsol filtered has had the low frequency componentsabove zero frequency and below 1 kHz removed. Gray bars indicate time periods where Fsol filteredis below 0.16.

75

Fsol is below 0.16 (arbitrarily chosen as a rough guide to distinguish “high” vs. “low”). Periodic

behavior in Fsol is easily discernible; the large scale oscillations between t = 0.241− 0.243 sec have

a frequency of ∼3 kHz. However, this periodicity in Fsol is somewhat irregular and appears to be

intermittent.

The intermittency of this ∼3 kHz feature is demonstrated by the spectrogram of Fsol shown in

Fig. 5.10. The spectrogram covers the entire recorded L-mode portion of shot 135042 with a window

of 256 time points (∼0.9 ms), giving roughly 1.1 kHz resolution. In constructing the spectrogram, a

sliding fast-Fourier-transform is used with Hanning windowing and 75% overlap. The spectrogram

shows a dominant mode that appears sporadically near 3 kHz. This mode fluctuates between 2-4

kHz, and tends to have relatively stationary durations of ∼1-2 ms. These characteristics are directly

apparent in Fig. 5.9a as well. Periodic bursts in Fsol and poloidal velocity at about 2.5 kHz start

near 0.2410 s and are clearly observable until 0.2420 s, then after 0.2425 s this periodicity revives.

During these time periods, Large scale fluctuations of the poloidal velocity appear to be about 180◦

out of phase with Fsol.

Figure 5.10: Spectrogram of Fsol for the L-mode portion (t ≈ 0.215−0.245) of shot 135042 plottedwith a linear color scale.

Shot 135021, and 135041-44 each have a feature in the spectrogram of Fsol with similar

qualities to shot 135042. This feature is generally centered between 2 and 5 kHz, and fluctuates

within a ∼2 kHz range around this center value. The feature is stationary for durations of 1-2

ms such as with shot 135042. The other shots (135022-23, and 135045-46) show a similar feature,

76

however it is less distinct in these shots and often does not persist through the entire shot.

The measured poloidal velocities shown in Fig. 5.9 also exhibit strong correlation with Fsol.

Periods of low Fsol (quiet periods) are correlated with positive peaks in poloidal velocity inside

the separatrix, and peaks in Fsol correspond to periods of nearly zero flow. Outside the separatrix

the reverse is true, with poloidal velocities tending toward zero during quiet periods and peaking

negative during turbulent bursts. The correlation coefficient at each of the three locations chosen

between Fsol and Vp for the time period shown is ∼0.6.

Correlation values for HOP-V tiles between successive images (which measure the “tracking

quality” of the velocimetry algorithm) for velocity vectors in the SOL tend to decrease during quiet

periods, but a majority of the vectors remain above an 80% correlation threshold. Therefore, it

appears that that the SOL velocities are not simply an artifact of low GPI signal in the SOL.

Reversals in poloidal velocity can be seen at the separatrix, with positive velocities observed

during quiet periods, switching sign as Fsol rises. The correlation between these quantities suggests

a relationship between the poloidal flow and the turbulent bursts. This may indicate that some

feature of poloidal flows acts to control the turbulence, or that the flow is generated by the release

of blobs. A causal relationship is, however, unclear at this point, and begs further inquiry.

5.7 Spatial Structure of 3 kHz mode

The spatial structure of the ∼3 kHz mode may be analyzed by first applying a bandpass

filter to the GPI signal around this dominant frequency. For t ≈ 0.241−0.243 s of shot 135042, the

filter is used to select a single mode near 2.4 kHz, corresponding to the frequency of peak spectral

power for GPI intensity ∼1 cm inside the separatrix. With the applied filter, a rotating mode is

clearly observed which exhibits long poloidal wavelength and radial size greater than 4 cm. The

mode appears in the images as an oscillating band centered 0.5 cm inside the separatrix with a

radial width of ∼2 cm that spans the full GPI view in the poloidal direction.

An alternative characterization of the spatial structure of this mode is illustrated by the

X-Y phase maps shown in Fig. 5.11. Figure 5.11a(5.11b) shows the X-Y map of the cross-phase

77

Figure 5.11: Map of phase differences of ∼2.4 kHz fluctuations for: (a) GPI signal with referenceGPI signal, (b) vpol with reference vpol, (c) GPI signal with vpol. Reference signals are at x ≈ 0cm,y ≈ 11 cm. Plots (a) and (c) have had the phase discontinuity remapped so that the contour plotsappear smooth. Plots cover the time range t ≈ 0.241 − 0.243 ms of shot 135042. The dashed-lineindicates the separatrix.

calculated from the cross-spectral density function of raw GPI signal (poloidal velocity) at each

point with a reference signal. Figure 5.11c shows the cross-phase of GPI signal with poloidal

velocity. The GPI fluctuations (Fig. 5.11a) undergo a phase shift of nearly π between locations

inside and outside the separatrix. However, in the poloidal direction at x ≈ −1 cm the phase

changes approximately linearly with position, and over the full GPI range it changes by ∼ π2 . The

poloidal wavenumber, kp can be obtained by a linear fit to the phase shift vs. y, resulting in

kp = 7.7 m−1 at x ≈ −1 cm. The poloidal velocity fluctuations (Fig. 5.11b) at x ≈ −1 cm have

a significantly longer (but not infinite) wavelength in the poloidal direction, with kp = 3.4 m−1.

The phase of the poloidal velocity also does not show the strong radial dependence seen in the GPI

intensity.

Figure 5.12 plots the phase of the ∼3 kHz oscilations versus radius and poloidal position for

periods of shots 135042-135045 where a rotating mode is visible in the bandpass filtered data. For

the radial dependence, the data is averaged poloidally before the phase is calculated, and the phase

versus y is taken ∼1 cm inside the separatrix. The phase of the GPI signal for these periods changes

rapidly from x=-2 to x=0 cm, while in the poloidal direction the phase has an approximately linear

dependence on position. From this linear dependence it can be seen that the poloidal wavelengths

78

Figure 5.12: Phase of ∼3 kHz mode plotted vs spatial cordinates for GPI signal and poloidalvelocity for periods of shots 135042-135045 with “rotating mode” visible in bandpass filtered. Timeperiods are 512 frames (1.8 msec) and begin at: 135042(black) t=0.241, 135043(gold) t=0.238,135044(green) t=0.230, 135045(red) t=0.236 s.

79

are long, with poloidal wavenumbers on the order of 10 m−1. As a function of radius, the phase of

the poloidal velocity remains small, but becomes more negative with increasing radius. This means

that the 3 kHz flow oscillations in the SOL lead the flow inside the separatrix, and this lag time

is ∼15 µs. For the poloidal direction, wavenumbers for the poloidal velocity are typically smaller

than the GPI signal by about a factor of 2.

Poloidal correlation lengths for the ∼3 kHz feature can be estimated from the coherence

function between two signals. Figure 5.13 shows the coherence and phase versus frequency for

poloidal velocity signals of shots 135042-44. The coherence function for two points separated by

∼12 cm in the poloidal direction is calculated from the cross-spectral and autospectral density

functions as outlined in section 11.6 of Bendat and Piersol [9]. The cross-spectral and autospectral

density functions are estimated by ensemble averaging estimates from 14 time blocks of 512 time

points (1.8 msec) each. Each shot shows a mode near 2.8 kHz that maintains high coherence and

small phase difference over ∼12 cm. All shots analyzed exhibit a peak in the coherence function in

the range of 2-4 kHz with phase differences less than π4 .

Figure 5.14 shows the coherence as a function of poloidal separation for poloidal velocity

signals. Coherence functions are estimated for increasing poloidal separations, and the Figure

compares the coherence of the 2.8 kHz mode to the average coherence value for frequencies between

20-30 kHz. The 2.8 kHz mode has a correlation length of 56 cm which is significantly longer than

the ∼4 cm correlation length for the 20-30 kHz band. Figures 5.14 and 5.13 suggests that this

poloidal flow oscillation near 3 kHz is a large scale, spatially coherent oscillation.

Figure 5.15 illustrates the radial dependence of the poloidal velocity power spectrum. The

spectra for the mean-subtracted poloidal velocity between t ≈ 0.240 − 0.243 ms are first norm-

squared, and then averaged poloidally. The figure shows the spectrum at three radial locations

at x=-3.9,0.6, and 3.5 cm. A strong feature near 2.5 kHz is visible at the same frequency in the

spectra at -3.9 and 0.6 cm, suggesting that the frequency of the mode does not depend on radius

and hence temperature profiles. In addition, This feature in the poloidal velocity spectra is seen to

extend a few centimeters into the SOL, but at 3.5 cm the feature is significantly reduced.

80

Figure 5.13: Coherence and phase plots versus frequency for poloidal velocity signals separated by∼12 cm poloidally at ∼1 cm inside the separatrix.

Figure 5.14: Plot shows coherence of poloidal velocity vs. y separation for 2.8 kHz mode (�) andbackground turbulence between 20-30 kHz (4) for poloidal velocity of shot 135042. Velocities aremeasured at ∼1 cm inside the separatrix. Solid lines are fits to exponential decays with correlationlengths of 56 cm for the 2.8 kHz mode and 4 cm for the background turbulence.

81

Figure 5.15: Autopower spectra of poloidal velocity at 3 radial positions for t ≈ 0.240− 0.243 s ofshot 135042. The feature at 2.5 kHz does not appear to shift in frequency.

In summary, a strong feature near 3 kHz is observed in both the GPI and poloidal velocity

spectra, and this feature exhibits a long poloidal wavelength in each case. In the radial direction, the

3 kHz flow oscillations outside the separatrix have a small negative phase shift relative to oscillations

inside the separatrix. This phase shift indicates that the 3 kHz flow in the SOL slightly leads the

flow inside the separatrix. Poloidal velocity fluctuations at 3 kHz also have a long correlation

length that is an order of magnitude greater than the turbulence correlation length. In addition,

the poloidal velocity fluctuations show no frequency variation between the separatrix and the inside

edge of the view (x ≈ −4 − 0), but the amplitude of the fluctuation decreases significantly inside

the SOL.

5.8 Flow shear and Reynolds stress calculations

Turbulent flow properties, such as the shearing rate, 〈dVp/dr〉, and the Reynolds stress,

〈δVpδVr〉, can be derived from the two dimensional flow field produced by the HOP-V algorithm.

Here, δV = V − 〈V 〉 and 〈· · · 〉 denotes a poloidal average unless otherwise noted. These quantities

are important to the evolution of the turbulent system, and may partly control the strength of

the turbulence and the transport of scalar quantities. The radial derivative of the Reynolds stress

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term is believed to drive the production of zonal flows via nonlinear energy transfer from drift wave

turbulence [31], thus leading to the suppression of drift wave turbulence and transport. Shearing of

poloidal velocity is also thought to control turbulent transport via vortex stretching [10, 148, 30].

To explore these relationships, Fsol is again plotted with poloidal velocity in Fig. 5.16, but

also included here are the shearing rate and the turbulent Reynolds stress derived from the velocity

field. Example traces of the Reynolds stress and poloidal velocity are taken ∼1 cm inside the

separatrix to illustrate the time dependence of these quantities. Due to the shearing of eddies, one

expects that flow shear localized to the area near the separatrix might suppress transport into the

SOL [148], thus Fig. 5.16 plots the shearing rate at the location of the separatrix. The plot covers

600 frames (∼2 ms) of shot 135042 approximately 3 ms before the L-H transition. In addition

to the correlated behavior of poloidal velocity with Fsol as described above, peaks in Fsol are

correlated with periods when the shearing rate is near zero, and quiet periods are coincident with

negative spikes in the shearing rate. For the time period shown, Fsol and the shearing rate are

well correlated with a correlation coefficient of 0.53. This is consistent with a scenario in which the

turbulence level is limited by the local shearing rate, as seen in recent simulations of the NSTX

edge (Figures 4 and 7 of Reference) [116]. Almost all of the shots analyzed have short periods of ∼2

ms where the correlation coefficient between Fsol and the shearing rate is greater than 0.4. Over

longer time periods, however, the correlation values are much weaker.

To better understand changes in the poloidal flow profile during the quiet-periods, we employ

a conditional averaging technique to produce averaged radial profiles of the poloidal flow during

periods of low Fsol and high Fsol. Figure 5.17 illustrates the differences in the conditionally

averaged poloidal flow profiles for ∼30 ms of shot 135042. In Fig. 5.17a the dashed(solid) line

is the poloidal flow profile averaged over times when Fsol is above(below) its mean value. Shots

#135021-23 and #135041-45 all show a shift in the mean flow profile similar to Fig. 5.17a. The

shape of the profile doesn’t appear to change significantly, indicating that the poloidal flow shear is

similar in both cases. However, the entire profile is seen to shift ∼1 km/s in the positive direction

during periods of low Fsol. The “error bars” in figure 5.17a indicate ±1 standard deviation for

83

Figure 5.16: Traces from shot 135042 of Fsol, poloidal velocity (km/s), shearing rate (MHz), andthe Reynolds stress (km2/s2). Poloidal velocity and Reynolds stress are taken at ∼1 cm inside theseparatrix, and the shearing rate is taken at the separatrix. Traces have been smoothed with a 3point boxcar average.

the low Fsol average. These indicate the level of fluctuation about the mean value, and do not

necessarily correspond to an uncertainty of the measured value. Also, The standard deviations

(Fig. 5.17b) inside the separatrix are ∼30-50% greater during these periods. Therefore, during

quiet periods we observe the mean flow becoming more positive, and, inside the separatrix, the

fluctuations in the flow have a greater amplitude. The skewness profiles (Fig. 5.17c) typically

do not show a significant difference during quiet periods, but some general characteristics may be

observed. Inside the separatrix the skewness is positive, while outside the separatrix it becomes

negative. Thus, inside the separatrix the distribution of measured poloidal velocities has a longer

tail in the direction of more positive velocities, while the more negative tail of the distribution

is longer outside the separatrix. This is consistent with our interpretation of Fig. 5.9 where we

observed strong positive peaks in poloidal flow velocity during quiet times and strong negative

peaks during bursts.

84

Figure 5.17: Poloidal velocity profiles for ∼30 ms of shot 135042 are plotted in (a). The averagesare over times where Fsol is greater than its mean value (dashed - bursty) and less than its meanvalue (solid - quiet). “Error bars” indicate ±1 σ(standard deviation) about the average value forthe low Fsol case, and indicate the level of fluctuation. Standard deviations are shown in (b), andskewness is shown in (c).

Recent SOLT (“Scrape-Off Layer Turbulence”) simulations [116] using parameters consistent

with NSTX L-mode discharges found a parameter regime with large intermittent spikes of particle

transport accompanied by quiescent periods of low particle transport. It was found that during the

quiescent periods the shearing rate had a negative value, but crosses zero shear during bursts. For

qualitative comparison, we include traces of Fsol, poloidal velocity, shearing rate (ξ), and Reynolds

shear stresses from a SOLT simulation in the bursty regime (Fig. 5.18). Similar to figure 5.16, we

see periodic bursts of Fsol at a frequency of 3.6 kHz accompanied by periods of low Fsol, quiet

periods. Fsol exhibits strong correlation with poloidal velocity, and bursts are well correlated with

the shearing rate, ξ approaching zero.

As described above, the Reynolds stress and it’s radial gradient play an important role in

the poloidal momentum balance, and regions of finite radial gradient of turbulent reynolds stress

can generate mean poloidal flow. Profiles of the Reynolds stress can be optioned from the 2-D flow

fields produced by the HOP-V algorithm. Here, Figure 5.19 plots radial profiles of the Reynolds

shear stresses for four shots averaged over 10 ms preceding the L-H transition. Interestingly, each

85

Figure 5.18: Traces of Fsol, poloidal velocity (km/s), shearing rate (ξ), and the Reynolds shearstress. Traces are from a SOLT turbulence simulation illustrating the bursty regime. The frequencyof bursts is 3.6 kHz.

profile has a local maximum near x = −1 cm, negative slope between -1 cm and 2 cm, and a section

of positive slope between -2 cm and -1 cm. The region of positive slope inside the separatrix is

coincident with the region of poloidal flow shear seen in Fig. 5.7. Error bars have been added to

the profile of shot 135042 assuming random, statistically independent uncertainties in the velocity

measurements of 0.5 km/s (∼0.5 pix/frame). This result suggests that turbulent reynolds stress is

acting to drive a net positive poloidal flow between x = −2 and x = −1 cm inside the separatrix,

and a negative poloidal flow in the SOL near x = +2 cm.

This conclusion should be taken with some skepticism, though. First, the above measure-

ments employ both an averaging over the limited GPI poloidal view and a time averaging procedure.

This assumes Taylor’s hypothesis, i.e turbulent fluctuations are assumed to be frozen into the flow

and swept past the measurement location so that space and time may be interchanged. Ideally,

one would conduct a “zonal” average of the Reynolds stress over a flux surface. Obviously this is

prohibitively difficult given diagnostic constraints, but it is not clear that the averaging performed

86

produces an adequate approximation of the flux surface average. Secondly, validation of Reynolds

stress from the HOP-V technique has not been performed. Comparison with probe measurements

of the Reynolds stress is an obvious path toward validation, but probe measurements, which are

very difficult to make in the edge of the Tokamak, are not available here. Furthermore, preliminary

results of work done on the linear plasma device CSDX to compare probe and imaging measure-

ments suggested that Reynolds stress measurements did not agree. This is still an open question

and an area of interest for future work.

Figure 5.19: Radial profiles of the Reynolds shear stress averaged over the poloidal direction and10 ms directly preceding the L-H transition for 4 shots. Error bars indicate estimated level ofuncertainty assuming uncertainties in the velocity measurments of 0.5 km/s.

5.9 Quiet-periods as Limit Cycle Oscillations

The behavior seen in Fig. 5.9 is in many ways qualitatively similar to the Limit Cycle

Oscillation (LCO) of predator-prey model of the L-H transition. The turbulence activity is seen to

be well correlated with the changes in poloidal flow, and the phase lag between the two quantities

is 180◦. It is important to note that the exact phase relationship is sensitive to the definition of

“turbulence amplitude” used. Here we use FSOL as a proxy for the turbulent activity. The results

presented above can be used to further evaluate the similarity with the predator-prey model. Of

87

particular interest are two questions: Is the observed 3 kHz flow a low frequency zonal flow or

GAM? Does the relationship between the turbulence and shear support a flow shear suppression

scenario? This section will attempt to address these questions.

As mentioned previously, zonal flows have a few identifying characteristics: n = m = 0

potential (flow) structure, and finite radial wavenumber that lies between the turbulence scales

and the system size described by the minor radius. This can be expressed as λr ≈ 10 − 50ρi.

Additionally, GAMs may be identified by their m=1 density perturbation and are predicted to

have a frequency of between 4-12 kHz in the NSTX edge [167]. The poloidal structure of the mode

can be inferred from Fig. 5.12 and compared with expectations for a zonal flow. The linear trend

of the 3 kHz poloidal velocity phase versus poloidal position indicates that the poloidal wavelength

of the mode is ∼ 1.2 m. While this is quite long, and indeed much longer than the length of the

GPI view, it is inconsistent with an m=0 mode. However, the pattern matching velocity algorithm

tracks changes in density contours, thus, if the 3 kHz feature is a GAM, the phase velocity of

the m=1 density perturbation could impact the poloidal structure of the flow measured by the

velocimetry algorithm.

The radial structure of the 3 kHz feature is plotted in Fig. 5.20 and Fig. 5.12. Two

observations can be made. First, The amplitude of the 3 kHz feature is non-zero in the SOL. This

would appear to be inconsistent with the zonal flow picture, because it is expected that zonal flow

would be heavily damped in this region. Furthermore, the radial profile of the amplitude and the

phase of the mode versus radius indicates that the radial scale of the mode is comparable to the

minor radius. Again, this is inconsistent with what is expected for a zonal flow.

The relationship between the shear and the turbulence was explored by Fig. 5.16 and Fig.

5.17. It was found that for some time periods a strong correlation between the shear and the

turbulence can be found (e.g. traces shown in Fig. 5.16), but over longer time periods, like that

considered in the conditional averaging of Fig. 5.17, the connection between the shear and the

turbulence is unclear. This relationship was examined further in [167], and similar results to those

discussed here were reported (i.e. some correlation between shear and turbulence observed). It

88

Figure 5.20: Radial profiles of the average poloidal velocity, and the amplitude of the 3 kHz flowfeature during the quiet period oscillations.

is possible, and indeed likely, that a more complex relationship exists between the shear and the

turbulence. Based on the simple dimensional argument of shear suppression presented above, one

would expect that turbulence would be affected when S = |dV/dr|(Lr/Lp)τ > 1. This would be

consistent with results presented in [167], however a detailed statistical analysis showing a significant

shift in turbulence amplitude or FSOL when the shear criterion is met was not presented.

In conclusion, a number of inconsistencies with the DWZF picture and the observations of

the quiet period oscillations exist despite good qualitative similarity on the surface. Still, the sim-

ilarities motivate further study. A future study would ideally be able to resolve the full 1D plus

time dynamics of the potential and turbulence present in the edge region. Also, long range corre-

lations or toroidally and poloidally separated potential measurements are needed to unequivocally

identify the zonal flow. Beyond this, measuring the nonlinear transfer of turbulent energy into the

zonal flow, and decoupling the mean shear and zonal shear effects on turbulence in experimental

observations are important pieces of understanding the L-H transition dynamics. Additionally, a

better understanding of the physical mechanism leading to the edge radial electric field well during

the transition is needed. As mentioned previously, experimental studies have made progress on

these points in the past several years [153, 95, 26], and theoretical work on an improved 1D model

89

of the L-H transition based on the initial predator-prey study is in progress [101].

Chapter 6

Precursor Fluctuations During Small ELMs in NSTX

The achievement of a high confinement mode, or H-mode, in ITER is widely considered

necessary to reach the operational goals of the project: namely, a burning plasma with a ratio

of fusion power generated to input power of 10 [37, 154]. The improved confinement in H-mode

operation is partly attributed to the development of an edge transport barrier (ETB) that restricts

the transport of particles and heat into the scrape-off layer (SOL), thus reducing losses to the wall

along open field lines [49]. The ETB then builds up a region of steep gradients in temperature and

density near the plasma edge known as the H-mode pedestal. Gradients in the edge pressure or

edge current profiles provide a source of free energy for a number of disruptive instabilities known as

edge localized modes, or ELMs [166]. ELMs are often explosive events that eject significant energy

into the SOL; upon contacting the plasma facing components, these energy bursts can exceed the

heat load limits of the material components and cause significant damage. Current projections

predict that ITER components will not be able to tolerate the impulsive energy densities of the

largest ELM events [165]. Therefore, it is clear that a thorough understanding of ELM physics and

a strategy for avoiding and/or mitigating ELMs is necessary for next-step fusion devices to succeed.

This chapter details a GPI study of edge localized mode (ELM) dynamics of the growth and

collapse period, and the characterization of a coherent precursor oscillation is presented. First, a

brief overview of ELM physics and observations is presented.

91

6.1 MHD stability

MHD instabilities are primary candidates for the driving mechanism behind ELMs because

of the ELM’s global impact on the plasma equilibrium, magnetic fluctuation signature, and short

timescales for ELM growth. To study the stability of an equilibrium in ideal MHD, the linearized

MHD momentum equation can be recast as an equation for the displacement of the plasma from

equilibrium

ρ∂2

∂t2ξ = F(ξ), (6.1)

where F(ξ) is the linearized force operator

F(ξ) = ∇(γp0∇ · ξ + ξ · ∇p0)−B1 × J0 −B0 × (∇×B1). (6.2)

The perturbed magnetic field is given by the frozen flux condition B1 = ∇× (ξ×B0). The change

in potential energy of the system is then given by

δW = −1

2

∫ξ · F(ξ)d3x. (6.3)

If δW > 0 for all displacements ξ then the plasma is stable, but the plasma is unstable if δW < 0

for any possible displacement. The displacement may be solved for by either solving the eigenmode

equation

ρω2nξn = −F(ξn), (6.4)

or by minimizing the Lagrangian

L = δK − δW, (6.5)

where δK =∫

12ρ|ξ|

2d3x. In practice this is exceedingly difficult to do analytically, and often is

quite difficult even numerically.

The change in potential energy δW can be expressed as the sum of two distinct contributions

δW = δWF + δWS , (6.6)

92

where δWF and δWS represent the contribution from the plasma volume, and surface. After

performing some non-trivial manipulations [8], these contributions can be expressed as

δWF =1

2µ0

∫Pγµ0p0(∇ · ξ)2 +B2

1⊥ +B20 [2ξ⊥ · κ + (∇ · ξ⊥)]2

−B0 · (B1⊥ × ξ⊥)µ0J0‖

B0− 2µ0(ξ⊥ · ∇p0)(ξ⊥ · κ)d3r, (6.7)

δWS =1

2µ0

∫S

(µ0p1 + B0 ·B1)ξ⊥ · ds, (6.8)

where the subscript ⊥,‖ indicates components perpendicular and parallel to the magnetic field,

and κ = b0 · ∇b0 is the local curvature of the equilibrium field. This form is useful for illustrating

the meaning of the various terms, though somewhat cumbersome to work with. The first three

terms are strictly positive, and so are stabilizing. The first term gives the stabilizing effect of

compressing a plasma with equilibrium pressure p0. The second and third terms are stabilizing

effects due an increase in magnetic field strength and the compression or bending of field lines. The

final two terms may be negative, and thus allow for instability. The fourth term is destabilizing

when a plasma displacement generates a magnetic force that reinforces the displacement, and is

driven by the equilibrium current. The final term gives the destabilization by the interaction of the

equilibrium pressure gradient with the magnetic field curvature. For a toroidal equilibrium, this

term is stabilizing on the inside of the torus and destabilizing on the outside where the radius of

curvature and pressure gradient point in the same direction. The stabilizing terms are minimized

by an incompressible perturbation, so these are often found to be the most unstable modes.

To examine the current driven instabilities more closely, we take the displacement to be

incompressible and ignore pressure effects so that the change in potential energy is given by

δW =1

2µ0

∫PB2

1 −B0 · (B1⊥ × ξ⊥)µ0J0‖

B0d3r +

∫vac

B2vac

2µ0d3r. (6.9)

In the large aspect ratio tokamak limit with circular cross-section [158], δW takes the form

δW = πR

∫ a

0

B21

µ0− Jz0(Br1ξθ −Bθ1ξr)dθrdr + πR

∫ b

a

B2vac

µ0dθrdr, (6.10)

where r is the minor radius, z = Rφ is the toroidal coordinate, and θ is the poloidal coordinate.

The plasma-vacuum boundary is at r = a, and a perfectly conducting wall is placed at r = b. The

93

θ and φ dependence may be decomposed into Fourier modes ξ = ξ exp[imθ− inθ]. Then, expanding

δW with ∇ · ξ = 0 and making use of the frozen flux condition to express B1 in terms of ξ gives,

after much simplification [158],

δW =π2B2

φ

µ0R

∫ a

0

[(rdξ

dr

)2+ (m2 − 1)ξ2

]( nm− 1

q

)2rdr

+π2B2

φ

µ0R

[ 2

qa

( nm− 1

qa

)+ (1 +mλ)

( nm− 1

qa

)2]a2ξ2a, (6.11)

where

q =# of toroidal orbits

1 poloidal orbit=rBφRBθ

(6.12)

is the safety factor and

λ =1 + (a/b)2m

1− (a/b)2m. (6.13)

Given Eqn. 6.11, we can draw a number of conclusions about the stability of this mode. First,

the integral over the plasma volume is strictly positive, so a plasma terminated by a conductor at

r = a will be stable to all perturbations of this type. Second, any mode with m/n < qa will be

stable. Typical q profiles are increasing functions of r, so this includes all modes with a resonant

surface inside the plasma. Finally, modes with m/n > qa can be unstable. The unstable modes

should be localized near the edge to minimize the stabilizing effects on the plasma interior, and

have resonant qr = m/n surface just outside the plasma (qr > qa) so that the surface contribution

to δW is negative. These modes are named the external kink or peeling instability due to their

helical nature and tendency to evolve in such a way as to “peel” off the outer flux surfaces of the

plasma [149].

Another important class of ideal MHD instabilities are the ballooning instabilities. These

instabilities are driven by the (ξ⊥ · ∇p0)(ξ⊥ · κ) term, and so are strongest in the bad curvature

region on the outside of the torus. Since the curvature is an essential feature of the instability,

a cylindrical treatment like that used for the kink instability is typically inadequate. For an

incompressible mode, a rough estimate of instability is given by balancing the stabilizing magnetic

94

energy term,

δWs =B2

1

µ0=

1

µ0|∇ × ξ ×B0|2 ≈

k2‖B2φξ

2

µ0, (6.14)

with the destabilizing pressure driven term,

δWd ≈ −1

Rc

dp0dr

ξ2. (6.15)

Setting these equal and taking k‖ = 1/qR, the distance along a field line from the inside to the

outside of the torus, and taking the radius of curvature to be approximately the major radius,

Rc ∼ R0, gives

−dp0dr∼

B2φ

µ0q2R0. (6.16)

A more detailed analysis [158, 22] of the high-n instability finds that magnetic shear, s = d ln q/d ln r

is stabilizing, and the stability boundary at moderate levels of shear can be approximated as

s = 1.67α where the normalized pressure gradient is given by

α = −2µ0Rq2

B2

dp

dr. (6.17)

The above discussion of ideal MHD instabilities finds that low-n kink modes are destabilized at

low edge pressure and high edge current, and high-n ballooning modes are destabilized at high edge

pressure and low edge current. It’s also possible for peeling-type instabilities and ballooning-type

instabilities to couple [59, 135], and the resulting intermediate-n instability appears at high edge

pressure and high edge current. This combined peeling-ballooning model of ideal MHD stability

[160] has been shown to be quite useful for understanding ELMs as we’ll discuss in the next section.

Beyond the Ideal treatment of these instabilities, non-ideal effects, such as resistivity, can also have

significant impact on the stability, and may be important for describing certain ELMing regimes.

For real cases, the stability of a given experimentally determined plasma equilibrium is quite difficult

to ascertain, and depends on the plasma magnetic equilibrium as well as profiles of pressure and

current.

95

6.2 An Overview of ELMs

A great many experimental and theoretical reviews of ELMs exist in literature [166, 143, 7,

110, 91, 75]. The focus of this section is to give a brief overview of ELM experiments, so the results

discussed later in this chapter may be put in context.

Experimental observations of ELMs in H-mode plasmas across many devices has lead to the

classification of several distinct ELM types [166]. These are:

• Type-I (Giant) ELMs seen in H-modes with high auxiliary heating and plasmas with

steep edge pressure gradients,

• Type-II (Grassy) ELMs seen in strongly shaped H-mode plasmas,

• Type-III ELMs seen in close proximity to the L-H transition and for low auxiliary heating.

Beyond these general classifications, a few machine specific ELM types have been observed (e.g.

Type-V ELMs of NSTX [92]).

Type-I ELMs can most readily be identified by their increase in frequency with heating power

[166]. This observation is consistent with a pressure or pressure gradient driven mode, like the ideal

MHD ballooning mode discussed above, limiting the pedestal growth. This is further reinforced

by the observation of a constant pressure limit in measurements of edge temperature and density

during Type-I ELMing regimes [143]. Magnetic fluctuations are also seen to increase during the

ELM event, and precursors have been observed 100-400 µs preceding the ELM in edge density,

temperature, and magnetics [110]. The precursor observations do not appear to be consistent

across all machines, however. ELM power loss is typically 5-15% of the pre-ELM pedestal stored

energy for Type-I ELMs [75].

Type-II ELMs are a class of high-frequency, small ELM originally observed in DIII-D and JT-

60U [143]. These ELMs typically occur at high edge pressure gradient, with high edge safety factor,

and strong elongation and triangularity. Because of these features, type-II ELMs are associated

with the second stability region of ideal MHD theory [166].

96

Type-III ELMs are seen soon after the transition to H-mode, while the edge temperature

and density are still relatively low compared to conditions for type-I ELMs [143]. These ELMs are

seen to decrease in frequency with increased heating power, and eventually disappear at sufficiently

high edge temperatures. Edge pressure gradients for type-III ELMs are not near typical ballooning

limits. These observations suggests that resistive effects may be important for these ELMs [75].

ELM power loss is typically < 5% of the pre-ELM pedestal stored energy. Magnetic precursors are

consistently seen preceding type-III ELMs. These precursors exhibit intermediate toroidal mode

numbers, n = 5− 10, and f ≈ 70kHz [166].

In recent years, the development of fast edge diagnostics has produced detailed descriptions

of the full ELM cycle including ELM filament structure [111, 108, 89]. One-dimensional imaging

systems viewing visible light fluctuations in the plasma edge have also been used to observe ELM

fine-structure on ASDEX [38] and ELM precursors on Alcator C-MOD [146], and two-dimensional

imaging systems have been used on MAST to examine ELM filament structure [83, 82]. In addition,

ELM filament structure has been examined using GPI observations in several ELMing regimes

[96, 93], and measurements of NBI heated H-modes with Type-III ELMs captured the birth of

filamentary structures in two-dimensions over ∼ 50 µs [96].

The theory of coupled ideal MHD peeling-ballooning modes has been very successful in de-

scribing observed stability boundaries for large, type-I ELMs [24, 134, 135, 132]. Peeling-Ballooning

stability calculations have shown the edge pedestal to be unstable to coupled, intermediate-n modes

at high edge pressure gradient and high edge current in DIII-D, Asdex Upgrade, JET, JT60-U, and

Alcator C-mod [132, 54]. At low aspect ratio, NSTX pedestals have been shown to be unstable

primarily to the kink/peeling instability [19] at low-n, and the pedestals typically exist far from the

high-pressure ballooning mode boundary. In addition to these linear stability studies, nonlinear

simulations of ELMs have also been carried out [36, 133, 65, 163, 102, 79, 112, 78].

97

6.3 NSTX observations of ELM precursors

In the remainder of the chapter, we’ll examine results from two-dimensional imaging of pre-

cursor modes preceding small ELM events in NSTX. Gas Puff Imaging (GPI) of visible light fluc-

tuations near the last closed flux surface has revealed precursor edge intensity fluctuations that are

wave-like in nature. These edge oscillations are seen to grow in amplitude preceding ELM events

and H-L back transitions, but they are also observed intermittently at low amplitudes throughout

H-mode operation. A magnetic signature is also observed that is concurrent and strongly correlated

with the edge intensity oscillations. The nonlinear evolution of the precursor mode and ELM crash

have been imaged in the plane perpendicular to the magnetic field using the GPI diagnostic, and

the two-dimensional structure and dynamics are analyzed and presented here. Precursor modes

were imaged primarily in near-threshold RF-heated H-modes and some Ohmic heated cases, but

they were absent in the similar, near-threshold NBI heated cases studied.

6.4 Operational Parameters and Plasma Conditions

The experimental observations discussed in this chapter were obtained during the 2010 run

campaign of NSTX. Discharges are deuterium plasmas with on-axis toroidal field Bt = 0.45 T and

a plasma current Ip = 900 kA. Table 6.1 contains specific details for each shot, and example traces

of Ip, radio frequency power Prf , and Dα light for a single shot are shown in Figure 6.1.

This database of shots was originally designed to probe the RF power threshold for the low

to high confinement mode transition in NSTX. The radio frequency (RF) heating system used is

a twelve antenna high harmonic fast wave (HHFW) heating system that can deliver up to 6 MW

of power at 30 MHz [117]. The database of shots considered in this study are RF heated H-mode

plasmas with 0.5-1.2 MW of input RF heating power operated at 180◦ phasing to optimize core

heating efficiency [62, 144].

The classification of ELM type is not straightforward in this operational regime. Low heating

powers and proximity to the L-H transition suggest that Type-III ELMs would be most likely,

98

Table 6.1: Shot database for this study including the shot number, timeframe of interest, toroidalmagnetic field, plasma current, and RF heating power

Shot # Timeframe [s] BT [T] IP [kA] PRF [kW]

141917 0.23-0.25 0.45 900 625141917 0.25-0.27 0.45 900 1200141918 0.22-0.24 0.45 900 625141919 0.23-0.25 0.45 900 625141920 0.24-0.25 0.45 900 625141922 0.24-0.27 0.45 900 700142000 0.22-0.25 0.45 900 1100142001 0.22-0.25 0.45 900 1100142002 0.22-0.25 0.45 900 1100142003 0.22-0.25 0.45 900 1100142006 0.25-0.27 0.45 900 1100

Figure 6.1: Time traces of RF heating power PRF , plasma current IP , Dα light, and line-integrateddensity N for typical shot from this study. Neutral beams are also used early in the shot for plasmaconditioning. The shaded region indicates the time period for traces plotted in Figure 6.3.

however a distinguishable ELM frequency is not observed. Additionally, the presence of L-H-L

dithers further complicates identification. Alternatively, ELM precursors observed in this regime

are most similar to precursors observed in Type-I ELMing regimes on NSTX, namely, precursors are

short-lived and exhibit intermediate toroidal mode numbers. Regardless of ELM type, the events

studied here are small, and the stored energy drop in most events is below the measurable limit.

99

However, the stored energy is observed to drop 3-5% during events which trigger H-L transition.

The primary observations used for this study are from the GPI diagnostic. For this campaign,

the GPI diagnostic used a Phantom v710 fast-framing camera to capture images at 400,000 frames

per second (2.5 µs per frame). Deuterium was used for the neutral gas species, and Dα (656

nm) line-emission was imaged using collection optics and optical filters. Images were recorded

at 64x80 pixel resolution covering roughly 25 cm by 30 cm of the edge region. The GPI view

is aligned with the magnetic field such that the horizontal (x) direction of the camera image is

approximately codirectional with the radial direction, and the vertical (y) direction is then the

generalized poloidal coordinate, perpendicular to both the magnetic field and the radial direction.

Therefore, the vertical direction is approximately the projection of the machine poloidal direction

into the plane perpendicular to the magnetic field.

6.5 Precursor Oscillations in GPI Intensity

Analysis of GPI observations from the shots listed in Table 6.1 reveals a periodic edge intensity

fluctuation seen to precede ELMs and ELM-induced H-L back transitions during RF heated H-mode

operation. These edge oscillations have a distinct, elongated mode structure and are visible up to

200 µs preceding the ELM event. While these fluctuations are seen to precede ELM events, low

amplitude oscillations are also seen intermittently throughout RF H-mode operation. In this section

we detail the GPI observations of the ELM precursors.

A series of image stills from the GPI diagnostic is displayed in Figure 6.2 to illustrate the

general features of the precursor phase leading to the ELM crash. Edge intensity oscillations with

a distinct elongated structure are visible preceding the ejection of plasma captured in frames (i)-

(j). During the precursor phase, frames (a)-(h), intensity peaks travel in the positive y direction

(in the electron diamagnetic direction), and as one peak leaves the camera view a second peak

enters from the bottom. The intensity structures appear to drift radially outward and become

increasingly deformed as they propagate through the camera view. This deformation is especially

evident in frame (h) directly preceding the filamentation. Eventually, an unidentified processes is

100

Figure 6.2: Multiframe image stills of an ELM event with precursor intensity fluctuations from shot141918. The time between frames is ∼7.5 µs. Distinct mode structure can be seen in precursoroscillations leading to the ejection of the filament in the last two frames. The approximate locationof the separatrix is indicated by the dashed line.

triggered precipitating the explosive ejection of plasma filaments into the SOL. Once in the SOL,

the filaments travel in the reverse direction (-y, ion diamagnetic direction). This reversal of the

propagation direction is commonly attributed to a change in the radial electric field from inside to

outside the separatrix. In many observed ELM events clear filamentation is evident such as in Fig.

6.2(j). However, the ELM crash is a complicated nonlinear process, thus the ejection of plasma

into the SOL does not always follow the clear time evolution shown in Fig. 6.2.

Two quantities may be defined from the GPI image sequence which will be of use in the

following analysis. The first of these is the integrated edge intensity Iedge, which is defined here

as the sum of the intensity ±2 cm around the radial location of the peak time-averaged intensity

profile. The second quantity is the ratio of light in the SOL to the images total intensity, FSOL.

This quantity is a measure of the relative intensity in the SOL, thus it is a good indicator of plasma

loss into the SOL such as ELM events, intermittent blobs, and turbulent losses during L-mode

101

Figure 6.3: Time traces of (a) scrape-off layer fraction FSOL and (b) integrated edge intensity Iedgecorresponding to shaded timeperiod in Fig. 6.1. Traces show edge intensity fluctuations precedingan ELM at 0.2425s, and an ELM-induced back transition at 0.245s. Low level fluctuations can alsobe seen near 0.244s.

operation. During an H-L transition the mean value of Fsol can change by .15-.20, while ELMs can

cause deviations greater than .40 for a large event. The precursor oscillations are easily observable

in time traces of either of these quantities as seen in Figure 6.3. Iedge fluctuations preceding an ELM

event can be seen at 0.2427s, and an ELM induced H-L transition is triggered at 0.245s. Low level

periodic fluctuations can be seen near 0.244s as well. These intermittent, low-amplitude fluctuations

are found throughout RF H-mode operation, and are similar in frequency and wavenumber to the

precursor oscillations, as will be discussed in the following section.

Figure 6.4 shows the 1D plus time dynamics of the precursor event, and it illustrates the

coherent nature of the precursor oscillations. The figure shows two-dimensional “slices” of the GPI

data with (a) one cut in x (radial) versus time at y=15.50 cm and (b) one cut in y (perpendicular)

versus time at x=10.5 cm. The fluctuating intensity pattern is easily discernible in (a) at time

0.2592s, and the intensity peaks appear to drift radially outward as the mode grows. A similar

pattern is evident in (b) where the tilt in the intensity pattern, typically referred to as streaks,

indicates propagation in the positive y (electron diamagnetic direction). A simple linear fit to the

102

Figure 6.4: Two-dimensional slices through GPI data with one cut at y=15.5 cm (a), and one atx=10.5 cm (b). Precursor fluctuations are very distinct brightness pulses which appear to moveupward, or in the electron diamagnetic direction indicated by the tilt of structures in (b). Structuresare also seen to drift radially outward as indicated by +x motion in (a).

intensity streaks gives an estimate of the perpendicular velocity of ∼13 km/s.

Figure 6.5: Traces of (a) Average perpendicular velocities, and (b) FSOL for many ELM eventsfrom the shots database. Velocities are measured ∼2 cm inside the separatrix. Timings are relativeto the peak FSOL for each event.

For a more refined velocity estimate, we employ the HOP-V code, a digital image velocimetry

103

algorithm based on pattern-matching techniques [105], to estimate the two-dimensional flow field

of the precursor intensity fluctuations. Poloidal velocities averaged in the vertical direction and

measured just inside the separatrix are plotted in figure 6.5 along with FSOL traces. The figure

shows the flow behavior during several observed ELM events with precursor fluctuations. Times

for figure 6.5 are relative to the peak in FSOL, so that the ejection of plasma occurs near t=0.0 ms

and precursor activity can be seen between -0.2 ms and 0.0 ms. Average perpendicular velocities

measured inside the separatrix during the precursor period typically range between +2-8 km/s,

but velocity estimates measured at the intensity maximum are in agreement with the streak fit

estimates. At the time of the ELM crash, measured radial velocities of intensity structures can

peak as high as 8 km/s as filaments are ejected into the SOL. Additionally, average perpendicular

velocities measured inside the separatrix are seen to briefly reverse direction during the crash, and

reach flows up to -4 km/s.

6.6 Wavenumber and Frequency Characterization of the Precursor Mode

Time-Frequency analysis of the integrated edge intensity Iedge is performed using the contin-

uous wavelet transform (CWT) [150]. This method is similar to the familiar technique of windowed

Fourier transforms, however the CWT uses a ’mother’ wavelet function that is scaled and translated

to measure the power contained in a signal at a given location in time-scale (frequency) space, thus

one obtains a time-frequency power spectrum analogous to the spectrogram of windowed Fourier

transforms. The advantage of the CWT is the inherent ability of the scaling operation to yield an

optimal product of the time and frequency resolutions at each frequency value. The variable reso-

lution allows the CWT to capture the fine details of a signal near singularities while still providing

efficient measurement of lower-frequency behavior.

The 6th order Morlet CWT of the Iedge signal from Fig. 6.3 is presented in Figure 6.6 along

with the accompanying FSOL trace. The ELM and H-L back-transition events are directly preceded

by significant increases in power in the 20-30 kHz band, and the low amplitude oscillations between

0.243 and 0.244 s also show significant levels of power near 25 kHz. Other shots in this collection

104

Figure 6.6: Wavelet scalogram of integrated edge intensity, Iedge accompanied by the time trace ofthe scrape-off layer fraction for shot 141919. The power spectrum shows significant power at the20 kHz scale during the ELM precursor fluctuations. The shaded region in the wavelet scalogramindicates where edge effects become important, and the white contour indicates the 95% significancelevel.

exhibit similar activity in the 20-30 kHz range, both preceding ELM events and at low levels

intermittently throughout H-mode operation. Power increases in the 3 kHz range are associated

with the timescale of the full ELM event including the period of increased fluctuations following

the crash.

Figure 6.7: Scatter plot of frequency of precursor fluctuations against the perpendicular wavenum-ber derived from GPI intensity fluctuations. Red triangles are events that lead to an ELM orback-transition, while blue squares are edge intensity fluctuations that do not lead to an ELM.

105

Short time window FFT analysis has also been used to corroborate the results of the CWT

analysis, and a linear fit to the cross-spectral phase versus poloidal position provides an estimate

of poloidal wavenumbers, ky for the precursor. Several short time periods from the shots listed in

Table 6.1 which exhibit periodic edge intensity fluctuations have been analyzed, and the results

are presented in Figure 6.7. In agreement with the CWT analysis, typical frequencies are in the

range of 20-30 kHz, and perpendicular wavenumbers are found to be between 0.05 and 0.21 cm−1.

Periods preceding ELM events and periods of low amplitude fluctuations were both analyzed and

no significant difference in wavenumber or frequency was found between the two sets of events.

Additionally, the points cluster about a phase velocity of ∼13 km/s, which is consistent with the

image velocimetry estimates. In this case, the pattern-matching velocimetry is tracking the motion

of individual intensity peaks and troughs because of the size of the wave relative to the size of the

image subframe. Therefore, it is expected that the velocimetry estimate would be consistent with

the phase velocity.

6.7 Quantification of Edge Deformation During Precursor Evolution

Figure 6.8: Example (a) image frame from shot 141917 and (b) xedge function overlayed on intensitycontours. Maximum radial excursion relative to EFIT separatrix location is plotted in (c), and theedge curvature, κ corresponding to this point is plotted in (d). The Dashed line indicates the timepoint of image (a).

−4−2024

0.2441 0.2442 0.2443 0.2444 0.2445t [s]

−0.4

−0.2

0.0

(a) (b)

(c)rmax−rsep

(d)

κ

106

During the evolution of the precursor the edge intensity profile becomes deformed, and por-

tions of the intensity profile balloon into the SOL as they propagate through the GPI view. The

radial excursion of the intensity fluctuations and the curved edge deformation are both observable

in Figure 6.2 frames (a)-(g). To quantify these features of the mode, we first define the plasma edge,

xedge(y) by taking a weighted average of the x coordinate at each y location using the cube of the

intensity as the weight for each point. This weighting insures that the edge location closely follows

the peak of the radial profile of the intensity, and this yields a definition for the “edge” which is

consistent with a visual estimate. When turbulence fluctuations are low, such as in H-mode, the

GPI neutral gas puff penetrates an approximately poloidally uniform distance in from the plasma

edge, so that xedge, to a rough approximation, traces out a flux surface. Thus we will consider

perturbations of xedge to be a proxy for perturbations of the outer flux surface. Figures 6.8(a)-(b)

illustrate an example of xedge. The edge curvature and maximum radial excursion may then be

derived from xedge.

The maximum radial excursion at any given time corresponds to the right-most point, or

point with largest x value, on the xedge curve, and the time evolution of this position relative to

the EFIT separatrix is plotted in figure 6.8(c). The EFIT separatrix location is typically uncertain

to ±1 cm at the location of the GPI view, and MHD activity generally associated with ELMs will

perturb the location of the separatrix. Additionally, the EFIT time resolution for this analysis is

4 ms, but over a 0.1 s period during H-mode the reconstructed position of the separatrix is slowly

varying and changes by no more than 3 cm. So, we believe it to be a reasonable estimate of the

equilibrium separatrix location about which the fast time-scale behavior fluctuates. Furthermore,

it is the most physically meaningful reference location available, so we choose to use it for relative

measurements of the radial position of the edge. It should be noted that the choice of reference

does not significantly change any of the conclusions.

The curvature of the edge is calculated using the mathematical definition: κ = x′′/(1+x′2)3/2,

and the necessary derivates are calculated from a quintic spline fit of xedge using two internal knots.

In this work, negative curvature indicates that the convex side is directed radially outward. Figure

107

6.8(d) shows a time trace of the curvature measured at the point of maximum radial excursion for

that time point.

Precursor activity can be observed in figures 6.8(c)-(d) between 0.2442 s and 0.2444 s, and

modulations of the radial excursion and the edge curvature are evident. These modulations coincide

with intensity peaks passing through the camera view as seen in figure 6.2. As the precursor mode

evolves, the radial excursion increase, and the curvature becomes increasingly negative indicating

significant deformation of the edge. Shortly following the image in 6.8(a), a filament forms and

is explosively ejected into the SOL near t=0.2444 s. After this time, a well-defined edge does not

exist until the turbulence is quenched.

Figure 6.9: Scatterplot of minimum edge curvature, κmin and maximum radial excursion relativeto the EFIT separatrix location for several ELM precursor events (diamond). Intensity fluctuationsnot leading to an ELM are also included (square).

Several precursor events are compared in Figure 6.9, and low amplitude events (such as in

Figure 6.3 at 0.2440) that do not precipitate an ELM are also included. The figure compares the

minimum (i.e. most negative) curvature κmin and the maximum radial excursion during each event.

Events directly preceding ELMs are found to reach closer to the EFIT separatrix location, and

curvature values for these events are more negative than events not preceding ELMs. Therefore, the

curvature appears to be an important characteristic of the underlying instability, and the increased

108

curvature for ELM unstable events suggests that this feature is important for the triggering of the

ELM crash. Still, it is unclear why the mode sometimes saturates at a low amplitude, and does

not always precipitate an ELM.

6.8 Magnetic Fluctuations

Signals from magnetic pick-up coils distributed in toroidal angle around the device have also

been analyzed, and precursor activity in the 20 kHz region is observed in the time derivative of

the magnetic field (B). Time traces of Iedge and a lowpass filtered B signal are plotted in Figures

6.10(a) and (b), respectively. These traces illustrate the similarities in the precursor behavior, and

indeed the B signal filtered below 200 kHz and Iedge signal are highly correlated as shown in Figure

6.10(c). The figure shows the time-lagged correlations for 200 µs long time segments preceding

the ELM crash, and the coil with the highest correlation is chosen for figures 6.10(b) and (c).

This particular coil is separated 30◦ in toroidal angle from the GPI view, and is located below the

midplane. Additionally, this coil is located near field lines that pass through the GPI view. The

absolute value of the correlation coefficient exceeds 0.8 at -10 µs, thus the magnetic fluctuations

measured at this position are delayed with respect to the intensity fluctuations. The high correlation

suggests that the precursor edge intensity fluctuations seen by GPI are electromagnetic in nature

as one would expect for typical MHD instabilities believed to produce ELMs.

Toroidal mode numbers are estimated from a toroidally distributed array of magnetic pick-up

coils. Time traces of B 0.3 ms in length and directly preceding ELM events are Fourier transformed,

and precursor modes are identified as peaks in the power spectrum. Toroidal mode numbers are then

determined from the phase of the precursor Fourier mode as a function of toroidal position of the

magnetic coil. This simple method assumes a global mode, and it does not account for the possibility

of a frequency modulation during the mode evolution. We find that toroidal mode numbers between

n = 5− 10 are typical for precursor behavior similar to what has been plotted in Figure 6.10. For

comparison, it is possible to estimate toroidal mode numbers from the perpendicular wavenumbers

estimated from the GPI data. If the structures are assumed to be aligned with the magnetic

109

Figure 6.10: Time traces of (a) Iedge and (b) low-pass filtered magnetic signals from shot 141917.The black trace has been lowpass filtered at 200 kHz, and the red trace has been bandpass filteredaround 20 kHz. Magnetics traces are strongly correlated with fluctuations in edge intensity asshown in the lagged correlation plot (c). Absolute values of the correlation coefficient reach 0.8 forperiods during the precursor activity.

field, then the perpendicular wavelength can be mapped to an equivalent toroidal wavelength given

the magnetic pitch angle. Using this mapping, toroidal mode number estimates from imaging are

between n = 4− 15 which agrees with the magnetic coil estimate.

6.9 Pedestal Characteristics

As discussed in 6.2, steep pressure gradients at the edge provide a source of free energy for

Ballooning type MHD instabilities. In H-mode, the pressure profile exhibits a pedestal structure

at the edge characterized by a steep rise to a pressure of hped over a width ∆ped. Figure 6.11 shows

a typical edge electron pressure profile for the near-threshold RF heated plasmas, and a modified

Tanh fit [55, 29] to the data. Here, ψn is a normalized flux coordinate defined as:

ψn =ψc − ψψc − ψs

,

where ψc is the flux at the core and ψs is the flux at the separatrix. Pedestal width ∆ped and

pedestal height hped are parameters of the fit function, and a comparison of these parameters

for different shots is included in Figure 6.11. Parameter uncertainties are estimated by manually

varying one parameter about the best fit, and then performing a reduced fit with that parameter

110

fixed. The error bars indicate the variation in the parameter that yields a change in the reduced

χ2 of the fit of ∆χ2 = 1.

Figure 6.11: Pedestal parameters for height hped and width ∆ped are extracted from a modified tanhfit to the electron pressure profile (inset) and a comparison of several shots is presented. Squaresare 0.6-1.0 MW RF shots, triangles are Ohmic H-mode shots, stars are ∼ 1.0 MW NBI shots, andthe diamond is a 4 MW NBI heated shot.

The comparison illustrated by Figure 6.11 indicates that Ohmic H-modes (triangles) have

similar edge electron pressure profiles as the near-threshold RF H-modes (squares). In fact, ELM

precursors quantitatively similar to those observed in RF heated plasmas were observed in several

Ohmic H-mode discharges. However, 34 events with the precursor magnetic signature were ob-

served in the 9 RF shots analyzed, while only 6 events were observed in 8 Ohmic shots. Pedestal

characteristics of low-power NBI heated shots were also similar to the ohmic and RF heated cases

within the uncertainties, and GPI data from 3 NBI heated shots near L-H threshold power were

analyzed for precursor behavior. Additionally, 10 NBI heated shots in the Type-I ELMing regime

were analyzed, but no short-lived coherent precursors were observed in the GPI data for any of the

NBI heated shots. It is possible, however, that in these cases GPI neutrals may not penetrate far

enough into the plasma to allow GPI to observe the precursor modes. To improve this comparison,

more GPI data from near-threshold NBI heated plasma discharges is needed

111

6.10 Concluding Remarks on the Nature of the Precursor Mode

The question of what underlying instability drives the precursor mode still remains. There is

some indication that micro-instabilities (such as drift-type instabilities or kinetic ballooning modes)

can play a role in the evolution of the pedestal during the inter-ELM period, but these modes are

significantly smaller than the precursors observed here. As discussed above, ideal MHD modes are

likely candidates because of the observed magnetic fluctuations and fast timescales associated with

the ELM crash. These are the low-n peeling mode, and the high-n ballooning mode. The ideal

ballooning mode appears at high edge pressure gradient which is incompatible with the pedestal

height and width measured during precursor modes. The peeling mode is destabilized by edge

currents, and so could be unstable at the low pressures seen during the precursor if high edge current

is present. These modes are low-n though, which is incompatible with the magnetic observations.

Coupled peeling-ballooning modes could appear at intermediate-n in agreement with the magnetic

fluctuations, but these modes are generally seen at high edge pressure and high edge current. Some

studies indicate that non-ideal effects, such as resistivity and diamagnetic stabilization of high-n

modes, could lead to the destabilization of an intermediate-n resistive ballooning mode [102, 79].

Precise assessment of the stability requires a detailed numerical calculation. This was pursued

for this study, but was eventually abandoned because the edge density and temperature profiles

could not be adequately constrained for the models. Hypothetically, a focused experiment could

operate in a regime where these ELM events could be observed repeatedly for longer time periods.

Then, multiple Thomson scattering profiles could be combined to help constrain the plasma profiles.

In addition, an improved radial resolution for Thomson scattering in the gradient region of the edge

would be beneficial.

Chapter 7

Directions for Future Work

This dissertation has presented a number of tokamak edge physics studies using the Gas Puff

Imaging diagnostic on NSTX. Chapter 3 compared edge turbulence and fluctuation measurements

between BES and GPI. Chapter 5 characterized flows in the edge preceding the L-H transition and

explored the similarities between the observations and the limit cycle oscillation. Finally, Chapter

6 presented observations of a coherent mode preceding ELMs. In light of these results, we now

consider avenues for further study.

One critical area of research for next generation tokamak reactors is ELMs, ELM mitiga-

tion, and ELM suppression. GPI, with its high time resolution, good spatial resolution and two-

dimensional coverage of the edge gradient region, is well positioned to play an important role in this

area. Studying the evolution of edge turbulence and fluctuations in between ELMs could improve

our understanding of pedestal physics and the physical mechanisms that determine the edge gradi-

ent, pedestal height, and width. Furthermore, GPI is well suited for studying changes in the edge

turbulence during ELM suppression with Resonant Magnetic Perturbations or inter-shot Lithium

evaporation, or ELM triggering via pellet injection.

Another important research thrust is in the comparison and validation of numerical codes

with GPI observations. Currently, ELM models that are used to predict stability and ELM onset

in various devices are predominantly linear models. The nonlinear stages of ELM development

are important to understand ELM triggering and the resulting Plasma-material interactions from

ELM filaments crossing the SOL and contacting the wall or divertor. This is especially important

113

for small ELM regimes, as these are candidate operating regimes for reactor scenarios. This is an

area where GPI observations in conjunction with numerical simulations, such as BOUT++, could

have a large impact. Other possibilities include validation of edge turbulence simulations in both

L-mode and H-mode plasmas. For almost every application, the close collaboration of numerical

simulation and experimental observation will be important for continued progress.

Many questions still surround the L-H transition and the formation of the edge transport

barrier. As discussed, recent studies have pointed to the importance of zonal flows, and the nonlinear

transfer of turbulent energy at the L-H transition is identified as a possible trigger mechanism.

There is still work to be done by experiment and theory to solidify this hypothesis. It is difficult

to see how GPI can play a role here because the quantity of primary importance is the potential.

Image-based velocity estimates can be used as a proxy, but more work needs to be done to verify

these measurements. It’s not yet clear if zonal flows play a role in all L-H transitions or only those

that exhibit a limit cycle oscillation.

In conclusion, many physical phenomena of great importance for next step fusion devices

exist in the plasma edge. For example, ELMs will continue to be a problem as plasma pressure and

confinement is pushed toward reactor relevant values. The physical mechanisms that determine

the H-mode pedestal directly impact the plasma performance and achievable fusion output. Access

to the H-mode is likely necessary to reach a burning fusion reactor, thus understanding the L-H

transition is crucial. GPI is well suited to address these concerns, and therefore can be an important

diagnostic for the next decade of fusion research.

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