Turbulence in Wendelstein 7 Advanced Stellarator Plasmas Measured by
Collective Light Scattering
Nils Plesner Basse
Optics and Fluid Dynamics Department Experimental Division E3
Risø National Laboratory Max-Planck-Institut f¨ur Plasmaphysik 4000 Roskilde 85748 Garching
Denmark Germany
Association Euratom 2001
Ørsted Laboratory
Niels Bohr Institute for Astronomy, Physics and Geophysics Denmark
1 Introduction - 5p 7
1.1 Motivation . . . 7
1.2 Method . . . 8
1.3 Results . . . 8
I Theory 10
2 Collective light scattering - 15p 11 2.1 Historical notes . . . 122.2 Introduction . . . 12
2.3 Scattering cross section . . . 13
2.4 Scattering theory . . . 14
2.4.1 Radiation source . . . 14
2.4.2 Single particle scattering . . . 14
2.4.3 Far field approximation . . . 15
2.4.4 Multiple particle scattering . . . 16
2.4.5 Phase separation . . . 16
2.5 The photocurrent . . . 17
2.6 Demodulation . . . 21
2.7 Density fluctuations . . . 22
2.7.1 Derivation . . . 22
2.7.2 An example . . . 24
3 Spatial resolution - 15p 26 3.1 The measurement volume . . . 26
3.2 Direct localisation . . . 27
3.3 Indirect localisation . . . 27
3.3.1 The pitch angle . . . 27
3.3.2 Single volume . . . 27
3.3.3 Dual volume . . . 32 2
4 Spectral analysis - 20p 33
4.1 Statistical quantities . . . 33
4.2 The autopower spectrum . . . 33
4.2.1 Single volume mixed phases . . . 33
4.2.2 Single volume phase separation . . . 33
4.3 The crosspower spectrum . . . 33
4.3.1 The crosspower amplitude . . . 33
4.3.2 The crosspower phase . . . 33
4.3.3 Simulation . . . 33
4.4 The autocorrelation . . . 33
4.5 The crosscorrelation . . . 33
II Experiment 34
5 Anomalous transport in fusion devices - 10p 35 6 The Wendelstein 7-AS stellarator - 15p 36 6.1 The magnetic field structure . . . 376.1.1 Nested flux surfaces . . . 37
6.1.2 Magnetic island formation . . . 37
6.2 Machine operation . . . 37
6.2.1 Heating methods . . . 37
6.2.2 Density control . . . 37
6.2.3 Standard configurations . . . 37
6.3 Plasma current . . . 37
6.3.1 The bootstrap current . . . 37
6.3.2 The Pfirsch-Schl¨uter current . . . 37
6.3.3 Inductive (external) current . . . 37
6.3.4 The rotational transform . . . 37
6.4 Confinement and transport (W7-AS specific) . . . 37
7 Experimental setup - 15p 38 7.1 Optical buildup . . . 38
7.1.1 Transmitting bench . . . 38
7.1.2 Measurement plane . . . 42
7.1.3 Receiving bench . . . 42
7.1.4 Diagnostic position . . . 43
7.2 Acquisition system . . . 43
7.2.1 Detectors . . . 43
7.2.2 Signal-to-noise ratio . . . 43
7.2.3 Absolute calibration . . . 43
7.2.4 Demodulation . . . 43
8 Investigated phenomena - 60p 46 8.1 Quasi steady-state . . . 47
8.1.1 Statistical analysis . . . 47
8.1.2 Autopower spectra . . . 47
8.1.3 Wavenumber spectra . . . 47
8.1.4 Crosspower spectra . . . 47
8.1.5 Scaling with plasma parameters . . . 47
8.2 Confinement bifurcations . . . 47
8.2.1 L- and H-mode experiments . . . 47
8.2.2 L-mode . . . 47
8.2.3 Quiescent H-mode . . . 47
8.2.4 Phase separation . . . 47
8.3 Fast confinement transitions . . . 47
8.3.1 Dithering H-mode . . . 47
8.3.2 L- and H-mode separation . . . 47
8.4 Slow confinement transitions . . . 47
8.4.1 Current ramp experiments . . . 47
8.5 High-β plasmas . . . 47
8.6 Detachment . . . 47
8.7 The ultra high density mode . . . 47
8.8 Electron temperature gradient? . . . 47
8.9 Density limit? . . . 47
9 Conclusions - 5p 48
A LOTUS setups, 1999-2001 55
B Dedicated experimental programs 58
2.1 Scattering geometry. Main figure: The position of a scatterer isrj andr0 is the detector position. Inset: The incoming wave vectork0 and scattered wave vectorksdetermine the observed wave vector k. . . 15 3.1 Magnetic field geometry. . . 29 7.1 Acquisition electronics. . . 44
5
7.1 Demodulation multiplication factors. . . 45
A.1 Experimental setups in the 1999 campaign - calibration factors. 55 A.2 Experimental setups in the 1999 campaign - lenses. . . 56
A.3 Experimental setups in the 2000 campaign. . . 57
B.1 Experiments performed on 27th and 29th of January 1999. Note: Shots 45230-44 had 4 mA detector current, 45275-88 had 7 mA detector current. . . 60
B.2 Experiments performed on 17th of May 1999. NOTE: ECRH deposition change 47192(HF launch, good)/47193(LF launch, bad). . . 61
B.3 Experiments performed on 14th and 16th of July 1999. Note: For shots 47932-46 laser exciter was on 9.5 mA, for shots 47974-76 on 6 mA. . . 62
B.4 Experiments performed on 26th of July 1999. . . 62
B.5 Experiments performed on 11th of August 1999. . . 63
B.6 Experiments performed on 13th of November 2000. . . 63
B.7 Experiments performed on 4th of December 2000. . . 64
6
Introduction - 5p
This thesis deals with measurements of fluctuations in the electron density of fusion plasmas. I will in the introduction outline the reasons these measurements are important for further progress and sketch the measurement principles. A brief outline of the obtained results will be presented in section 1.3 along with an overview of the thesis structure.
1.1 Motivation
If one were to make a survey of where we are, what we know and what we do not know about fusion plasmas, turbulence would certainly be an area marked ’Here Be Monsters’. The cross-field transport (perpendicular to the main toroidal magnetic field) assuming that only binary particle collisions contribute is called the neoclassical transport. This transport level includes effects associated with toroidal geometry. However, in general the measured transport is several orders of magnitude larger than the neoclassical one, especially for the electrons. This phenomenon has been dubbed anomalous transport and is subject to intense studies on most fusion devices.
It is generally believed that turbulence in the plasma creates fluctuations visible in most plasma parameters. Therefore a concerted effort has been devoted to the study of fluctuations and their relation to global (and local) plasma confinement quality.
The simplest modus operandi for the analysis of the importance of fluctuations with respect to confinement is to plot the amplitude of the fluctuations versus plasma confinement. But this approach often leads to more confusion than clarity, since it is frequently a fact that the fluctuation amplitude decreases while the confinement decreases or vice versa. If the measurements are frequency resolved, one can study the power in different
7
frequency intervals to determine whether certain bands are linked to confinement.
A step up in sophistication is to cross correlate measurements of fluctuations in different parameters, for example electron density and poloidal magnetic field. But even if a correlation exists, this does not mean that cross-field transport results; if the measurements are out of phase, the net transport will be zero.
Finally, one can calculate crosspower spectra between different fluctuating quantities if they are sampled using a common clock. This method yields the ’true’ transport level versus frequency.
1.2 Method
Most of the measurements presented in the thesis were made using a CO2
laser having a wavelength of 10.59 µm. The laser light scatters off bunches of electrons and is therefore called collective scattering.
Measurements of fluctuations in fusion plasmas using collective scattering were first performed in the nineteen seventies, made possible by the appearance of stable monochromatic laser sources.
Subsequent important players.
Pros and cons of the method.
Localisation, direct and indirect.
Non-intrusive.
State at the time of my Ph.D.
1.3 Results
Overall theme: Confinement transitions and their possible relation to fluctuations.
Fast L-H transitions, slow current ramp transitions associated with rational surfaces, high beta and detachment transitions.
First on stellarator (Heliotron-E, W7-A), stellarator/tokamak comparison important.
’Old’ 1 beam localisation, ’new’ 2 beam localisation.
Flexibility, angles, high wavenumber.
The thesis is composed of two main parts:
The first part (containing chapters 2 through 4) deals with the theoretical aspects of collective light scattering (chapter 2), spatial localisation
(chapter 3) and spectral analysis (chapter 4).
The second part (containing chapters 5 through 8) treats anomalous transport in fusion devices (chapter 5), the W7-AS stellarator (chapter 6), the experimental setup (chapter 7) and experimental findings (chapter 8).
Finally, the main conclusions are put forth in chapter 9. A bibliography and two appendices complete the thesis.
Theory
10
Collective light scattering - 15p
In this chapter we will investigate the theoretical aspects of scattering in detail. The main result will be the derivation of an expression for the observed photocurrent.
The reader may wonder why such a large portion of the thesis will be used treating what is standard scattering theory. The reason is that I have read through all material covering this subject I could find; I found that none of the existing sources contains a clear derivation beginning with the basics and ending with the final results. The purpose of the present chapter is to provide such a derivation.
To prepare the chapter I have used numerous sources, both easily available and harder to find. To list the sources simply in alphabetical order:
1. Antar [1], spatial bandpass filter
2. Antar et al. [3], phase separation theory 3. Demtr¨oder [14], description of Gaussian beams 4. Elbek [18], scattering cross section
5. Gr´esillon et al. [26], photocurrent
6. Holzhauer et al. [33], photocurrent and wavenumber resolution 7. Honor´e [35], far field approximation
8. Hutchinson [37], scattering classification and cross section 9. Menicot [45], demodulation
10. Slusher et al. [59]
11. Truc et al. [65], beam profile 11
2.1 Historical notes
J.C.Maxwell predicted that light is an electromagnetic phenomenon (1865) and H.Hertz confirmed this by observing electromagnetic waves (1886).
Lord Rayleigh (born J.W.Strutt) calculated the scattering of an
electromagnetic field by a perfectly conducting sphere of radius r0 much smaller than the wavelength of the wave λ0 (1871). We will always work in the Rayleigh limit, where r0 ¿λ0.
The late 1950’s saw the advent of lasers; this provided stable sources of monochromatic radiation.
First observation of density fluctuations in a fusion device was made by C.M.Surko and R.E.Slusher in the Adiabatic Toroidal Compressor (ATC) tokamak [63].
2.2 Introduction
I would like to touch upon a few subjects relating to the type of scattering that is observed. First of all a classification of scattering is useful:
• If one were to describe scattering of an electromagnetic field off a particle quantum mechanically, the description would be of photons bouncing off the particle. Thomson scattering: Negligible change in mean particle momentum during collision with the photon
(h
Ã
ω¿mc2). Compton scattering: The case where photons are so energetic that their momentum cannot be ignored. As we work with a wavelength λ0 = 10.59 µm in the infrared range, the photon energy is much smaller than the rest mass of the electron. Therefore we will restrict ourselves to consider classical Thomson scattering.• Since the ions are much heavier than the electrons, their acceleration and hence radiation is usually sufficiently small to be ignored. So the electrons do the scattering.
• The Salpeter parameter αS = 1/kλD [57] determines whether the scattering observed is incoherent (αS<1) or coherent (αS>1). Here, k is the wavenumber observed and λD =p
ε0T /ne2 is the Debye length. Basically, incoherent scattering is due to scattering off single electrons, while coherent scattering is due to scattering off a bunch of electrons; this is also known as collective scattering and is the limit we are observing with the diagnostic.
To sum up, we are dealing with collective Thomson scattering.
Four elements go into the process of scattering:
1. The incident radiation (the laser beam).
2. Set of scatterers (electrons).
3. Reference beam.
4. The detector.
In this chapter we describe the first 3 parts; a description of the detectors used is to be found in chapter 7 which also contains a detailed description of the practical implementation of the scattering diagnostic.
2.3 Scattering cross section
The power per unit solid angle (Ωs) scattered at an angle ζ by an electron is given by
dP dΩs
=P0r2qsin2ζ, (2.1)
where P0 is the incident laser power. The differential scattering cross section is then defined as
dσ dΩs
= dP dΩs
/P0 =
µ µ0e2 4πmq
¶2
sin2ζ =r2qsin2ζ (2.2) The classical electron radius re is given by inserting the electron mass into the expression for rq and is
re = µ0e2 4πme
(2.3) Equation 2.2 shows us that ions do not contribute appreciably to scattering because of their small scattering cross section due to large mass (compared to the electrons). The classical electron radius can be rewritten using the static polarisability (assumed identical for each particle j: αj ≡α=p/ε0E)
α= µ0e2
mek20 (2.4)
to become
re= k20α
4π (2.5)
Using that dΩs = 2πsinζdζ we get σ =
Z
dσ = 2πr2e Z π
0
sin3ζdζ = 2πr2e(4/3), (2.6) which one could interpret as an effective size of the electron for scattering.
2.4 Scattering theory
2.4.1 Radiation source
Our incident laser beam has a direction k0, where k0 =ω0/c, and a
wavelength λ0 = 10.59 µm. For a linearly polarised beam, the electric field is given as in Equation 2.7, where E0(r) = E0u0(r)ei(k0·r). E0 is a vector whose direction and amplitude are those of the electric field at maximum.
E0(r, t) =Re{E0(r)e−iω0t} (2.7) Assuming Gaussian beams, the radial profile near the waist w will be of the form u0(r) =e−(r2⊥/w2), where r⊥ is the perpendicular distance from the beam axis.
The frequency of the laser radiation ω is much higher than the plasma frequency ωp =p
ne2/ε0me. This means that the refractive index of the plasma
n =q
1−ωp2/ω2 (2.8)
is close to one, or that refractive effects are negligible [55]. This is a
significant advantage compared to microwave diagnostics, where raytracing calculations must assist the interpretation of the measurements.
2.4.2 Single particle scattering
For a single scatterer having index j located at position rj (see figure 2.1), the scatterer radiates an electric field at r0 (the detector position) as a result of the incident beam field. This field is given in Equation 2.9, where nj is alongr0−rj and approximately perpendicular to E0(rj).
Es(r0, t) =Re{Es(r0)e−iω0t} Es(r0) =
½k20α 4π
eik0|r0−rj|
|r0−rj|nj ×[nj × E0(rj)]
¾
(2.9) The scattered field is simply the radiation field for an oscillating dipole having a moment p. Therefore the above expression for the scattered electric field is often called the dipole approximation. It is an
approximation because the equation is only valid in the nonrelativistic limit. For very energetic electrons the relativistic corrections become significant, see e.g. [37].
origin
detector
r kj( )0
r’
r r n’- ( )j j
scattering region
k0 ks k
Figure 2.1: Scattering geometry. Main figure: The position of a scatterer is rj and r0 is the detector position. Inset: The incoming wave vector k0 and scattered wave vector ks determine the observed wave vector k.
2.4.3 Far field approximation
Two assumptions are made:
1. The position where one measures (r0) is far from the scattering region 2. The opening angle of the detector is small,
leading to the validity of the far field approximation. This means that we can consider the scattered field from all j particles in the scattering volume
to have the same direction denoted n0 parallel to nj. We further assume that the coordinate system origin is close to the scattering region. This means that rj ¿r0 and allows us to expand to first order:
|r0−rj| 'r0−rj ·n0 (2.10) Therefore we can simplify equation 2.9 to become
Es(r0, t) =Re{Es(r0)e−iω0t} Es(r0) =
½k02α 4π
eik0r0
r0 u0(rj)eirj·(k0−ks)n0×[n0 × E0]
¾
, (2.11)
where ks=k0n0. k=ks−k0 is the wave vector selected by the optics.
2.4.4 Multiple particle scattering
The scattered field at the detector due to several particles can be written as a sum
Es(r0, t) =Re{Es(r0)e−iω0t} Es(r0) = k02α
4π eik0r0
r0 X
j
u0(rj)n0×[n0× E0]eirj·(k0−ks) (2.12) In going from a single particle scattering description to more particles, we will approximate the position of the individual scatterers rj by one common vector r. The particles will have a density distribution n(r, t). We write the scattered field as an integral over the measurement volume V:
Es(r0, t) = k20α 4π
eik0r0 r0
Z
V
u0(r)n0×[n0× E0]n(r, t)e−ik·rd3r (2.13)
2.4.5 Phase separation
Since the theory behind phase separation is extensively described in section 2 of [3], we will here only give a brief recapitulation of the basics.
The observed signal is interpreted as being due to a large number of
’electron bunches’, each moving in a given direction. The light scattering signal i is being described as
i(t) =
Nb
X
j=1
ajeiφj =AeiΦ, (2.14)
where Nb is the number of bunches, while aj and φj is the amplitude and phase of bunch number j, respectively. The criterion for determination of direction is
∂tΦ>0⇒k·U >0⇒fluctuationskk
∂tΦ<0⇒k·U <0⇒fluctuationsk −k, (2.15) where Φ =k·Ut and U is the average velocity.
2.5 The photocurrent
The incident optical power reaching the detector can be found integrating the Poynting vector over the detector area A
S(t) = 1 µ0
Z
A
(E×B)·d2r0 = 1
µ0c Z
A|ELO(r0, t) +Es(r0, t)|2d2r0 = 1
µ0c Z
A|ELO(r0, t)|2+|Es(r0, t)|2 + 2×Re{E∗LO(r0, t)Es(r0, t)}d2r0 (2.16) What we are interested in is the last term of the equation, namely the beating term
SB(t) = Z
A
sB(r0)d2r0 = Z
A
2
µ0cRe{E∗LO(r0, t)Es(r0, t)}d2r0 (2.17) Assuming a detector quantum efficiency η leads to the photocurrent
iB(t) = eη h
Ã
ω0Z
A
sB(r0)d2r0 e
NEP Z
A
sB(r0)d2r0 (2.18) The electric field of the LO beam along n’ at the detector is given as
ELO(r0, t) = Re{ELO(r0)e−i(ω0+ω∆)t}
ELO(r0) =ELOuLO(r0)eik0n0·r0 (2.19)
In the above equation we have assumed that kLO =ks =k0n0. Now we can derive an expression for sB(r0)
sB(r0) = 2
µ0cRe{E∗LO(r0, t)Es(r0, t)}= 2
rε0
µ0
Re{Es(r0)· ELO∗ (r0)eiω∆t}, (2.20) where ω∆ is the beat frequency. The photocurrent due to an ensemble of scatterers at the detector position r0 (replacing iB by ik) is
ik(t) = eη h
Ã
ω0Z
A
sB(r0)d2r0 = 2Re
½ 1 µ0c
Z
A
[E∗LO(r0, t)Es(r0, t)]d2r0
¾
= 2Re
½ 1 µ0c
Z
A
h
ELO∗ u∗LO(r0)e−ik0n0·r0eit(ω0+ω∆) k02α
4π eik0r0
r0 Z
V
u0(r)[n0×[n0× E0]n(r, t)e−ik·re−iω0t]d3r
¸ d2r0
¾
= 2Re
½k02α 4π
1 µ0ceitω∆
Z
V
·iλ0
iλ0
Z
A
eik0r0
r0 u∗LO(r0)ELO∗ e−ik0n0·r0d2r0
¸
e−iks·r[n0×[n0× E0]eik0·ru0(r)n(r, t)d3r]ª
= 2Re
½ ik02α
4π λ0
µ0ceitω∆ Z
V ELO∗ u∗LO(r)e−iks·rE0u0(r)eik0·rn(r, t)d3r
¾
= 2Re
½ iπα
λ0
rε0
µ0
eitω∆ Z
V ELO∗ u∗LO(r)E0u0(r)e−ik·rn(r, t)d3r
¾
, (2.21) since
k02α 4π
λ0
µ0c = πα λ0
rε0
µ0
n0×[n0 × E0] = E0 (2.22) The central quantity in equation 2.21 is the Kirchhoff-Sommerfeld
reconstructed field 1 iλ0
Z
A
eik0r0
r0 u∗LO(r0)ELO∗ e−ik0n0·r0d2r0 =u∗LO(r)ELO∗ (2.23)
which is the radiated field from a known monocromatic field distribution on a diaphragm A. This radiated field (the antenna or virtual LO beam) propagates from the detector to the scatterers. Equation 2.23 (which we used for the derivation of the current) holds when the antenna beam
coincides with the LO beam. For this to be true three conditions have to be met:
1. The quantum efficiency η of the detector area is uniform.
2. The LO beam power is incident on a small portion of the detector area to avoid edge diffraction.
3. The phase shift between the LO and the antenna beams is small.
The expression for the current now becomes
ik(t) = 2 eη
h
Ã
ω0rε0
µ0
λ0Re
½
ireeiω∆tE0ELO∗
Z
V
n(r, t)u0(r)u∗LO(r)e−ik·rd3r
¾
, (2.24) where ELO∗ and E0 hereafter are to be considered as scalars since the laser field and the LO field are assumed to have identical polarisation.
Introducing a shorthand notation for the spatial Fourier transform
(n(t)U)k= Z
V
n(r, t)U(r)e−ik·rd3r
U(r) =u0(r)u∗LO(r), (2.25) we note that
Z
V
n(r, t)U(r)e−ik·rd3r=n(k, t)? U(k) n(k, t) =
Z
V
n(r, t)e−ik·rd3r U(k) =
Z
V
U(r)e−ik·rd3r, (2.26) where ? denotes convolution. We arrive at
ik(t) = 2 eη h
Ã
ω0rε0
µ0
λ0Re[ireeiω∆tE0ELO∗ (n(t)U)k] (2.27)
Defining
α≡ eη h
Ã
ω0rε0
µ0
λ0reE0ELO∗ (2.28) Equation 2.24 in its final guise is
ik(t) = i[αeiω∆t(n(t)U)k−α∗e−iω∆t(n(t)U)∗k] (2.29) Note that the e−ik·r-term in (n(t)U)k constitutes a spatial band pass filter (k is fixed). Three scales are involved:
• Fluctuations occur at scales much smaller than
λ= 2π/k ⇒k·r ¿1⇒e−ik·r ≈1. The Fourier transform becomes the mean value of the density fluctuations (which is zero).
• Fluctuations occur at scales similar to λ= 2π/k; this leads to the main contribution to the signal.
• Fluctuations occur at scales much larger than
λ= 2π/k ⇒k·r À1⇒e−ik·r is highly oscillatory. The mean value will be roughly equal to that of e−ik·r which is zero.
The scattered power resulting from the interference term can be written by defining a constant
ξ≡ rε0
µ0
λ0reE0ELO∗ (2.30) and replacing α with this in Equation 2.29
Pk(t) = h
Ã
ω0eη ik(t) = i[ξeiω∆t(n(t)U)k−ξ∗e−iω∆t(n(t)U)∗k] =
2Re[iξeiω∆t(n(t)U)k] (2.31) If E0 and ELO are real numbers (meaning that ξ is real) we can go one step further and write
Pk(t) = 2ξRe[ieiω∆t(n(t)U)k] = 8λ0re
πw2
pP0PLORe[ieiω∆t(n(t)U)k] (2.32) assuming that P0/LO= πw42qε
0
µ0|E0/LO2 | (for a given U, see subsection 2.7.2).
2.6 Demodulation
The task now is to extract real and imaginary parts of (n(t)U)k. We construct two signals that are shifted by π/2:
j1(t) =Re[eiω∆t] = 1
2(eiω∆t+e−iω∆t) = cos(ω∆t) j2(t) = Re[ei(ω∆t+π/2)] = 1
2(ei(ω∆t+π/2)+e−i(ω∆t+π/2)) = sin(ω∆t) (2.33) Now two quantities are constructed using equations 2.29 (divided into two equal parts) and 2.33:
id,1 = ik(t)
2 j1(t) = i
4[αei2ω∆t(n(t)U)k+α(n(t)U)k− α∗(n(t)U)∗k−α∗e−i2ω∆t(n(t)U)∗k]
id,2 = ik(t)
2 j2(t) = i
4[αei2ω∆teiπ/2(n(t)U)k+αe−iπ/2(n(t)U)k−
α∗eiπ/2(n(t)U)∗k−α∗e−i2ω∆te−iπ/2(n(t)U)∗k] (2.34) Low pass filtering of these quantities removes the terms containing the fast 2ω∆ expression. The result is that
id,complex = [id,2−iid,1]lpf = 1
2(Re[α(n(t)U)k]−i(−Im[α(n(t)U)k])) = α
2(n(t)U)k (2.35) Now we have (n(t)U)k and can analyse this complex quantity using spectral tools. The alternative to heterodyne detection is called homodyne (or video) detection. There are two advantages that heterodyne detection has compared to homodyne (direct) detection:
1. The LO beam provides an amplification factor to the detected signal (see equation 2.32)
2. It leaves the complex (n(t)U)k intact multiplied by a wave having frequency ω∆; in direct detection the electric field complex number is transformed into a real number and the phase information is lost.
The frequency sign of the scattered power tells us in which direction the fluctuations are moving.
2.7 Density fluctuations
2.7.1 Derivation
The current frequency spectral density measured is
Ik(ω) = lim
T→∞
|ik(ω)|2 T ik(ω) =
Z T
eiωtik(t)dt (2.36)
Using 2.29 this can be written
Ik(ω) = |α2|
T {|(n(ω)U)k|2+|(n(−ω)U)k|2} (n(ω)U)k =
Z d3r
Z T
n(r, t)U(r)ei(ωt−k·r)dt n(k, ω) =
Z T
n(k, t)eiωtdt, (2.37) assuming that n(k, ω) and n(k,−ω) are independent (i.e. no mixed terms).
Note that we have dropped the ω∆ terms; it has previously been explained how we filter these high frequencies away. Now we are approaching an analytical expression for the weighted mean square density fluctuation. The time fluctuating part of n(r, t) is
˜
n(r, t)≡n(r, t)− lim
T→∞
1 T
Z T
n(r, t) dt (2.38) This definition enables us to express the weighted mean square density fluctuation as
hn˜2iU T = RT
dtR
˜
n2(r, t)|U(r)|2d3r T R
|U(r)|2d3r (2.39)
The subscript means averaging over the profile function U(r) and a time interval T. We can transform this via Parseval’s theorem
Z T
dt Z
|n(r, t)U˜ (r)|2d3r= Z dω
2π
Z d3k
(2π)3|(˜n(ω)U)k|2 (2.40) to the wave vector-frequency domain
h˜n2iU T =n0
Z dω 2π
Z d3k
(2π)3SU(k, ω) SU(k, ω) = |(˜n(ω)U)k|2
n0TR
|U(r)|2d3r, (2.41) where n0 is the mean density in the scattering volume. SU(k, ω) is the measured spectral density also known as the form factor. Conventionally, this is given as
S(k, ω) = lim
T,V→∞
|n(k, ω)˜ |2 n0V T
˜
n(r, t) = Z dω
2π
Z d3k
(2π)3n(k, ω)e˜ −i(ωt−k·r) (2.42) Combining Equations 2.41 and 2.37 we get
SU(k, ω) +SU(k,−ω) = Ik(ω) n0|α2|R
|U(r)|2d3r (2.43) The term with positive frequency corresponds to density fluctuations
propagating in the k-direction, while negative frequency means propagation in the opposite direction [65].
The wavenumber resolution width is
∆kr3 ≡
·Z
|U(r)|2d3r
¸−1
(2.44) Finally - after working our way through a maze of equations - we are at the goal; our final expression for the mean square density fluctuations is
hn˜2iU T =
Z d3k (2π)3
hn˜2ikr
∆kr3 hn˜2ikr = 1
|α2|£R
|U(r)|2d3r¤2
Z ∞
−∞
dω
2πIk(ω) (2.45)
The frequency integration is done numerically, while a wave vector integration can be done by measuring Ik for different wave vector values.
2.7.2 An example
When the beam profile U(r) is known, quantitative expressions for the density fluctuations can be calculated. The following assumptions are made:
• Antenna beam corresponds to LO beam.
• Beams have Gaussian profiles.
• Beams are focused in the measurement region with identical waists w.
• Forward scattering.
Furthermore, the function U(r) is assumed to be
U(r) = u0(r)u∗LO(r) = e−2(x2+y2)/w2 for |z|< L/2
U(r) = 0 for |z|> L/2, (2.46) where L is the measurement volume length and the beams are along z.
The wavenumber resolution width ∆kr3 becomes 4/(πw2L) and we find the wavenumber resolution itself by calculating
U(k) = Z
V
U(r)e−ik·rd3r= Z L/2
−L/2
e−ikzzdz
·Z ∞
−∞
e−(w22x2+ikxx)dx
¸ ·Z ∞
−∞
e−(w22y2+ikyy)dy
¸
= 2
kz
sin µkzL
2
¶ ·rπ 2we−k
xw2 2 8
¸ ·rπ 2we−
k2 y w2
8
¸
(2.47) allowing us to define the transverse wavenumber resolutions ∆kx,y = 2/w (e−1/2) and a longitudinal wavenumber resolution ∆kz = 2π/L (sine term zero). We further obtain an expression for the main (and LO) beam power
P0 = rε0
µ0
Z ∞
−∞|E02|e−4(x2+y2)w2 dxdy= πw2
4 rε0
µ0|E02|, (2.48)
In= e2hηP
Ã
ω0LO = e2PLONEP1 and PLO = πw42qε0
µ0|ELO2 |. Using equation 2.45 for this example we get
hn˜2ikr= 1 (2π)3
µh
Ã
ω0eη
¶2
1 λ20r2eL2
1 P0PLO
Z ∞
−∞
dω
2πIk(ω) = 1
(2π)3 h
Ã
ω0η 1 λ20r2eL2
1 P0
Z ∞
−∞
dω 2π
Ik(ω) In
(2.49)
Spatial resolution - 15p
3.1 The measurement volume
The time-independent field from the two Gaussian beams creating the measurement volume can be written
u(r) =
s 2P πw2(z)e−
x2+y2 w2(z)+ik0z
µ
1+ x2+y2
2(z2 R+z2)
¶ +iφ(z)
(3.1) Here,
w(z) =w0 s
1 + µ z
zR
¶2
(3.2) is the beam radius at z and zR is the Rayleigh range
zR = πw02 λ0
, (3.3)
which is the distance from the waist to where the beam radius has grown by a factor √
2. The phase is given by
φ(z) = arctan³zR z
´ (3.4)
Note that we use the complete Gaussian description here instead of the simple form used in chapter 2.
An excellent treatment of the measurement volume has been given in [28];
therefore we will here restrict ourselves to simply quoting the important results and approximations below.
26
3.2 Direct localisation 3.3 Indirect localisation
3.3.1 The pitch angle 3.3.2 Single volume
Localised autopower
We introduce the instrumental selectivity function ρ(z, α) =e−
³α−θp(z)
∆α
´2
(3.5) ,where α is the angle between the observation wave vector and the major radius, ∆α= ∆kk = kw2 is the relative wavenumber resolution and θp is the pitch angle; in our case rather the horizontal pitch angle, which we define as Arctan³
BR
Bϕ
´. Using this instrumental function, the scattered power
Ik(ω, α)∝ Z
S(k, ω, z)ρ(z, α)dz (3.6) The above integrand times the differential step in the z direction is the power scattered by each volume element up through the plasma.
Below we will derive the expression for the scattered power (equation 3.6).
We will ignore constant factors and thus only do proportionality
calculations to arrive at the integral. This equation will prove to be crucial to the understanding of the observed signal and the limits imposed on localisation by the optical setup.
∗ ∗ ∗ Approach 1: Taken from [45].
The starting point is the current spectral density (equation 2.37) Ik(ω)∝
Z Z
hn(r, ω)n(r−r0, ω)iU(r)U(r−r0)e−ik·r0d3r0d3r (3.7) and the local spectral density (as opposed to the global spectral density in equations 2.41 - 2.43)
S(k, ω,r)∝ Z
hn(r, ω)n(r−r0, ω)ie−ik·r0d3r0 (3.8)
Brackets indicate time averaging. The inverse Fourier transform of the local spectral density gives us
hn(r, ω)n(r−r0, ω)i ∝ Z
S(κ, ω,r)eiκ·r0d3κ (3.9) ,where κ is the fluctuation wave vector, allowing us to rewrite the current spectral density
Ik(ω)∝
Z Z Z
S(κ, ω,r)U(r)U(r−r0)ei(κ−k)·r0d3κd3r0d3r (3.10) We assume that the beams are bisected by z and that the local spectral density only varies along this line:
S(κ, ω,r) =S(κ, ω, z) (3.11) Assuming that our U’s are given as before, we obtain
Ik(ω)∝ Z
dz Z
d3κS(κ, ω, z) Z
dx0dy0dz0ei(κ−k)·r0dxdye−2/w2(2x2+2y2−2xx0−2yy0+x02+y02)∝ Z
dz Z
d3κS(κ, ω, z) Z
dx0dy0dz0ei(κ−k)·r0e−1/w2(x02+y02) (3.12) From geometrical considerations in rectangular coordinates (x,y,z) (see figure 3.1) we find that
−(κ−k)·r0 = (kcosα−κ⊥cosθp+κksinθp)x0 +(ksinα−κ⊥sinθp−κkcosθp)y0
−κ⊥zz0
=k1x0 +k2y0−κ⊥zz0 (3.13) The abbreviations are put into our equation
Ik(ω, α)∝ Z
dz Z
d3κS(κ, ω, z) Z
dx0dy0dz0e−(1/w2x02+ik1x0)e−(1/w2y02+ik2y0)eiκ⊥zz0(3.14)
Z y
x
k k
^
k||
a q
p
qp
Figure 3.1: Magnetic field geometry. k= (kcosα, ksinα,0),κ= (κ⊥cosθp− κksinθp, κ⊥sinθp+κkcosθp, κ⊥z).
The assumptions that the minor radius a¿L and κk ¿κ⊥ mean that Z L/2
−L/2
eiκ⊥zz0dz0 ≈δ(κ⊥z)
S(κ, ω, z) =S(κ⊥, ω, z)·δ(κk) (3.15) Performing the integrations over primed coordinates we get
Ik(ω, α)∝ Z
dz Z
d3κS(κ⊥, ω, z)e−w2(k21+k22)/4 (3.16) Setting κk equal to zero
k21 +k22 =k2+κ2⊥−2kκ⊥cos(α−θp) (3.17) and integrating over d3κ = dκ⊥ dκk (i.e. replacing κ⊥ by k) we arrive at
Ik(ω, α)∝ Z
S(k, ω, z)e−w2(2k2[1−cos(α−θp)])/4dz (3.18) Finally, assuming that the angles α and θp are small, we can expand the function in the exponent as
2k2[1−cos(α−θp)]≈2k2[(α−θp)2]/2 =k2(α−θp)2 (3.19) so that we obtain the final expression
Ik(ω, α)∝ Z
S(k, ω, z)e−w2k2[(α−θp)2]/4dz (3.20)
This concludes our derivation.
∗ ∗ ∗ Approach 2: Taken from [4].
Basic signal:
n0(k, t) = Z
V
eik·x˜n(x, t)e(θ2+φ2)/w2dx (3.21) ,where the volume element position x = (r, θ, φ) and
1/w2 = 1/w2P B+ 1/wAN2 . We assume that the density fluctuations ˜n can be written as
˜
n(r, θ, φ, t) = 1 (2π)2
Z
V
˜
n(r,Kθ,Kφ)e−i(Kθ·r+Kφ·r)dKθdKφ (3.22) ,where K is the fluctuation wave vector and kis the analysing wave vector.
These wave vectors can be written
K=Kθcosξ(r)ˆθ+Kθsinξ(r) ˆφ
k=kcosαθˆ+ksinαφˆ (3.23) Inserting the expression for the density fluctuations and the wave vectors into equation 3.21 we get
n0(k, t)∼ 1 2a(2π)2
Z a
−a
dr Z ∞
0
dθ Z ∞
0
dφ Z
dKθ
Z dKφ
e(θ2+φ2)/w2e−i(Kθcosξ−kcosα)θe−i(Kθsinξ−ksinα)φn(r, K˜ θ, Kφ, t) (3.24) Note for equation 3.24: Where does the 2a factor originate? I have changed the sign of the imaginary exponential functions and added the missing θ and φ.
We now assume that Kφ'0 and perform the integrations over θ and φ in equation 3.24:
n0(k, t)∼ Z
dr Z
dKθe−w2[(Kθcosξ−kcosα)2+(Kθsinξ−ksinα)2]/4n(r, K˜ θ, t) (3.25) Note for equation 3.25: Integrating only over zero to infinity and not minus infinity to plus infinity one would get additional error functions in
performing the integration?
Assuming that the angles α andξ are both close to zero, we can expand the sines and cosines of equation 3.25 to first order and arrive at
n0(k, t)∼ Z
e−w2(Kθξ−kα)2/4dr Z
e−w2(Kθ−k)2/4n(r, K˜ θ, t)dKθ (3.26) The last integral of equation 3.26 acts as a spatial bandpass filter, leading to the selection rule Kθ 'k, which allows us to write equation 3.26 as
n0(k, t)∼ Z
e−w2k2(ξ−α)2/4n(r, k, t)dr˜ (3.27) This concludes the derivation. The relationship between the measured signal intensity |n0(k, t)|2 and the real turbulence intensity δn(r, t)2 is
|n0(k, t)|2 ∼ Z
ST(r)dr
ST(r) =δn(r, t)2e−w2k2(ξ−α)2/2 (3.28) Fluctuation profiles
Performing α scans, the ALTAIR team [65] arrived at the following expression for the squared density fluctuations integrated over all frequencies where fluctuations were observed:
δn/n=Ic+Ie(r/a)ζ (3.29) ,where r/a is the normalised minor radius, which corresponds to reff/a in the stellarator geometry. Ic is the core level of turbulence
(δn/n(r/a= 0) =Ic), while Ie controls the edge turbulence amplitude in conjunction with Ic (δn/n(r/a= 1) =Ic+Ie). The third parameter ζ determines how fast the turbulence intensity increases towards the edge plasma.
Initially, the parameters found for L-mode plasmas were: Ic = 6×10−5, Ie = 0.015 (Ic/Ie= 0.004) and ζ = 7±2 [16]. However, there is some confusion regarding the exact functional form of the turbulence level, since the equation line 4, page 72 in [16] has a typing error (the parenthesis should be squared).
Most recent results on Tore Supra L-mode plasmas yield the following results: Ic = 0.7%, Ie = 10% and ζ = 8 [4]. Is this really in percent (absolutely calibrated)?!
The above expression for the density fluctuation intensity as a function of plasma radius is to be inserted into equation 3.20 (S(k, ω, z) = δn2) or equation 3.28.
3.3.3 Dual volume
Localised crosspower
Spectral analysis - 20p
4.1 Statistical quantities
4.2 The autopower spectrum
4.2.1 Single volume mixed phases 4.2.2 Single volume phase separation
4.3 The crosspower spectrum
4.3.1 The crosspower amplitude 4.3.2 The crosspower phase 4.3.3 Simulation
4.4 The autocorrelation 4.5 The crosscorrelation
33
Experiment
34
Anomalous transport in fusion devices - 10p
Summarise observations of fluctuations and their connection to transport
35
Chapter 6
The Wendelstein 7-AS stellarator - 15p
6.1 The magnetic field structure
6.1.1 Nested flux surfaces
6.1.2 Magnetic island formation
6.2 Machine operation
6.2.1 Heating methods
Electron cyclotron resonance heating Ion cyclotron resonance heating Neutral beam injection heating
6.2.2 Density control
6.2.3 Standard configurations
6.3 Plasma current
6.3.1 The bootstrap current
6.3.2 The Pfirsch-Schl¨ uter current 6.3.3 Inductive (external) current 6.3.4 The rotational transform
6.4 Confinement and transport (W7-AS
specific)
Experimental setup - 15p
In this chapter the practical realisation of the density fluctuation diagnostic LOTUS is described. The 3 main parts of the optical system - transmitting bench, measurement plane and receiving bench - are described in section 7.1 along with details of the diagnostic position. In section 7.2 I then proceed to describe the detectors and acquisition electronics in detail.
7.1 Optical buildup
The 3 following subsections containing the detailed description of the optical setup are organised as a step-by-step chronological explanation of the optical components.
7.1.1 Transmitting bench
When describing the transmitting system, we initially encounter the radiation source, which is a CO2 laser, built by Ultra Lasertech and
modified by Risø. The laser delivers roughly 20 W cw (continual wave) and is operated in the fundamental TEM00 mode. The laser wavelength λ0 is 10.59 µm. Mode suppression calculations have been made, resulting in the insertion of an aperture of adjustable size into the laser cavity just before the output mirror. The standard diameter used is 7.8 mm. The typical values of the exciter voltage and current is 25 kV and 9.5 mA, respectively.
The cavity length is 1380 mm, the radius of curvature of the end mirror 3000 mm with a flat output mirror connected to a PZT (piezo electric) element. The gain medium is contained in a glass tube with Brewster windows mounted on each end.
Several modifications to the laser system had to be made in the course of
38
my project, including replacement of high voltage resistors in the exciter and the installation of a new laser stabilisation scheme due to inadequacies of the in-built system.
The beam waist w0 at the output mirror has a nominal value of 2.25 mm.
∗ ∗ ∗
One can calculate the beam waist between the lenses of the system by assuming that the lenses have a confocal spacing. This means that two lenses having focal lengths f and f0 should be placed with a distance of f +f0 between them. When quoting actual focal lengths, I will do so in mm. Thus, assuming confocal lens spacing (and Gaussian beams), one obtains that the beam waist wbefore and after a lens with focal length f is related by Equation 7.1.
wafter lens = λ0f πwbefore lens
(7.1) One then steps through the optical system and calculates the waist between each pair of lenses at their focal lengths. The most important waist sizes are those in the plasma and at the detectors.
We need an additional two simple formulas to proceed with the optical layout; the first one is
tanθ/2'θ/2 =d/2f (7.2)
,where d is the distance between two parallel beams before the lens andθ is their crossing angle at the focal point after the lens. Further, using figure 2.1, we find that
tanθs 'θs=k/k0 (7.3)
,which relates the wavenumber observed using the diagnostic k to the laser wavenumber k0 ≡2π/λ0 and the scattering angle θs. The approximations in equations 7.2 and 7.3 leading to the right-hand equations are valid for small angles. We are now properly equipped to continue the descrption of the transmitting table.
All lenses in the setup are ZnSe AR (anti-reflection) coated.
1. Shutter: Enables one to block the beam.
2. ZnSe beam sampler: Diffracts 0.1 percent power to one side of the main beam. The diffracted beam is focused into a detector to monitor the laser mode spectrum.
