Download Ion Power Balance in Neutral Beam Heated Discharges on the NSTX

I ON P OWER BALANCE
IN
N EUTRAL B EAM
H EATED D ISCHARGES
ON THE
NATIONAL
S PHERICAL TORUS E XPERIMENT (NSTX)
PATRICK W ILLIAM ROSS
A D ISSERTATION
P RESENTED
OF
IN
TO THE
P RINCETON U NIVERSITY
C ANDIDACY
OF
FACULTY
D OCTOR
OF
R ECOMMENDED
BY THE
FOR THE
D EGREE
P HILOSOPHY
FOR
ACCEPTANCE
D EPARTMENT
OF
A STROPHYSICAL S CIENCES
P ROGRAM
IN
P LASMA P HYSICS
A DVISORS : D.A. G ATES , R.B. W HITE
J UNE 2010
c Copyright by Patrick William Ross, 2010.
All Rights Reserved
Abstract
In this dissertation, ion power balance is studied in the National Spherical Torus Experiment (NSTX). In order to fully understand all the terms in the power balance equation, an
experiment was conducted in which the input power of the NSTX discharges was modulated using the neutral beam input power. Individual neutral beam sources were turned on
or off to increase or decrease the input power. The modulations were designed to create a
localized perturbation in the fast ion phase space distribution. Measurements were taken
with diagnostics on NSTX including the neutral particle analyzer (NPA), the NSTX Thomson scattering diagnostic (MPTS) and the charge exchange diagnostic (CHERS), which
measure the ion and electron density and temperature.
Reconstructions of the plasma equilibrium were made using the LRDFIT code and
power balance was calculated using the TRANSP transport code. The TRANSP code
inputs the electron and ion temperatures and densities, and calculates the various power
balance terms.
Excess ion heating is shown to exist for some neutral beam heated discharges on NSTX.
When all three neutral beam sources are on, the total power outflow through the various
loss terms exceeds the input power calculated through classical heating of the ions by the
slowing down of the neutral beams. The excess heating is approximately 0.5 MW of power,
which represents 8% of the total power put into the plasma from the neutral beams.
This excess heating occurs at the same time as an increase in the amplitude of high
frequency compressional Alfvén eigenmodes (CAEs) with frequencies of 0.5-1.4 MHz.
The CAE amplitudes coincide both temporally and spatially with the onset of the excess
heating. The high-k microwave scattering diagnostic and magnetic pickup coils were used
to estimate the amplitudes of these modes. Full orbit calculations were then performed
using these modes, which demonstrate that thermal ions can be heated by the CAEs.
The Edge Neutral Density Diagnostic (ENDD) was developed to aid in the understanding of ion power balance. The ENDD uses a 2-D charge-coupled device camera to meaiii
sure neutral hydrogen line emissions from the plasma, which are input into a collisionalradiative model to determine the edge neutral density profile. The data from the ENDD
was used as an input into the TRANSP calculations.
The NPA was used in an attempt to study the fast ion population. However, limitations
in the NPA design prevented it from measuring the full fast-ion distribution. This dissertation discusses the limitations of the NPA and suggests future improvements in diagnostics
for ion power balance measurements.
This work was supported by the Department of Energy, contract DE-AC02-09CH11466.
iv
Acknowledgements
This research would never have been completed without a host of people assisting me.
First, I would like to thank David Gates who has always been there, spotting the holes in
my arguments and guiding me in the right direction. I learned early on to ignore his advice
at my own peril. Also, I would like to thank Roscoe White, who let me bounce ideas off
him before patiently and repeatedly showing me my mistakes.
I would also like to thank Jon Menard and Michael Bell who were kind enough to be
readers for this dissertation. Their corrections and suggestions helped shape this into a
much stronger thesis. I also want to thank Stan Kaye for his help with TRANSP, as well as
his comments and help in strengthening my arguments.
I would also like to thank Lane Roquemore, Doug Labrie, Vlad Soukhanovskii, Gretchen
Zimmer and the NSTX technicians who helped with my diagnostic design and calibration.
Many of them spent many late hours helping me with my in vessel work, for which I would
like to thank them.
I also want to thank Barbara Sarfaty. I appreciate all that she has done and cannot
express my thanks to her enough. I also owe a debt of gratitude to Tom Jenkins and Sterling Smith, who were both good friends during my tenure here at PPPL. They set a great
example for me, and were great friends as well.
Finally, I would like to thank my wife, Angela, and my son, Justin. Justin has always
been a little trooper when Dad had to work. Angela has been there to motivate me and keep
me going when I wanted to give up. And she has always been willing to take care of our
family even when I have had to work long into the night. She is my inspiration and I love
her.
v
vi
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
1 Introduction
1
1.1
Fusion energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Magnetic Fusion and Tokamaks . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Overview of National Spherical Torus Experiment . . . . . . . . . . . . . . 11
1.4
Physics Issues on NSTX . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5
Past Work on Ion Power Balance . . . . . . . . . . . . . . . . . . . . . . . 17
1.6
Scope of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Ion Power Balance
20
2.1
Analytic Derivation of the Power Balance Equation . . . . . . . . . . . . . 20
2.2
Measuring Power Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3
2.2.1
Input Power Pin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2
Stored Energy: Ẇ
2.2.3
Neoclassical Heat Transport: PN C . . . . . . . . . . . . . . . . . . 28
2.2.4
Ion-Electron Coupling: Pie . . . . . . . . . . . . . . . . . . . . . . 29
2.2.5
Diagnostic Requirements for Power Balance Analysis . . . . . . . 30
. . . . . . . . . . . . . . . . . . . . . . . . . . 26
Excess Ion Heating in NSTX . . . . . . . . . . . . . . . . . . . . . . . . . 30
vii
3 Measurements of Ion Power Balance
34
3.1
Experimental Determination of Power Balance . . . . . . . . . . . . . . . 34
3.2
Diagnostics on NSTX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.1
Thomson Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2
CHERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.3
MSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.4
Other Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3
Electron and Ion Temperature Changes . . . . . . . . . . . . . . . . . . . . 46
3.4
Errors in the Excess Heating Calculations . . . . . . . . . . . . . . . . . . 52
4 MHD Modes and Ion Power Balance
57
4.1
Theory of Ion Heating by Fast-particle Driven Modes . . . . . . . . . . . . 57
4.2
Fast Particle-driven MHD modes in NSTX . . . . . . . . . . . . . . . . . . 59
4.3
Analytic Evaluation of Compressional Alfvén Modes (CAEs) . . . . . . . . 63
4.4
Modeling CAEs and Calculating Ion Heating . . . . . . . . . . . . . . . . 71
4.4.1
Determining the Amplitudes of the MHD Modes . . . . . . . . . . 73
4.4.2
Implementing CAEs in GYROXY . . . . . . . . . . . . . . . . . . 86
4.5
Mode Heating Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6
Applying a Maxwellian Distribution to GYROXY Heating . . . . . . . . . 94
4.7
Comparing Measured and Calculated Heating . . . . . . . . . . . . . . . . 96
4.8
Excess heating in Future Magnetic Confinement Devices . . . . . . . . . . 98
4.8.1
Excess Heating and Neutron Production . . . . . . . . . . . . . . . 101
5 Edge Neutral Density Diagnostic
104
5.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2
Edge Neutral Density Diagnostic Design . . . . . . . . . . . . . . . . . . . 107
5.3
Edge Neutral Density Diagnostic Analysis . . . . . . . . . . . . . . . . . . 111
5.4
Results of the Edge Neutral Density Diagnostic . . . . . . . . . . . . . . . 113
viii
5.5
Errors in the Edge Neutral Density Diagnostic Analysis . . . . . . . . . . . 115
6 Neutral Particle Analyzer and Ion Power Balance
120
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2
Calculating NPA Viewing Pitch . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2.1
Converting NPA to Cylindrical Coordinates . . . . . . . . . . . . . 125
6.3
Summary of Coordinate System Transformations . . . . . . . . . . . . . . 127
6.4
NPA Measurements on NSTX . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5
Conclusions from NPA Measurements . . . . . . . . . . . . . . . . . . . . 131
6.6
Overcoming NPA Limitations . . . . . . . . . . . . . . . . . . . . . . . . 134
7 Summary and Future Work
135
7.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A Magnetic Ripple in NSTX
139
A.1 Overview of Magnetic Ripple . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.2 Ripple Loss in NSTX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.3 Implementing Ripple in ORBIT . . . . . . . . . . . . . . . . . . . . . . . 147
A.3.1 Analytic Approximation to |δB| . . . . . . . . . . . . . . . . . . . 147
A.3.2 Fortran PsuedoCode . . . . . . . . . . . . . . . . . . . . . . . . . 148
A.4 Results of Guiding Center Calculations . . . . . . . . . . . . . . . . . . . 148
A.5 Guiding Center vs. Full Lorentz . . . . . . . . . . . . . . . . . . . . . . . 152
A.6 Implementing Ripple in GYROXY . . . . . . . . . . . . . . . . . . . . . . 155
A.7 Results of the Ripple Calculation in GYROXY . . . . . . . . . . . . . . . 156
A.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B Edge Neutral Density Diagnostic Design and Calibration
159
B.1 ENDD Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
ix
B.2 Calibrating the Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
B.3 Data Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
B.3.1
Abel Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
B.3.2
Onion Layer Inversion . . . . . . . . . . . . . . . . . . . . . . . . 166
x
Chapter 1
Introduction
1.1 Fusion energy
The goal of fusion energy research is to provide a clean, limitless source of energy through
fusion reaction. Nuclei are heated or compressed and held together for long enough for
nuclei to spontaneously fuse. There are several potential reactions which can yield fusion
energy. Potential fusion reactions include
D + T → α(3.5M eV ) + n(14.1M eV )
(1.1)
D + D → T (1.01M eV ) + p(3.03M eV )
(1.2)
D + D → 3 He(0.82M eV ) + n(2.45M eV )
(1.3)
T + T → α + 2n + 11.3M eV
(1.4)
D +3 He → 4 He(3.6M eV ) + p(14.7M eV )
(1.5)
p +11 B → 3α + 8.6M eV
(1.6)
The advantage of reactions 1.5 and 1.6 is that they are “aneutronic”; that is, they are
reactions that occur without the generation of neutrons, which lead to radioactive waste.
1
Figure 1.1: Reaction rates of various fusion fusion reactions, averaged over a Maxwellian
distribution. With the highest reaction rate, D-T provides the most promising fuel for fusion
energy.
The disadvantage is that the reaction rate for these reactions is lower than for deuteriumtritium reactions and would require a higher temperature or higher density to achieve net
energy production. (See figure 1.1) As a result, the focus of the worldwide fusion program
is the deuterium-tritium (D-T) reaction in equation 1.1.
Since the D-T reaction has the highest cross section for most temperatures (see figure
1.1), that is the reaction that is the subject of most fusion research. One of the main goals
of plasma physics research is to obtain self-heated reactions, where the products from the
reactions provide the energy necessary to sustain them. The input power then goes to zero.
This condition of self-sustaining reactions is known as “ignition”. The ratio of the fusion
2
power to the input power is known as Q. When ignition is achieved, Q approaches infinity.
Q≡
Pf
→∞
Pin
(1.7)
where Pf is the power obtained from fusion reactions and Pin is the external power required
to maintain the burn. This means that the power from the fusion reaction that is being used
to heat the plasma is greater than the power being lost from the plasma by various methods.
Assuming that the plasma heating is coming from the fusion alpha particles, this means
Pf (α)
∼1
Pl
(1.8)
where Pl is the power that leaves the plasma through loss mechanisms such as diffusion.
The lost power is proportional to the density and temperature. For an isothermal plasma,
Pl =
R
3nkB T dV
τE
(1.9)
where n is the density, kb is Boltzmann’s constant and T is the temperature (assuming
Ti = Te ) and τE is the energy confinement time. This makes equation 1.8 become
Pf (α) =
R
3nkB T dV
τE
(1.10)
during ignition. The fusion power can be calculated as
Pf =
Z
n1 n2 hσvif (Ef )dV
(1.11)
where Ef is the energy released per fusion reaction, n1 and n2 are the ion densities of the
fusion reactants and hσvi is the rate coefficient of the fusion reaction. If the densities of the
3
reacting species are equal, this can be approximated as
Pf =
n2
hσvif (Ef ).
4
(1.12)
Equating this with equation 1.10 gives rise to the conditions required in magnetic confinement plasma known as Lawson’s criterion [1].
According to Lawson, for D-T reactions to be sustainable,
ne T τ E ≥
12kB T 2
Ef hσvi
(1.13)
where ne is the electron density, T is the plasma temperature, τE is the energy confinement
time of the plasma, kB is Boltzmann’s constant, Ef is the energy of the fusion products, σ
is the fusion cross section for a desired reaction, v is the relative velocity, and hi represents
an average over a Maxwellian distribution.
For the D-T reaction in a pure isothermal, isobaric plasma with T in the range 10-40
keV, the Lawson Criterion simplifies to
ne T τE ≥ 6 × 1021 keV s/m3 .
(1.14)
If the condition that Ti = Te is relaxed, then the lost power is proportional to the total
R
pressure Pl = 32 nkB (Ti + Te )/τE dV , where hi indicates the volume average of the
quantity and equation 1.14 becomes [17]
hn2i Ti2 iτE
& 3 × 1021 keV s/m3 .
hn(Te + Ti )i
(1.15)
Equation 1.15 is interesting in that it shows the advantage of a direct ion heating mechanism. A large electron temperature makes equation 1.15 more difficult to satisfy, due to
an increased power loss at fixed fusion power. If the ion temperature exceeds the electron
4
Figure 1.2: The tokamaks bend the magnetic field into a toroidal shape. The magnetic
geometry alters the confinement compared to a straight solenoid by causing particle drifts
due to non-uniform magnetic fields.
temperature, then the power loss is less and equation 1.15 is easier to satisfy.
1.2 Magnetic Fusion and Tokamaks
Two main approaches are being taken toward achieving ignition. One method, known as
inertial confinement, attempts to increase density by compression the fuel using lasers, xrays, or heavy ion beams. Magnetic confinement uses magnetic fields to confine the fusion
fuel as a plasma.
One method of confining the plasma with magnetic fields is to bend the field into a
toroidal shape known as a tokamak. (see figure 1.2)
The confined plasma is heated through various methods. This corresponds to increasing
T and τE in equation 1.14. One common method applied on various tokamaks is the use
5
of high energy neutral beams. Unlike ohmic heating using a large transformer, this method
has the potential to be steady state, and also has the advantage of simultaneously fueling
the plasma if deuterium or deuterium-tritium atoms are used in the beams.
The curved magnetic field alters the confinement for plasma particles.Particle confinement in magnetic fields is a diffusion-like process. For a diffusion-like process, the number
density of particles satisfies a diffusion equation of the form [2]
∂n
∂ ∂n =
D
.
∂t
∂x
∂x
(1.16)
D is a diffusion coefficient given by
D=
(∆x)2
.
2∆t
(1.17)
where ∆x is the average step-size (variation in position due to collisions) and ∆t is the
time between steps. In a straight, uniform magnetic field (such as is found in a solenoid),
the step-size is given by the gyro-radius. A particle’s gyro-radius is given by
ρ≡
mv⊥ c
v⊥
=
ωc
ZeB
(1.18)
where v⊥ is the perpendicular velocity, ωc is the cyclotron frequency, m is the mass of the
particle, e is the charge of a proton, B is the magnetic field, c is the speed of light, and Z is
the charge state of the particle. The time step is normally given by the collisional processes
in the plasma:
∆t =
1
,
νie
(1.19)
where νie is the ion-electron collision frequency and is a function of temperature, density
and mass. (see reference [3]). Diffusion in a uniform magnetic field is known as “classical”
6
diffusion. The classical diffusion coefficient for ions is
1
Dclass = ρ2i νie .
2
(1.20)
Bending the magnetic field into a torus, however, alters the step size by causing particle
drifts. Bending the magnetic field creates a magnetic field gradient, with the magnetic field
strongest near the machine’s center (not the plasma core), and decreasing proportionally to
major radius. The gradient causes particles to drift, with the particle drift is given by [4]
vD =
µB + vk 2
(n̂ × ∇B)
ΩB
where µ(≡ 21 mv⊥ 2 /B) is the magnetic moment, Ω (≡
ZeB
)
mc
(1.21)
is the gyro-frequency, and n̂
is the direction of the vacuum magnetic field. The direction of the drift due to the gradient
will always be in the same direction along the direction of the toroidal axis, either up or
down depending on the direction of the vacuum magnetic field and the sign of the particle’s
charge Ze.
The gradient causes the particle to drift off the magnetic field it rotates around. As the
particle travels poloidally, it travels above and below the magnetic surface it was orbiting.
If the perpendicular energy is sufficiently high, the particle can reflect back from the higher
magnetic field on the inboard side of the vessel. The result is that if the collision frequency
is low, some particles execute “banana” orbits. (see figure 1.3) The width of the banana
excursion is
ρq
1/2
(1.22)
rBT
,
R0 BP
(1.23)
r
,
R0
(1.24)
Λ'
where q is given by
q=
is the inverse aspect ratio
=
7
Figure 1.3: A poloidal cross section showing “banana” orbits executed by some particles
(solid red curve). The nominal flux surface of the particles is the dotted red line. The
banana orbit increases the step-size of the diffusion process.
BT and Bp are the toroidal and poloidal magnetic fields, r is the minor radius and R 0 is the
major radius(see figure 1.2).
The banana orbit increases the step size for diffusion. The step size is equal to the
banana width. Thus, in a torus, the diffusion coefficient for a plasma with low collisionality
becomes
Dneo =
νie ρ2i q 2
.
23/2
(1.25)
The extra factor of −1/2 is due to the fact that not all of the ions are trapped and execute
banana orbits, reducing the diffusion coefficient.
8
Figure 1.4: The collision frequency ν can strongly affect the diffusion coefficient. At
low collisionality, the diffusion is dominated by banana effects. At high collisionality, the
particles collide before the banana orbit is made, and the step size is determined by the
gyro-radius.
The increase in diffusion due to neoclassical effects is
q2
Dneo
= 3/2 .
Dclass
(1.26)
If the collision frequency is very high, the particles do not have time to execute banana orbits. This means that equation 1.26 no longer holds. Instead, the step size for
very high frequencies reverts to the classical calculation of the gyro-radius. This region is
known as the Pfirsch-Schluter region of transport [5]. At intermediate frequencies, there is
a transition between these collisionality regimes, creating a plateau region for the diffusion
coefficient. (see figure 1.4)
For a complete review of neoclassical transport, see reference [4].
9
Tokamak configurations also have an effect on the stability of the plasma. For example, as the density increases, it can cause the formation of magnetohydrodynamic (MHD)
modes, which increase the transport in the plasma preventing further increases in density.
The density limit is described by the Greenwald limit [6]:
n̄(1014 cm−3 ) <
I(MA)
πa2 (cm)
(1.27)
where a is the minor radius in cm, and I is the plasma current in MA.
MHD theory can be used to calculate the limit on the ratio β of plasma pressure to the
magnetic field:
β≡
2µ0 hpi
B2
(1.28)
where the pressure is proportional to the temperature and density
P = nkT.
(1.29)
In a tokamak, the limit on the volume average β can be shown to be
R
2µ0 P dV
I
= hβi ≤ c
2
B V
aBT
(1.30)
where c is a constant based on geometry (≈ 3 for conventional tokamaks) with units
mT
.
MA
This β limit is known as the Troyon limit. [7] Above this limit, pressure driven instabilities
arise in the plasma which prevent further increases in density and temperature.
The cross-sections in equation 1.12 are approximately proportional to T 2 for temperatures in the range of 10-40 keV. Equation 1.12 can then be written as
Pf ∼ n 2 T 2
10
(1.31)
In terms of β, the fusion power can be written as
Pf ∼ β 2 B 4 .
(1.32)
Equation 1.32 reveals that for a given magnetic field strength, fusion power is increased as
the square of β. Since the maximum β is determined by the geometric properties of the
confinement device, as well as magnetic field and plasma current dependence, it is desirable
to design a machine that allows for a maximum β, and hence, maximum fusion power for
a given magnetic field.
1.3 Overview of National Spherical Torus Experiment
The National Spherical Torus Experiment (NSTX) is a magnetic magnetic confinement device with a toroidal shape, similar to a tokamak. [8] Unlike conventional tokamaks, a spherical torus uses a low aspect ratio (≡ a/R in figure 1.2) to achieve a higher β[≡ 2µ 0 hpi/B 2 ].
[9] The ST may also contribute to a reactor development path by providing a volume neutron source for component testing. [10, 11] Reactor configurations for ST’s have also been
proposed. [12–14] The high β makes them attractive options. The ST configuration allows
the maximum field line length in the region of good field curvature. (see figure 1.5)
NSTX is designed to create a plasma with the following parameters: [8]
• Low aspect ratio: R/a ≥ 1.26
• Plasma current (Ip ) ≥ 1 MA
• Vacuum toroidal magnetic field (BT ) ≤ 5.5 kG
• Neutral beam heating (PN B )≤ 7 MW
• High harmonic fast wave heating (HHF W ) ≤ 6 MW
11
Figure 1.5: The ST configuration maximizes the time that the plasma spends in the region
of good curvature. The left hand figure is a conventional aspect ratio tokamak. The ST, on
the right, allows the plasma to spend more time in the good curvature region. Reproduced
from [15]
• Close fitting copper shell to stabilize pressure driven MHD instabilities for maximum
β
• Pulse length up to 1.5 s
• Bp /Bt ≈ 1
A diagram of the NSTX device is shown in figure 1.6
NSTX also has the ability to form a variety of plasma shapes. Unlike early tokamaks
which had a circular-cross section plasma, NSTX has the ability to create highly shaped
plasmas, studying the plasma performance at both high triangularity (δ ≡
elongation (κ ≡
W
)
H
d
)
a
and high
(see figure 1.7). Figure 1.8 shows NSTX highly shaped discharges.
Plasma performance is being evaluated as a function of these parameters and should
provide insight into the optimal plasma shape for future machines.
12
Figure 1.6: A cross-sectional diagram of NSTX. Reproduced from [15].
13
Figure 1.7: Definition of elongation and triangularity in a plasma. Reproduced from [16].
14
Figure 1.8: NSTX has the ability to create highly shaped plasmas. The discharges can be
either high elongation or high triangularity or both. Panel (a) shows a highly elongated
plasma (κ = 3.0). Panel (b) shows a highly triangular plasma (δ = 0.65)
15
1.4 Physics Issues on NSTX
The unique geometry and properties of ST’s allows NSTX to explore plasma regimes that
are not available in conventional tokamaks. [9] Some new parameters include:
• High βT (≤ 40%)
• Small plasma size relative to gyro-radius (a/ρi ∼ 30-50)
• High qedge (∼ 10) (ratio of toroidal to poloidal turns of a magnetic field line)
• High elongation (κ ≤ 3.0)
• High triangularity (δ ≤ 0.7)
• Large super-Alfvénic beam ion population (vf ast /vA ∼ 4-5)
These new regimes open the possibility of new physics issues. Ion and electron confinement are important issues that are being explored. Several diagnostics have been implemented to look at transport and confinement issues, with the majority of the diagnostics
focusing on the electrons.
Since the cross-sections of the reactions in figure 1.1 are strong functions of ion temperature, it is important to understand ion power balance. As shown in equation 1.15,
if heating mechanisms are available to directly heat the ions rather than the electrons, it
would be useful to incorporate this mechanism in future machines. Thus it is important to
understand the input and loss terms in ion power balance in order to achieve and sustain
ignition.
It is possible to use the diagnostic tools available to explore ion confinement issues and
power balance. Diagnostics are available on NSTX which measure the electron and ion
temperatures, electron density, magnetic fluctuations, and various other aspects related to
ion confinement.
16
1.5 Past Work on Ion Power Balance
Past analyses of ion power balance have shown interesting effects on ion heating and confinement [17]. Previous experimental results were obtained in which the total ion loss
terms exceeded the power to the ions from classical calculations of the neutral beam slowing down [18–20]. Since this heating was above the heating from the neutral beams, this
implies an excess heating source. The excess heating was seen to be large, and in some
cases, on the order of the total neutral beam heating power [17]. It was observed that this
excess heating occurred during the period of maximum neutral beam heating. Each neutral beam on NSTX can provide up to 2 MW of neutral beam power to the plasma. The
maximum heating occurred when all three neutral beams were used simultaneously.
It was also observed that during the period of maximum heating, high frequency MHD
modes were observed in NSTX. Although these modes are ubiquitously present in the
plasma, the amplitude of the modes increases with increasing beam power. These modes
were identified as compressional Alfvén eigenmodes. [21] The modes have a frequency
less than the ion cyclotron frequency (ωci > ω). More about these modes will be discussed
in section (4.2).
To test the theory that the heating could be caused by the MHD modes, high frequency
perturbations were put into a simplified 1-D slab model and the effects on the thermal
plasma were observed. In the slab model, heating occurred that could explain the heating
observed in the plasma. [17]
1.6 Scope of this Thesis
This purpose of this thesis is to demonstrate that excess heating can occur on NSTX and to
test the hypothesis that the high frequency MHD modes can be responsible for the heating
observed. Ion power balance was explored more completely than in the previous studies.
Enhancements to several diagnostics, as well as the implementation of new diagnostics
17
have made it possible to more accurately analyze the power balance terms. The diagnostics include the Thomson scattering diagnostic, which measure the electron temperature
and density and the Charge-Exchange recombination spectroscopy diagnostic, which measures ion temperature and density. The Thomson scattering diagnostic was expanded in
2007 from 20 radial channels to 30 radial channels. The CHERS diagnostic was expanded
in 2008 to include measurements of poloidal rotation. Recently implemented diagnostics include a high-k scattering diagnostic, designed to measure electron turbulence in the
plasma. Combined with other diagnostics such as Mirnov coils, the Neutral Particle Analyzer, a more complete picture can now be constructed of the ion power balance. To test
the theory from reference [17], it was necessary to alter the fast ion distribution, which was
accomplished by modulating the neutral beams during the discharge. The goal was to alter
the high frequency modes that were theorized to be responsible for the extra heating.
After modulating the modes, it was necessary to demonstrate that the modes could, in
fact, be responsible for the extra heating. The previous work demonstrated heating in a slab
model. This work extends the calculations into a toroidal geometry using characteristics
of the modes observed in NSTX. The mode amplitudes are estimated using the high-k
diagnostic on NSTX.
As part of the ion power balance analysis, a new diagnostic was developed to measure
the edge neutral density profile. The Edge Neutral Density Diagnostic (ENDD) provides
neutral density information about the outboard midplane of NSTX discharges. This information is necessary for understanding results of other diagnostics and for accurately
simulating the plasma in transport codes.
The ENDD uses a 2-D camera to obtain an image of the Dβ emission of the plasma. A
collisional-radiative model is then used to calculate the neutral density. This is important
in interpreting results from the NPA as well as calculating power lost through collisions
external to the plasma.
One other power balance term was investigated. Ripple loss is the result of a finite
18
number of toroidal field coils in a confinement device. Ripple loss calculations are typically performed in the guiding center approximation, which neglects gyro-radius effects.
However, in NSTX, beam ions have a gyro-radius that is a significant fraction of the minor
radius. At the outset of this study, it was unclear what effects a full gyro-orbit calculation
would have on ripple loss of beam particles. Full gyro-orbit calculations were performed
and compared with ripple loss from guiding center calculations.
Chapter 7 contains a summary of results as well as conclusions and suggestions for
future work.
19
Chapter 2
Ion Power Balance
2.1 Analytic Derivation of the Power Balance Equation
The understanding of power balance is an important aspect of magnetic confinement.The
power balance equation is obtained by taking the second moment of the Boltzmann equation. Before doing this, however, it is useful to discover a relationship between the distribution function f and a test function g. The distribution function f represents the number
density of particles in a volume element of phase space at a given time. The test function g
is a generalized test function that we will choose later to explore different properties of the
distribution.
Assuming that f is a distribution function f (r, v, t), and g is a quantity of interest
g(r, v, t), then in general we can calculate the average value of the test function g by integrating over velocity space. [22]
n(r, t)g(r, t) =
Z
dv g(r, v, t)f (r, v, t).
(2.1)
where g(r, t) is the average of g, given by
g(r, t) ≡
R
dv g(r, v, t)f (r, v, t)
R
.
dv f (r, v, t)
20
(2.2)
This can be interpreted as the average value of quantity g at r, t times the density of particles
that coordinate in phase space.
The full distribution function, ftot (r, v, t) is separable by species. It can be written as
ftot (r, v, t) = felec (r, v, t) + fion (r, v, t)
(2.3)
The Boltzmann equation describes the evolution of the distribution function. It is given
by
∂f ∂f F
∂f
+ v · ∇r f +
· ∇v f =
+
.
∂t
m
∂t coll
∂t source
(2.4)
where v is the bulk velocity, F is the sum of all forces acting in the particles, and m is
the mass density.
Since the distribution function consists of two additive parts (the electrons and the ions),
equation 2.4 is also separable by species. The ion component is identical to the full Boltzmann equation except that it uses only the ion component of the distribution function, f ion .
Continuing to follow the analysis by Rose and Clark [22], we take moments of the
Boltzmann equation. The moment is found by integrating the ion distribution function over
all velocities. Using our test function g, the moment of the ion Boltzmann equation is
h ∂f
i
F
i
dv g(r, v, t)
+ v · ∇ r f i + · ∇v f i =
∂t
m
Z
Z
∂f ∂f + dv g(r, v, t)
.
dv g(r, v, t)
∂t coll
∂t source
Z
This can be expanded to
21
(2.5)
∂
∂g(r, t)
[ni (r, t)g(r, t)] − ni (r, t)
+ ∇r · ni (r, t)vg(r, t)
∂t
∂t
ni (r, t)
−ni (r, t)∇r · vg(r, t) −
F(r, t) · ∇v g(r, t)
mi
Z
h ∂f (r, v, t) i
h ∂f (r, v, t) i
i
i
+
.
= dv g(r, v, t)
∂t
coll
∂t
source
(2.6)
We use the test function g to obtain the moments. Setting g = 1 yields the familiar
continuity equation:
∂ni (r, t)
+∇r ·[ni (r, t)v(r, t)] =
∂t
Z
dv
h ∂f (r, v, t) i
i
∂t
coll
+
Z
dv
h ∂f (r, v, t) i
i
∂t
source
. (2.7)
Letting g = mv gives the momentum transport equation, which we write in the standard
form:
mi ni
∂v
∂t
+ (v · ∇)v = ni q(E + u × B) − ∇ · Pi − mi Sv
(2.8)
where u is the bulk plasma velocity, Pi is the pressure tensor, and S is the source term
representing the rate of particle creation.
To study power balance, we need the 2nd moment and we derive the energy (power)
equation by choosing
g=
m 2
v
2
(2.9)
which gives
n
∂ ni mi 2 i
mi vv 2 = j · E +
v + ∇r ·
∂t
2
2
Z
where ni is the ion number density, and mi is the ion mass.
dv
mi v 2 ∂f 2
∂t coll
(2.10)
In this equation, the first term is the rate of change of energy density with time and
the second term is the leakage of energy from a closed surface bounding the element of
interest due to bulk flow of the plasma. Since we are examining a steady state toroidal
plasma with axisymmetry and the bulk flow is toroidal, the divergence of any bulk flow
is zero. The first term on the right hand side is the work done on particles by the electric
22
field. MHD fluctuations can lead to local electric fields which could heat the plasma. This
will be discussed below. The second term on the right hand side is the energy per unit time
added to the volume element due to collisions. Although ion-ion collisions do not transport
heat, the collisional term has three components of interest for ions: the heating term from
the neutral beam, the diffusive heat flux term and the electron-ion coupling term. Thus, we
can write equation 2.10 as
∂ nm 2 v =
∂t 2
Z
mv 2 ∂f +
dv
2
∂t f lux
Z
mv 2 ∂f dv
+
2
∂t ie
Z
mv 2 ∂f dv
(2.11)
2
∂t N B
The collisional term comprises many aspects of power balance. For example, this is the
term that contains the neutral beam heating power. The beam particles charge exchange or
are ionized by electron impact, then collide with electrons and ions, collisionally transferring heat to them.
Another component of the collisional term is the electron-ion heat transfer term. If the
ions are hotter than the electrons, the collisions with the electrons will cool down the ions
and heat up the electrons and power will flow from from the ions to the electrons. This
is frequently the case during neutral beam heating on NSTX. Instead, if the electrons are
hotter, power flows from the electrons to the ions, and this term in the ion power balance is
a heating term. The electron-ion heat transfer term will be explained below.
The collision term also includes neoclassical heat losses, which can be modeled as
PN C = ∇ · nχi ∇Ti ,
(2.12)
where χi is the ion heat transport diffusivity. This diffusive heat flow comes from scattering
of the thermal ions in velocity space by collisions with electrons, and is the result of a
gradient in temperature.
The j · E is modeled using numerical equilibrium solvers, where E is the electric field
from Faraday’s law using the time-evolving axisymmetric equilibrium field. It is impor23
tant to point out that small-scale fluctuation, such as those from non-axisymmetric MHD
instabilities are not included in the equilibrium solvers. These reconstruction codes use
the Mirnov coils located around NSTX, which are used to determine the magnetic fields.
The Mirnov coils are integrated to obtain the magnetic field. The equilibrium solvers use
data obtained from the coils to reconstruct the plasma by solving the Grad-Shafranov equation [23, 24].
Some equilibrium solvers also implements constraints that the electron temperature be
constant on a flux surface, and that the magnetic field (and hence the plasma current) match
measurements from the Motional Stark Effect (MSE) diagnostic, which measures magnetic
field pitch in the plasma, giving a limit on the plasma current. Rotational effects are also
included in LRDFIT calculations.
2.2 Measuring Power Balance
While equation 2.11 contains all of the information of the ion power balance, it is useful to
recast the terms into a more intuitive form.
If we imagine a volume of space inside the plasma, we can study the ion power balance of the thermal ions by examining the power flow to and from this volume. A simple
example is illustrated by the first term in equation 2.11. This term represents the rate of
change of energy density with time. However, in terms of what is measurable, it is much
more instructive to think of the change in stored energy (Ẇ ), which is calculated directly
by measuring the temperature and density of the plasma.
By focusing solely on the local ion power balance, we can write the various terms in
equation 2.11 in terms of measurable quantities. In rewriting the terms, it is useful to
classify the different aspects in three different groups: input power, lost power, and change
in stored energy.
24
In the simplest form, we can then write the ion power balance equation as:
Pin = Pout + Ẇ
(2.13)
where Pin is the input power to ions in a volume element of plasma from neutral beams,
Pout are power loss mechanisms from a small volume, and Ẇ is heating that occurs inside
the volume of interest, as a result of a change in temperature or density in the volume. In
order to understand the Pout term, it is necessary to consider all of the loss mechanisms.
Pout = Pie + PN C + Pother
(2.14)
where Pother contains all the other loss mechanisms, including radiative losses and rotational effects on the power balance through drag and other effects, as well as other loss
mechanisms such as direct particle loss or wave-particle coupling. For a more complete
analysis of the effects of rotation on power balance, see reference [25], although it is important to remember that this analysis was done for a large aspect ratio torus and is not
entirely applicable to the low aspect ratio found in NSTX.
Putting these terms into equation 2.13 gives
Pin = Ẇ + Pie + PN C + Pother .
(2.15)
Note on the notation: Various aspects of power balance have been worked out at different times by different people. The result is a bit of confusion regarding the notation.
Traditionally, in neoclassical theory literature, QN C refers to the surface averaged heat
flux, that is, energy flowing through a surface. The reason behind this notation is clear: it is
conceptually easier to think of power flowing through a surface due to neoclassical effects.
The other terms, however, are easier to think of as changes in power density due to various
other effects.
25
In order to avoid confusion, I refer to all power densities as P , while heat fluxes are
referred to as Q. This makes the neoclassical power density
PN C = ∇ · Q N C .
(2.16)
2.2.1 Input Power Pin
For this experiment, the primary source of heating was the NSTX neutral beam system,
which injects neutral deuterium beams into the plasma [8, 26]. The neutral beams heat the
plasma by collisionally slowing down on the plasma. The input power from the neutral
beams is modeled as classical slowing down theory of a neutral beam. Numerical codes
model the neutral beams by Monte Carlo deposition of particles, and then evolve the distribution in time [27, 28].
Figure 2.1 shows the total ion heating calculated for discharge 128820. The total ion
heating is nearly equal to the heating due to neutral beams, with compressional heating
providing only a small amount of heating. This means that we can approximate the ion
heating as Pnb .
2.2.2 Stored Energy: Ẇ
In order to understand the stored energy term, it is necessary to look at the energy of a
single particle. The translational energy of a particle is given by the equation
3
1
E = mv 2 = kT
2
2
(2.17)
where k is Boltzmann’s constant and T is the temperature. The total energy contained in a
finite volume of plasma is found by summing over all the particles.
N
X
mi v 2
i
2
i=0
26
(2.18)
Figure 2.1: Ion heating as calculated by the TRANSP transport code integrated to ψ = 0.95.
The neutral beam heating makes up the majority of the total ion heating.
27
and the energy density is simply given by the particle density times the average energy per
particle:
3
nmv 2
W = nkT =
2
2
(2.19)
To change this equation from energy to power, take the time derivative of equation 2.19,
3∂
∂ nmv 2
∂W
= Ẇ =
nkT =
∂t
2 ∂t
∂t 2
(2.20)
which gives the first term of equation 2.11. To calculate the total power contained inside
the volume, one would simply integrate this term over the volume.
2.2.3 Neoclassical Heat Transport: PN C
Energy can be carried in a manner similar to the diffusion of particles. In this case, the
neoclassical heat flux (QN C ) can be calculated. Using a pitch angle scattering operator, it
is possible to calculate the neoclassical heat conductivity, χi [4]. Although the derivation
is too complicated to reproduce here, the neoclassical conductivity coefficient is given by
χi = 0.661/2 ρ2iθ νi
(2.21)
where (≡ r/R) is the inverse aspect ratio, ρiθ is the Larmor radius in the poloidal field
Bθ , given by
2
ρ2iθ = vthi
/Ω2iθ = 2mi c2 kTi /e2 Bθ2
(2.22)
and νi is the collision frequency, given by
4
ni Zi4 e4 lnΛ
νi = π 1/2 1/2
3/2
3
m i Ti
where Z is the charge state (=1 for deuterium), and Λ is the Coulomb logarithm.
28
(2.23)
To convert the heat flux term into a power balance term, we take the divergence.
PN C = ∇ · Q N C
(2.24)
For a more complete derivation, including a description of the collision operator, see
reference [4].
2.2.4 Ion-Electron Coupling: Pie
The ion-electron coupling term arises as a result of the difference in temperature between
the ion and electron thermal populations. The direction of heat flow depends on which
species has the higher temperature. As would be expected, if the electrons have a higher
temperature, then heat flows from the electrons to the ions, with the reverse being true for
a larger ion temperature. Assuming only one ion species, energy is conserved with respect
to this operation. Thus,
Pie = −Pei .
(2.25)
Coupling between the ion and electron populations is a relaxation process. That is, if
you have a population of ions at one temperature and a population of electrons at a different
temperature, then eventually the system would come to a steady state with a temperature
between the two populations. In NSTX, the two temperatures never equilibrate, since heating sources are acting on the plasma, such as neutral beam heating and heat flux.
The coupling term can be calculated by solving the change in energy of an ion by
collisions with electrons and then summing over full distributions of particles. The reader
is referred to section 8.3 of reference [22], where the ion-electron coupling is derived in
depth. The ion-electron coupling term is shown to be
3
Pie ≡ nνie (Ti − Te ).
2
29
(2.26)
Pnb
Pie
PN C
Ẇ
Reconstruction
MPTS
x
x
x
x
x
CHERS
x
x
x
x
x
MSE Magnetics ENDD
x
x
x
x
Table 2.1: The main power balance terms can be analyzed using measurements from only
a few diagnostics. To complete the analysis, the Grad-Shafranov reconstruction must also
be performed.
where νie is the ion-electron collision frequency (see reference [3])
2.2.5 Diagnostic Requirements for Power Balance Analysis
The advantage of writing the power balance equation in the form of equation 2.15 is that the
diagnostics on NSTX can be used to measure the terms in equation 2.15. The stored energy
term (W ) is measured by measuring the ion temperature (Ti ) using the CHERS diagnostic
and the electron density (ne ) using the Thomson scattering (MPTS) diagnostic. In the
power balance analysis, it is the time rate of change in stored energy ( Ẇ ) that matters.
The heat transfer from ions to electrons (Pie ) also uses the CHERS and MPTS diagnostics. The neoclassical heat loss term (PN C ) is calculated using the ion temperature gradient
and electron temperature and density, obtained from the Thomson Scattering and CHERS
diagnostics. Thus, by measuring only a few plasma parameters, it is possible to model the
global and local power balance.
2.3 Excess Ion Heating in NSTX
Equation 2.15 can also be applied separately to the electrons. Some terms are complementary with the ion power balance equation. For example, any energy lost by the ions to the
electrons, must then appear as energy gained by the electrons in the electron energy balance
30
Figure 2.2: The power balance equation can be solved locally in the plasma. At a given
volume, most of the terms of the power balance can be measured and excess heating can
be calculated.
equation.
If we can identify all of the terms giving energy to the ions and all of the places where
that energy goes, the two sides of equation 2.15 will match.
The ion power balance is a local quantity. If the magnetic field strength and direction
and the ion and electron temperatures, densities, and gradients are known at a given location
in the plasma, it is possible to solve equation 2.15 locally. (See figure 2.2) If we postulate
that there is ion heating which we will denote Pexcess beyond the term Pnb calculated from
the classical collisional slowing down of neutral beams on the ions, then the left side of
equation 2.15 becomes
Pin = Pnb + Pexcess
(2.27)
Pnb + Pexcess = Ẇ + Pie + PN C + Pother
(2.28)
making equation 2.15
31
The Pother term includes other power loss terms such as radiation and rotational effects, as
well as MHD induced heat losses and thermal diffusion from turbulence.
It is possible to put a lower bound on the excess heating term by neglecting terms that
are not measured and are small compared to other terms. These include P rot , Prad , charge
exchange losses and MHD induced losses, as well as any diffusion losses greater than
neoclassical. Another loss mechanism included in the Pother term is ripple loss. However,
an analysis of ripple loss on NSTX is given in Appendix A, which demonstrates that in the
conditions of these experiments, the ripple loss is a negligible effect. By neglecting these
terms, equation 2.28 becomes
Pnb + Pexcess > Ẇ + Pie + PN C .
(2.29)
Assuming that we know the heating from the neutral beams (Pnb ) from classical slowing
down calculations, we can then solve for the excess heating.
Pexcess > Ẇ + Pie + PN C − Pnb
(2.30)
It is important to remember that we are only considering the ion power balance. The term
Pnb , for example, represents only the slowing down of the beam ions that heats the thermal
ions. Power from the beams that heats the electrons would appear in the electron power
balance equation. Also remember that equation 2.30 is actually a lower bound on the excess
heating. Keeping the neglected loss terms from equation 2.28 would increase the excess
heating required to meet the power balance equation.
As stated above, improvements in the diagnostic capabilities on NSTX have led to a
more complete study of the excess heating on NSTX. These diagnostics are explained in
more depth below. With these new and improved diagnostic capabilities, it is possible
the put more accurate bounds on the excess heating required to solve the power balance
equation. The magnetic diagnostics are used to perform a Grad-Shafranov equilibrium
32
reconstruction, and the temperature and density data are then used with the equilibrium in
the TRANSP transport code to analyze the power balance and calculate excess heating.
The TRANSP code is a self-consistent transport code [31, 32]. It reads in the magnetic
equilibrium reconstructions from LRDFIT [33] or EFIT [34], and reads several other input
parameters from the plasma. These inputs include neutral beam information, Rogowski coil
information for plasma current, and electron and ion temperatures. TRANSP then performs
self-consistent Monte Carlo calculations of the fast ion distribution function, including
guiding-center drift orbiting, and collisional and atomic physics. TRANSP power balance
calculations include several loss processes and the calculations are performed in a full
geometry. Using these distribution functions, TRANSP then calculates the electron and
ion power balance.
In order to do the analysis, TRANSP divides the plasma into a series of flux zones,
based on the equilibrium. Inside each flux zone, the quantities are assumed to be uniform
in the poloidal direction. When the power balance analysis is performed, the quantities are
integrated over the flux surface to calculate the total power in each term.
33
Chapter 3
Measurements of Ion Power Balance
3.1 Experimental Determination of Power Balance
Although equation 2.15 contains all of the information of the power balance in a volume of
plasma, it is difficult to predict exactly how a change in the input power into the plasma will
affect each of the terms on the right hand side. In order to fully understand all the terms
in the power balance equation, an experiment was conducted in which the input power
of the NSTX discharges was modulated using the neutral beam input power. Individual
neutral beam sources were turned on or off to increase or decrease the input power. The
modulations were designed to create a localized perturbation in the fast ion phase space
distribution. This perturbation allowed us to distinguish the large terms from the small
terms in equation 2.15.
When an additional beam source turns on, the plasma total stored energy begins to
increase, i.e. Ẇ increases. However, the other terms do not react so quickly. Until the ion
temperature increases, Pie and Pnc , which depend on temperatures and their differences, do
not change immediately. Thus, modulating the neutral beams allows for an investigation
into certain terms in the power balance.
It was previously observed that during certain discharges, there were periods of time
34
RM aj
Ip
N BP ower
Btor
1.01 m
900 kA
2-6 MW
4.5 kG
Table 3.1: Typical plasma parameters used in the ion power balance study.
when the inferred stored energy and loss terms (Ẇ , Pie , PN C ) exceeded the input power
from classical neutral beam heating calculations. [17] In such cases, an excess heating term
is required to satisfy the inequality in equation 2.29, i.e.
PN B ≯ Ẇ + Pie + PN C
(3.1)
A lower bound on the excess heating can be calculated as
Pexcess = Ẇ + Pie + PN C − PN B
(3.2)
If MHD modes or other effects increase the power loss, excess heating above the lower
bound calculated by equation 3.2 would be required to satisfy the power balance equation.
In the experiment designed to investigate whether an excess heating term is required,
neutral beams A and C were turned on early in the discharge. (see figure 3.1) These beams
were on for approximately 0.3 s prior to the modulation of the third source. This was to
allow the plasma to reach a thermal steady state. Neutral beam B was then turned on to
perturb the plasma. The neutral beam power and plasma current for discharge 128820 are
shown in figure 3.2. The modulations were timed to coincide with a period free of largeamplitude, low-frequency MHD modes (See figure 3.3). Figure 3.4 shows an expanded
view of the time around the turn-on of the third beam. The power balance analysis in
TRANSP was performed before and after the third source turned on.
Figure 3.5 shows a cross section of the plasma equilibrium immediately before the third
beam turns on. Table 3.1 lists the basic plasma parameters for the series of discharges.
35
Figure 3.1: Top-down view of NSTX showing neutral beam trajectories. Source A has
a tangency radius of 70 cm. Source B has a tangency radius of 60 cm. Source C has a
tangency radius of 50 cm. The beams can be independently switched on and off. The
power delivered by an individual source is controlled by its ion acceleration voltage which
is fixed for a given plasma shot but can be varied over a limited range from shot to shot.
36
Figure 3.2: Neutral beam (red) and current (blue) waveforms for discharge 128820. Each
beam adds 2 MW of power to the plasma.
37
Volts
0.06
200
150
0.04
0.40
0.40
\bdot_l1dmivvhn1_raw for shot 128820; #FFTs=4096
0.35
0.35
100
50
0
0.30
0.5
0.30
-0.5
freq (kHz)
0.45
0.45
sec
0.50
0.50
0.55
0.55
0.60
6
5
4
3
2
0.60
Power
0.08
0.02
0
Figure 3.3: The beam modulations occur during a period of time that is free from large
amplitude low frequency MHD. The lower panel shows the raw Mirnov signal (black) and
the neutral beam power (red). The upper panel shows the Fourier transform of the Mirnov
signal for signals below 200 kHz.
38
Figure 3.4: Blow up of the third beam turn-on time. The power balance analysis is compared before and after the third beam turns on (black lines).
39
Figure 3.5: Poloidal flux contours of LRDFIT09 reconstruction of shot 128820 at t = 397
ms. This shows the plasma equilibrium immediately prior to the turn-on of the third neutral
beam source.
40
Over a series of 15 discharges, the modulations were performed on two different time
scales. For some discharges, the beams were modulated for 30 ms, with a duty cycle of
50%. This time scale was chosen because it is on the order of the beam slowing down time,
but less than the total energy confinement time, which for these plasmas is 40-50 ms. The
goal of these fast modulations was to alter the fast ion population and any corresponding
heating of the thermal ion population, without allowing the thermal population to equilibrate. This would allow the diagnostics to observe changes in the power balance before the
energy could be lost from diffusion.
In other discharges, the beams were modulated for 60 ms, with a duty cycle of 50%.
This longer time frame allowed the thermal ion population to respond more fully. In each
case, the power balance was analyzed and the results were compared before and after the
third beam turn-on.
3.2 Diagnostics on NSTX
Because the power balance analysis relies so heavily on plasma diagnostic capabilities, a
discussion of the important NSTX plasma diagnostics is necessary.
3.2.1 Thomson Scattering
The primary diagnostic providing electron density and temperature information on NSTX
is the Multi-point Thomson Scattering (MPTS) diagnostic, which measures laser light scattered by the plasma. [35] The MPTS system provides time-resolved radial profiles of both
the electron density (ne (R, t)) and electron temperature (Te (R, t)). The MPTS diagnostic on NSTX provides density and temperature information at 30 radial locations. The
Thomson scattering diagnostic uses pulsed lasers to scatter light off electrons. The time
sampling time of the MPTS system is approximately 10 ns, with the beam pulses spaced
16.7 ms apart. See reference [36] for a more complete explanation of Thomson scattering.
41
Figure 3.6: View of the MPTS measurement locations mapped onto the poloidal flux contours of a typical plasma equilibrium on NSTX.
Figure 3.6 shows the locations of the MPTS channel measurements.
3.2.2 CHERS
The Charge Exchange Recombination Spectroscopy (CHERS) diagnostic is critical in evaluating the ion power balance in NSTX. (On other machines, the diagnostic is called CXRS
or CER.) Charge exchange relies on the presence of neutral particles in the plasma colliding with plasma ions and exchanging an electron. However, the ion or neutral atom created
is not in its ground state. Instead, the exchange leaves it in an excited state. As the electron
falls to the ground state, a photon is emitted.
Although the thermal equilibrium concentration of neutrals in the NSTX plasma would
be very small, the neutral beams provide a source of neutral atoms with which the plasma
ions can charge exchange. The NSTX CHERS diagnostic measures the emissions from
carbon ions. Carbon emission is used because of the difficulty in using hydrogen emission.
42
The carbon is an impurity, which comes from carbon being ejected from carbon tiles which
line the vessel. Hydrogen emission is dominated by emission in the edge of the plasma,
making it difficult to measure hydrogen emission from the plasma core. The transition used
for the NSTX CHERS diagnostic is the emission from C VI (wavelength 529.0 nm). (see
figure 3.7 for CHERS viewing locations)
As with Thomson scattering, the thermal Doppler broadening of the emission is used
to determine the temperature for the particle distribution. Calculations are then made in
TRANSP relating the temperature of the impurity ions (carbon) with the bulk plasma ion
temperature (such as deuterium). It is also possible to measure the carbon impurity density
from CHERS, using the absolute line intensity. To get an estimate of the deuterium density,
the CHERS diagnostic calculates
ni = n e − Z c nc .
(3.3)
This value, which involved the assumption that carbon is the dominant impurity species, is
used as an input to TRANSP.
3.2.3 MSE
It is important to model the plasma accurately in order to calculate the power balance.
Doing so requires an accurate measure of the internal magnetic field. In practice, this
is difficult. The vacuum toroidal field can be calculated because the current through the
toroidal field coils is applied by external power supplies, and can therefore be known with
accuracy ≈ 1%. Assuming the geometry of the coils is known, the Biot-Savart integration
can then be performed to determine the vacuum field.
However, to maintain a toroidal MHD equilibrium, both toroidal and poloidal currents
must flow in the plasma, modifying the applied magnetic fields. It is difficult to measure the
plasma current in the plasma itself to determine the profile of the poloidal field. Rogowski
43
Figure 3.7: View of CHERS measurement locations on NSTX, mapped onto the poloidal
flux contours of a typical plasma equilibrium.
44
Figure 3.8: Overhead view of CHERS and MPTS viewing locations. Reproduced from
[37].
45
coils could be used outside the plasma to determine the total plasma current, but they cannot
provide internal information about the distribution of the current.
A solution to this problem has been found in the Motional Stark Effect (MSE) diagnostic on NSTX. [30] The MSE diagnostic relies on the Stark effect [38] which causes the
splitting of spectral lines in the presence of electric fields, induced by the motion of ions
in the magnetic field. In the plasma, atoms are collisionally excited and then fluoresce.
Accurate measurement of the polarized emission can result in a measure of the local magnetic field pitch angle (ratio of the poloidal to toroidal magnetic fields). For a complete
description of the NSTX MSE diagnostic, see reference [30]
3.2.4 Other Diagnostics
Several other diagnostics are used in this study. Among those used for power balance
are Mirnov pickup coils, the Edge Neutral Density Diagnostic (ENDD), and the Neutral
Particle Analyzer (NPA). The ENDD and the NPA will be discussed in detail later.
Mirnov coils work by measuring the voltage induced by a changing magnetic field. By
integrating the voltage through the whole discharge, it is possible to obtain the magnetic
field. Variations in the Mirnov coil voltage correspond to changes in the magnetic field.
Including the fluxes and coil currents, it is possible to perform the magnetic reconstruction.
3.3 Electron and Ion Temperature Changes
The electron and ion temperatures were measured for each of the discharges. In each
case, after the third neutral beam source turned on, the ion temperature increased, although
the percentage increase changed from discharge to discharge. Figure 3.9 shows the ion
and electron temperature immediately before and after the third beam turned on at 400
ms in discharge 128821. The ion temperature increases significantly, with the central ion
temperature increasing approximately 60% within 20 ms after the beam turns on. This
46
Figure 3.9: Electron and ion temperatures in NSTX discharge 128821. When the third
beam turns on at 0.4 s, the electron temperature increases significantly for the next 20 ms.
The electron temperature, however, remains nearly unchanged over the same period.
interval is shorter than the slowing down time of the beam, which was around 30 ms for
these plasmas. The CHERS diagnostic is integrated over a period of 10 ms. This means
that the temperature at a given time is an average of the ion temperature during that 10
ms window. The electron temperature, however, did not change nearly as much as the ion
temperature, increasing approximately 10% over the same time period.
Electron and ion temperatures alone are not enough to understand ion power balance.
As an example, if the third beam turn-on caused a significant increase in electron transport without affecting the ion transport, then the electron temperature might remain nearly
unchanged while the ion temperature might increase, satisfying equation 2.15 without requiring excess heating. Thus, we use the TRANSP code to calculate the terms in the ion
47
power balance to determine if any excess heating is required to satisfy equation 2.29.
Terms in the ion power balance for NSTX discharge 128820 as calculated by TRANSP
are shown in figure 3.10. The loss terms are shown in figure 3.11. The TRANSP code
performs flux surface average power balance. The flux surface average is then integrated
over the volume of each flux region. The figure shows the cumulative integral, which is
the sum of each flux region integral. Thus, only the value at the edge of the cumulative
curve matters, since it represents all required excess heating, and is unaffected by internal
redistributions of power. Before the third beam turns on, the total cumulative input power
exceeds the calculated output. This difference can be attributed to heat loss greater than
neoclassical, as well as the additional loss terms that were neglected to arrive at equation
2.30.
When the third beam turns on, power balance is again calculated. This time, the total
output plus loss terms exceed the total input power from the classical neutral beam calculations. It is important to note that figure 3.10 is the volume integrated power balance. The
integral is cumulative over the volume from the inside out, so only the power difference at
the edge of the plasma matters.
For these analyses, power balance calculations outside of major radius 137 cm are neglected. Beyond this radius, the electron and ion temperatures fall below approximately
200 eV and the relative errors in the CHERS and MPTS diagnostics become high in the
edge of the plasma. This results in large relative errors, particularly in P ie , which is particularly susceptible to low electron temperature. Since the electron temperature is already
low at the plasma edge, the diagnostic errors make the calculation unreliable.
The excess heating profile also provides information about the excess heating mechanism. With a given profile, it is possible to look for sources of heating that could match it.
The excess heating profile for NSTX Discharge 128820 is plotted in figure 3.12. Since the
TRANSP code calculates the flux zone average of the power balance terms, the power balance terms must be volume integrated to calculate the total power in each term. The large
48
Figure 3.10: The cumulative volume integrated power balance. The plasma center is 100
cm, and the plasma edge is approximately 145 cm. The red line represents the total ion
heating as calculated by TRANSP. The blue line includes Ẇ , Pie , and Pnc . Before the 3rd
neutral beam source turns on 400 ms (top frame), the power balance equations shows loss
greater than heating. After the third beam turns on (bottom frame), the loss terms exceed
the beam heating, implying another heating mechanism.
49
Figure 3.11: Power balance loss terms for discharge 128820. Most of the increase from
the total comes from an increase of power flow from the ions to the electrons. These terms
are volume integrated out to 137 cm. The diamond symbols indicate the times at which the
electron temperature is measured.
50
Figure 3.12: The excess ion heating profile. The top frame shows the excess ion heating
profile. The bottom frame shows the profile multiplied by the volume of each flux region,
which takes into account the larger volume near the edge of the plasma.
volume in the edge zones means that the volume integrated power deposition is largest near
the edge.
As figure 3.12 shows, the excess heating is not uniform through the plasma. The heating
is strongly peaked at around 134 cm. Outside of this radius, the excess heating rapidly
decreases.
Using classical slowing down theory, TRANSP calculated that 31% of the power deposited from the neutral beams into the plasma is transferred directly to the ions through
beam-ion collisions. However, when the additional ion heating is taken into account, it
appears that 42% of the total heating from the beams goes to the ions. This extra power
must come either from power that would have been deposited on the electrons or power
that would have been lost to the wall.
51
Figure 3.13: Variation in ion heating from Monte Carlo simulation. The average over this
time is shown as the dashed line. The Monte Carlo variation is approximately 50 kW,
showing that the calculated excess heating is not a result of Monte Carlo numerical error.
3.4 Errors in the Excess Heating Calculations
TRANSP calculations show that excess heating is required to satisfy the power balance
equation. This heating correlates strongly both temporally and spatially with compressional
Alfvén eigenmodes observed in the plasma, which might be an indication that the CAE’s
are responsible for the heating. However, in order to have confidence in the calculations of
the ion power balance, it is necessary to examine the errors in the calculations. Just as with
the power balance equation, it is possible to analyze the input and loss terms separately.
The neutral beam is the main source of input power in these plasmas. This means
that the variation in neutral beam input power could lead to errors in the power balance
calculations.
Figure 3.13 shows the total ion heating as calculated by TRANSP. The total neutral
beam power in this figure is 6 MW. The variability in the Monte Carlo simulation is approximately 50 kW. The excess heating for most discharges in this experiment is in the
52
EFIT02
LRDFIT06
LRDFIT09
Magnetics MSE
x
x
x
x
x
E field correction
x
x
Te-iso Rotation
x
x
x
Table 3.2: Different magnetic reconstruction require different inputs. LRDFIT09 was predominantly used in this analysis. The magnetics are the Mirnov coils. The MSE diagnostic
measures magnetic field pitch angle. The E field corrections includes effect of of radial
electric fields. Rotational effects include frictional effects.
range of 600 kW - 1.1 MW. Thus, the variability in the classical ion heating ranges from
4-8% of the excess heating.
Systematic errors can also cause errors in the power balance analysis. If codes used in
this analysis are based on incorrect assumptions or if the inputs from diagnostics contain
systematic errors, TRANSP will fail to provide an accurate power balance analysis.
Another source of error is variation in the reconstructions. Several different options
are available for reconstructions. The reconstructions used in this analysis were LRDFIT09. Table 3.2 compares the LRDFIT09 with other reconstructions options. LRDFIT09
involves the most complete plasma analysis. The inclusion of rotation causes a shift in the
the reconstruction profiles. Centrifugal forces cause the plasma to push outward. Profiles
calculated during the reconstructions also shift outward. If the outward shift was not taken
into account, the reconstructed profiles such as the pressure profile would not match the
data measured by the Thomson scattering and CHERS diagnostics. Since TRANSP incorporates both the reconstruction and the individual diagnostic measurements, it is important
to ensure that the reconstruction profiles match the measured data.
Variations in excess heating due to the reconstructions are shown in figure 3.14. Figure
3.15 shows the differences in LRDFIT06 and LRDFIT09 reconstructions. LRDFIT06 is
shown in black, and several contours of the LRDFIT09 are shown in pink. The LRDFIT09
reconstruction has a larger central shift, and is further out on the edge. These shifts result
in differences in the excess heating calculations shown in figure 3.14
Errors in the loss terms such as neoclassical heat transport and losses to electrons are
53
Figure 3.14: Variations in excess heating from different plasma reconstructions integrated
to 137 cm. Variations within each band are from variations in the ion and electron temperature diagnostics.
54
Figure 3.15: LRDFIT06 and LRDFIT09 reconstructions of discharge 128820. The LRDFIT09 reconstruction includes the effects of rotation on the plasma profile. The differences
in the reconstructions result in differences in the excess heating calculations.
55
dependent on the plasma temperature and density, as well as errors in the neoclassical
model itself. For the purpose of this analysis, the Hinton-Hazeltine neoclassical model of
transport is assumed to be correct. [4] However, in addition to affecting the neoclassical
transport, error in the measurements of electron and ion temperature and density also affect
the calculations of the amount of heat deposited from the beams into the plasma. In order
to evaluate the variability in the beam heating and loss terms, the input electron and ion
temperatures and densities were varied randomly over the error bars from the diagnostics.
The variation in excess heating from the TRANSP calculations is shown as the error bars
in figure 4.4. For the majority of the discharges, the amount of excess heating lies outside
the error bars, indicating that the apparent heating is not due to errors in the loss terms.
56
Chapter 4
MHD Modes and Ion Power Balance
4.1 Theory of Ion Heating by Fast-particle Driven Modes
In the previous chapter, excess heating was shown to exist for several discharges with neutral beam heating. However, since overall energy must be conserved, it is necessary to
investigate the source of this heating.
A mechanism was previously proposed that could explain the observed excess heating.
[17] Gates, et al. proposed that MHD modes could be responsible for stochastic heating
of the plasma ions. It was proposed that high frequency compressional Alfvén eigenmodes
(CAEs), could cause perturbations in the thermal ions’ orbits. It is possible that the orbit
perturbations could modify the particles’ energies, similar to a surfer speeding up by riding
on the front of a wave. The ions would gain energy at the expense of the wave. Thus, a
source of free energy would be needed to continue heating the ions.
To investigate this, full orbit calculations were performed in slab geometry. The magnetic perturbations included in the calculations were similar in frequency to those observed
in NSTX as will be discussed below. When the MHD modes were included in the calculations, the particle energy increased steadily with time. (See figure 4.1)
The magnitude of the magnetic perturbations was also varied in these calculations. (See
57
Figure 4.1: Particle heating calculation from modes in a slab geometry. The perpendicular
and parallel energies for an ensemble of particles is plotted vs. time. The modes used in
this geometry were similar in frequency to those observed in NSTX. (Reproduced from
reference [17])
58
Figure 4.2: Heating rate as a function of magnetic perturbation strength (arbitrary units).
The heating rate increases roughly as the square of the perturbation amplitude. (Reproduced from reference [17])
figure 4.2) The calculated rate of heating increased roughly as the square of the magnetic
field energy. The heating is primarily in the perpendicular direction. The heating in the
parallel direction is the result of pitch angle scattering collisions.
While the slab geometry model indicates that stochastic ion heating due to high frequency modes is possible, it is important to investigate the heating in a toroidal geometry.
4.2 Fast Particle-driven MHD modes in NSTX
High frequency (500-1500 kHz) MHD modes have been observed nearly ubiquitously in
NSTX in neutral beam discharges [21]. The amplitudes of these modes increase with increasing neutral beam power. During the discharges analyzed in the preceding section, the
59
MHD modes were observed to intensify as the third neutral beam source turned on. Additionally, corresponding to the turn on of the third beam, some of these modes exhibited a
bursting behavior.
These bursting modes are very short, lasting 0.01-0.1 ms, and are spaced approximately
1 ms apart. During the time between bursts, the Mirnov coils detect no significant perturbations. The modes then rapidly increase then rapidly decrease in amplitude. The average
duty cycle for these modes is approximately 3%.
Since we are interested in the effects of high frequency modes on thermal ions, we need
to isolate the effects of the high frequency modes. As previously seen in figure 3.3, the neutral beam modulations were performed during a period with very little low frequency (<100
kHz) MHD activity. The low frequency MHD activity has been correlated with increased
transport in NSTX [39, 40]. In order to avoid the possibility of these low frequency modes
affecting the transport, the third neutral beam was not modulated until the low frequency
MHD activity ceased.
Using the definition for excess heating defined in the previous chapter, it is possible to
calculate excess heating for each discharge. The required excess ion heating for a series of
nearly identical discharges is shown in figure 4.4. These discharges all used three neutral
beams, similar to discharge 128820 shown in figure 3.2. The discharges were H-mode
discharges with 900 kA of plasma current.
As shown in figure 4.4, excess heating is not always required, even when the discharges
have similar parameters. The variation in excess heating and Mirnov signal amplitude
could be due to the evolution of the plasma, including the current profile, or other plasma
properties, such as radiation from contaminants.
The majority of these discharges required some amount of excess heating. The figure also shows that the heating is correlated with the Fourier transform of the Mirnov
coil signal. Figure 4.5 shows the Fourier transform of the Mirnov coil signal for discharge 128820. The Mirnov signals for figure 4.4 were integrated over the frequency range
60
\bdot_l1dmivvhn1_raw for shot 128820; #FFTs=4096
1200
0.08
1000
0.06
freq (kHz)
800
600
0.04
400
0.02
200
0
0.35
0.40
0.45
0.50
0.55
0.60
0.55
6
5
4
3
2
0.60
Volts
0.5
-0.5
0.30
0.35
0.40
0.45
sec
0.50
Power
0
0.30
Figure 4.3: The high frequency MHD modes become evident when the third neutral beam
source turns on at t=0.40 s. The modes around 500 kHz exhibit a bursting behavior that
may be associated with the occurrence of the excess heating.
61
Figure 4.4: Excess heating is required for a series of discharges. The heating was integrated
out to a radius of 137 cm. Note that 25% of the discharges (4 of 16) do not require excess
heating to satisfy power balance, which could be due to increased transport or loss. The
error bars indicate the variation due to varying the temperature and density inputs within
the diagnostic error bars in TRANSP.
62
Figure 4.5: Fourier Transform of Mirnov coils showing high frequency MHD activity at
410 ms. Figure 4.4 was made by summing over the amplitudes from 4 × 10 5 − 1.5 × 106
Hz.
4 × 105 − 1.5 × 106 Hz. Above a certain threshold of high frequency MHD activity, the
amount of excess heating required increases with increasing MHD signal. Figure 4.5 shows
the Fourier transform of the Mirnov coil signal for discharge 128820.
4.3 Analytic Evaluation of Compressional Alfvén Modes
(CAEs)
In order to compare the modes with the excess heating, it is necessary to examine the CAE’s
eigenmode equation. [41, 42] The electric field of the CAE has the form
63
√2θ √2(r − r ) 0
E θ = E 0 φm
φs
ei(nφ−ωt)
Θ
∆
iωc
Er =
Eθ
ω
(4.1)
where E0 is the amplitude, φ is the toroidal angle, θ is the poloidal angle, m, n, and s are
the poloidal, toroidal, and radial mode numbers, ωc is the cyclotron frequency and ω is the
mode frequency. The mode peak location is given by r0 . In this equation,
p
2σi (1 + σi )
∆2
=
a2
m(1 + σi )(1 + 0 )
r0 2
1
∆2
=
−
(2s
+
1)
a2
1 + σi
a2
(4.2)
(4.3)
φy (x) are the Chebyshev-Hermite functions
x2
e− 2
φy (x) = p
√ Hy (x),
y!2y π
(4.4)
dy −x2
e ,
dxy
(4.5)
Hy (x) are the yth order Hermite polynomials given by
Hy (x) = (−1)y ex
2
Θ is given by
Θ2 =
and 0 is given by
s
4(1 + 0 ) ∆ ,
(2s + 1)0 r0
0 =
r0
r0 + R 0
64
(4.6)
(4.7)
Figure 4.6: Fit of equation 4.8 to normalized electron density. The location of the CAE
mode peak amplitude depends weakly on a fit to the electron density.
where R0 is the major radius of the magnetic axis. The term σi is found by fitting the
electron density to the equation
r 2 σi
ne (r) = n0 1 − 2 ,
a
(4.8)
where a is the normalized minor radius. (See figure 4.6) The magnetic field of the wave
can then be solved from the wave equation.
In order to compare the excess heating with the radial mode profile, we examine the
terms that do not contain a radial dependence. For s = 0 modes, the Hermite polynomial
65
reduces to 1, and the equations can be simplified to
Eθ = E 0 e−
(r−r0 )2
∆2
(4.9)
ei(nφ−ωt)
r0 2
1
∆2
=
− 2
a2
1 + σi
a
(4.10)
where all of the non-radially dependent terms in the first equation are collected into the E 0
term and ∆ is still defined above, and the radial electric field Er =
ωci
Eθ
ω
After fitting the electron density profile to obtain σi , r0 is then calculated and equation
4.9 can be plotted. Figure 4.7 shows the normalized radial mode profile and figure 4.8 show
the poloidal eigenfunction for discharge 128820. To find the region in which the modes can
propagate, it is necessary to solve the dispersion relation (ω 2 = vA2 (kr2 + kθ2 + kφ2 ), where
k’s are the wave numbers in each direction [41]. Solving this for k θ2 + kφ2 gives
ω2
− kφ2 = kr2 + kθ2 .
vA2
(4.11)
The modes can propagate in regions where kr2 + kθ2 > 0. Outside of this region, the modes
are evanescent. Figure 4.9 shows the poloidal cross-section of the region where the modes
can propagate.
To directly compare the radial heating profile to the radial mode amplitude, we overlay
the two normalized profiles in figure 4.10.
From this figure, it is apparent that the square of the radial eigenmode is similar in shape
to the excess ion heating profile, consistent with the prediction of the excess heating theory.
However, the profiles, as determined by fitting the calculation shown in figure 4.9, do not
perfectly match with the heating calculated by TRANSP. The discrepancy in the radial
location of the peak could be due to approximations made in derivations of the eigenmode
equation above.
Note that the heating calculation and mode profile calculation were performed inde-
66
Figure 4.7: Typical radial eigenfunction of the compressional Alfvén eigenmode.
67
Figure 4.8: Poloidal cross section of a typical CAE. The total magnetic field perturbation
is plotted. The radial dependence can also be seen. This mode is an m=4 mode.
68
Figure 4.9: Poloidal cross-section showing the region in which CAEs can propagate. The
CAEs can only propagate in regions where kr2 + kθ2 > 0, according to equation 4.11.
69
Figure 4.10: Overlay of normalized radial profiles of high frequency CAE modes and the
normalized excess ion heating. The CAE mode characteristics were determined from fitting
the calculations in figure 4.9, with the mode peak at ψ = 0.8 and ∆ = 0.2. The mode
amplitude is shown in blue and the square of the calculated amplitude is shown in red. The
mode has a similar shape to the amplitude squared, matching the theory from Gates. [17],
although it peaks further out than the heating.
70
pendently, although some of the inputs are the same (such as the electron density). The
calculations of the excess ion heating were performed in TRANSP, which does not model
the effects of MHD modes. While the mode equation does depend on the electron density,
it was derived analytically and matched to the propagation calculation, independently of
the TRANSP excess heating calculations.
The modes are present even when excess heating is not inferred. During these time
periods, the modes could be heating the ions, but it is impossible to verify, since the excess
heating power term from the previous chapter represents only a lower bound on the heating
occurring in the plasma. If the losses were large (for example, if the transport was much
greater than neoclassical), it could overshadow any heating that occurred as a result of the
modes.
4.4 Modeling CAEs and Calculating Ion Heating
In order to determine if the CAE modes could be responsible for ion heating, it was necessary to model the effects of the modes on the thermal ion population. This requires
performing full gyro-orbit calculations in the three dimensional magnetic field inside the
vacuum vessel.
GYROXY was written by R.B. White and is a full gyro-orbit code [43, 44]. Individual
particles are stepped forward in time using a 4th order Runga-Kutta method according to
the Lorentz force law:
~ + ~v × B).
~
F~ = m~a = q(E
(4.12)
In GYROXY, it is possible to add time dependent perturbations to the equilibrium field.
For this study, the modes are simulated in GYROXY using frequency information obtained
from the Mirnov coils. The frequencies of the modes are determined by taking the Fourier
transform of the Mirnov coil signal. By including several of the strongest modes measured
by the Mirnov coils, it is possible to simulate the presence of the modes and approximate
71
Figure 4.11: A mode with toroidal mode number n=8. The toroidal mode number can be
determined by fitting the Mirnov phase and location data. Each mode must be analyzed
separately to obtain its mode number. The red lines indicate the best phase fit to the different Mirnov coils.
their effect on the ions in the plasma.
In order to accurately model the effects of the modes, it is necessary to know toroidal
and poloidal mode numbers. The toroidal mode number can be determined from Mirnov
coils located at various toroidal points on the machine. By fitting the phases of the individual mode components from the array of Mirnov coils, it is possible to obtain toroidal mode
numbers.
Each mode must be examined individually to determine its toroidal mode number. (see
figure 4.11) When analyzing the modes for heating, each mode is treated independently
and the modified B field is the sum of the the equilibrium field and the time varying pertur-
72
bations from the modes. Poloidal mode numbers will be discussed below.
4.4.1 Determining the Amplitudes of the MHD Modes
Mirnov coils are the most common tools for detecting MHD modes. By taking the Fourier
transform of the digitized time series data from the coils, it is possible to determine the frequency of the modes. By comparing the phases of Mirnov coils placed around the machine,
it is possible to determine toroidal mode numbers. Theoretically, it should be possible to
determine poloidal mode numbers by the same mechanism. An array of Mirnov coils
placed poloidally could provide that information.
One shortcoming of the Mirnov coils is their inability to determine the location of
the magnetic perturbation. The coils are located outside the plasma and only detect the
perturbation there, which will have decayed.
If we were able to measure the magnetic perturbation amplitude for a reference perturbation and also measure the Mirnov coil amplitude, it should be possible to estimate the
amplitude of CAEs in other discharges by measuring the Mirnov signal.
B̃ = M̃
B̃ref
Mref
(4.13)
where M̃ is the measured Mirnov amplitude for a given discharge, and B̃ref and Mref are
the magnetic field perturbation and Mirnov signal amplitude for a reference discharge.
While a direct measurement of the magnetic fluctuations is not possible, it is possible
to measure the density fluctuations associated with the CAEs using an interferometer. By
calculating how the density changes as the magnetic field changes, it is possible to calculate
the magnetic field perturbations.
To find the relationship between magnetic field and density perturbations, begin with
the continuity equation for ideal MHD. Assuming no source of particles, the continuity
73
equation is
∂n
+ ∇ · (n~v ) = 0.
∂t
(4.14)
The total derivative of the local density is given by
∂n
dn
=
+ ~v · ∇n
dt
∂t
(4.15)
∇ · (~v n) = ~v · ∇n + n∇ · ~v .
(4.16)
Expanding out ∇ · (~v n) gives
Substituting equations 4.14 and 4.16 into equation 4.15 gives
∂n
+ ~v · ∇n = −n∇ · ~v .
∂t
(4.17)
If we assume a B field perturbation goes like
B̃ ∝ eiωt eik·~r
(4.18)
∇ · B̃ = ~k · B̃ = 0
(4.19)
~
then
Writing the velocity in terms of displacement gives
~v =
∂ ξ~
∂t
(4.20)
Let the density consist of an equilibrium part and a perturbed part.
n = n0 + ñ
74
(4.21)
Since all perturbed quantities vary as eiωt , the equilibrium terms cancel and equation 4.17
becomes
ñ = −ξ · ∇n − n∇ · ξ~
(4.22)
The density in the second term is the equilibrium density. Since, to first order, the density
has only a radial variation, the first term in the right-hand side involves only the radial
derivative.
ñ = −ξr
∂n
− n∇ · ξ~
∂r
(4.23)
The lowest damped CAE modes have k⊥ kk . Since, by definition,
~ = 0,
k~⊥ · B
(4.24)
B̃ ∼
= B̃|| .
(4.25)
this implies
The field lines are “locked into” the plasma, and the magnetic field perturbations are in the
parallel direction. For these modes,
B̃k
.
∇ · ξ~ =
B
(4.26)
The displacement is calculated from the E × B drift. In general, the drift in a magnetic
field due to a force is given by
vf =
1F×B
.
q B2
(4.27)
where F is the applied force, q is the charge of the drifting particles. Since the predominant
electric field is the poloidal field Eθ , the displacement is
ξr =
iEθ c
.
Bω
75
(4.28)
where the frequency corresponds to the frequency of the perturbative field.
Equation 4.23 then becomes
ñ
iEθ ∂ln n B̃k
=−
−
n
Bω ∂r
B
(4.29)
In the case where ω ωci , Eθ becomes negligibly small, and equation 4.29 becomes
B̃k
ñ
=−
n
B
(4.30)
In NSTX for deuterium,
ωci =
eB
≈ 4.5 × 103 B
mc
(4.31)
On the magnetic axis, where typically B=4.5kG,
ωci ≈ 2 × 107
rad
s
(4.32)
Out where these modes peak, the magnetic field is approximately 3.5 kG, which makes the
ion cyclotron frequency
ωci = 1.6 × 107
rad
s
(4.33)
The high frequency modes have frequencies in the range
f = 0.5 − 1.2 × 106 Hz → ω = 3.1 − 7.5 × 106 Hz
(4.34)
ω
≈ 0.2 − 0.5
ωci
(4.35)
This means that at very high frequencies (≈1.2 MHz), the condition for the approximation
in equation 4.29 is questionable, but over most of the range considered, it is reasonable.
All of the subsequent calculations are based on this approximation.
It is possible to measure the integrated density perturbations with an interferometer.
76
The high-k microwave scattering diagnostic system on NSTX [45] can be used as an interferometer. It measures 280 GHz (approximately 1 mm wavelength) scattered signal from
the plasma to obtain a fluctuation spectrum. By mixing the scattered signals with a reference beam, it is possible to shift the spectrum of the scattered signal to be centered at zero
frequency. Coherent perturbations in the plasma, such as from a CAE, appear as symmetric peaks in the power spectrum (both positive and negative frequency). By adjusting the
optics of the high-k system, it is possible to measure perturbations at various locations. For
these discharges, the high-k microwave beam was directed into one of the detector channels
at a tangency radius of 124 cm (see figure 4.12).
In order to confirm that the signals observed by the high-k diagnostic are the perturbations of interest, it is necessary to compare the Fourier transforms from the two signals.
(see figure 4.13) Besides the broad mode of the Mirnov (lower) spectrum at around 600
kHz, the Mirnov coils and the high-k channel spectra show very similar signals. This is
strong evidence that the perturbations in density observed by the high-k system come from
the MHD modes observed by the Mirnov coils.
To use the high-k channel as an interferometer, it is necessary to compare the shifted to
unshifted signal amplitudes. Compared to a reference beam, the electric field for the beam
passing through the plasma is given by
E = E0 ei(φ+ωt)
(4.36)
where φ is the phase shift induced by different index of refraction of the plasma, and the
eiωt part is common to both beams. If the plasma was in equilibrium (no MHD modes
existed), then the phase shift would vary slowly on the timescale of the evolution of the
density profile. The CAE modes add a short-timescale perturbation to the phase shift.
φ = φ0 + φ̃
77
(4.37)
Figure 4.12: Overhead view of the high-k microwave beam and detector. The optics can
direct the microwave beam to various tangential radii. For these discharges, the microwave
beam was directed into one of the detector channels. Reproduced from [46].
78
Figure 4.13: Spectral plots showing the high-k channel 1 (top) and Mirnov (bottom) spectrograms. The two diagnostics show similar spectra including start times which correspond
to the turn on of the third neutral beam source. Both Fourier transforms were performed
over 1 ms time windows.
79
where φ̃ is the phase shift due to the mode fluctuations. The measured signal amplitude
goes as
A = A0 eiφ = A0 ei(φ0 +φ̃)
= A0 ei(φ0 ) ei(φ̃)
(4.38)
(4.39)
Since φ0 is a constant, this can be rewritten as
= A1 ei(φ̃) .
(4.40)
Assuming that φ̃ is small, this can be expanded to
A ≈
A1 (1 + iφ̃)
(4.41)
(1) (2)
where term (1) is the unshifted part and term (2) is the shifted part. The phase shift can
then be calculated as the ratio of the unshifted to shifted components. Since the high-k
scattering system mixes the signal with a reference beam, it shifts the unperturbed part
of the spectrum to zero frequency, and the perturbations appear as symmetric peaks. The
phase of the perturbation can be measured by comparing the amplitude of the peaks with
the zero frequency peak.
The high-k power spectrum is shown in figure 4.14. In this power spectrum, it is possible to analyze a peak to determine the phase shift. The amplitude of the dominant fluctuation peak is 27 dB below that of the zero frequency signal. Thus,
φ̃2 = −27dB = 10−27/10 = 2.0 × 10−3
(4.42)
φ̃ = 4.4 × 10−2
(4.43)
80
Figure 4.14: The high-k power spectrum. The symmetric peaks around 1 MHz are interferometric peaks The Fourier transform was performed over a window of 0.5 ms (7.5 × 10 6
Hz, 4096 points). The dominant peaks are -27 dB below the zero frequency peak, which
corresponds to an amplitude of 4.4 × 10−3 .
81
For an interferometer the phase shift is given by,
φ̃ = k
Z
(4.44)
Ñ dl
where k is the wave number of the beam and Ñ is the perturbed index of refraction. For an
electromagnetic wave in a plasma the dispersion relation is given by
(4.45)
ω 2 = k 2 c2 + ωp2 ,
where ωp is the plasma frequency. The index of refraction N is
N=
kc
.
ω
(4.46)
Solving for N gives
N=
r
1−
ωp2
1 ωp2
1 ωp2 λ2
≈
1
−
=
1
−
ω2
2 ω2
2 (2πc)2
1 4πe2
1 4πe2
hni −
hñi.
N =1−
2 m(kc)2
2 m(kc)2
(4.47)
(4.48)
where hni and hñi are the average equilibrium and perturbed densities. The second term
relates to the phase shift of the beam through the equilibrium plasma. The third term relates
to the plasma perturbation. This makes the phase shift
φ̃ = k
Z
1 4πe2
ñe dl = 4.48 × 10−15
2 m(kc)2
Z
ñe dl
(4.49)
where dl is the path integral. By using the phase shift obtained from the power spectrum in
equation 4.43, the integrated density perturbation becomes
Z
ñe dl = 1.0 × 1013 cm−2
82
(4.50)
Mode #
3
4
5
6
7
8
9
Scaling factor
0.108
0.080
0.224
0.293
0.277
0.190
0.133
Table 4.1: Calculated scaling factors of the line integrated density perturbations. The calculation was performed along the high-k diagnostic line of site (1.24 m).
where the units of ñ are in cm−3 . Solving for the normalized density perturbation gives
Z
R
ñe dl
1.0 × 1013
ñcalc dl = R
=
= 1.7 × 10−3 ,
6.0 × 1015
ne dl
(4.51)
where the integrated equilibrium density comes from the discharge being analyzed at the
appropriate time. In this example, the integrated density comes from discharge 130335, at
t=430 ms.
Equation 4.51 gives the integrated density perturbation along the line of sight through
the plasma. In order to estimate the perturbations from the modes, it is necessary to simulate
the line integral. By simulating a mode with a given mode number, it is possible to calculate
the integral along the line of sight of the high-k diagnostic.
Since the mode frequency is higher than the time window of the Fourier transform of the
diagnostic, it is necessary to evolve the mode to get an average amplitude. For example,
integrating a CAE MHD mode with an amplitude A and toroidal mode number n = 8
(matching the toroidal mode number of the reference mode) gives
Z
ñcalc dl = 0.19A
(4.52)
The scaling factors in table 4.1 were obtained by integrating the CAE eigenmode with
different toroidal mode numbers. The integration was performed over a variety of phases
83
Figure 4.15: Top-down schematic of NSTX plasma showing the high-k diagnostic line of
sight. In this figure, an n=7 mode is shown, along with the mode amplitude along the line
of sight (bottom panel). Each mode number must be analyzed separately.
84
(relative to the diagnostic line-of-sight), and the average was taken. To calculate the amplitude of the magnetic field perturbations in the plasma, we divide the integrated density
obtained from the diagnostic by the integration factor.
R
B̃k
ñ
dl
0.0017
R meas =
=
0.19
B
ñcalc dl
(4.53)
B̃k
= 0.0089
B
(4.54)
where the 0.19 is the scaling factor from the reference n = 8 mode. (see figure 4.14) This
gives
Each mode can be analyzed from the high-k spectrum. However, the high-k diagnostic
was not available during the majority of the discharges analyzed. Instead, a reference
discharge was used correlate the Mirnov coils with the amplitudes measured by the high-k
diagnostic to calibrate the Mirnov coils. Then the Mirnov coils were used to analyze the
discharges in the power balance experiment.
Each mode must be individually analyzed. The scaling factor from the line integration
depends on the toroidal mode number. For example, if the mode number of figure 4.15
were increased, the line of sight would pass through more peaks and troughs which would
change the integrated signal. Most NSTX plasmas during the beam heated discharges have
similar profiles, with 0.25 < σi < 0.35. Since the location of the mode peak is a weak
function of the fit, and the radial profile of the mode is a weak function of σ, the effects of
the density profile are minor.
The toroidal mode number is measured using Mirnov coils. However, no suitable coils
exist to measure the poloidal mode numbers. Theoretical calculations of the CAE modes
correlate the mode frequency with the poloidal and toroidal mode numbers. [47], [48] The
frequency is given by
2
ω '
vA2
n2
s2 + 2+ 2
r2
R
Lr
m2
(4.55)
where vA is the Alfvén velocity, L is the characteristic radial width of the effective potential,
85
R is the major radius, r is the minor radius, and m, n, and s are the poloidal, toroidal and
radial mode numbers.
The CAEs observed have frequencies in the range of 500 kHz-1.4 MHz. The Alfvén
velocity is given by
vA =
B
.
(4πni mi )1/2
(4.56)
In NSTX plasmas in the region of interest, vA ≈ 1 × 108 cm sec−1 . Assuming s = 0,
ω
'
vA
r
m2
r2
+
n2 ' 0.03 − 0.08m−1
R2
(4.57)
The plasma has a major radius of 100 cm and a minor radius of approximately 34 cm to the
peak of the mode. In order to obtain a poloidal mode number, the Alfvén speed was calculated from equation 4.57, using the highest toroidal mode number from a given spectrum,
and the corresponding frequency, and setting the poloidal to m = 2. The minimum mode
number is chosen to because the eigenmode equations were solved in the limit of large m.
For the spectrum analyzed, vA was calculated using f =866.7 kHz, n =7. Using the value
of vA calculated from this mode, the poloidal mode numbers of the rest of the modes were
calculated. See table 4.2 for a list of toroidal and poloidal mode numbers used in these calculations. The table also shows the variation in peak amplitudes of the Fourier transform
due to varying the time window over which the transform was performed. This uncertainty
leads directly to uncertainty in the mode amplitude. For the heating calculations below, the
Fourier transforms were performed over a 1 ms window.
4.4.2 Implementing CAEs in GYROXY
When these modes were put into GYROXY, heating of particles with characteristics of
thermal ions was observed.
The heating only occurred when the modes were present. When the modes were removed from the plasma, no heating occurred and energy was conserved (see the red line in
86
Frequency
(kHz)
958.3
1073.0
974.1
716.6
737.3
786.1
866.7*
1 ms
0.5 ms
2 ms
FFT
FFT
FFT
Amplitude Amplitude Amplitude n m
0.0449
0.0356
0.0355
6 4
0.0277
0.0385
0.0248
5 5
0.0254
0.0230
0.0186
5 5
0.0166
0.0110
0.0087
5 3
0.0158
0.0175
0.0090
6 2
0.0149
0.0145
0.0089
6 2
0.0117
0.0148
0.0067
7 2
Table 4.2: CAE modes used in heating calculations. These modes are the strongest modes
from NSTX shot 128820 at t=400 ms, calculated over a variety of Fourier transform windows. * = The mode used to calculate vA , by setting m=2.
Figure 4.16: Radial mode eigenfunctions used in GYROXY calculations. The modes used
were from discharge 128820 at t=0.405 s and matched to the mode propagation calculation.
Each trace shows a different frequency of mode used (see table 4.2). The magnetic axis is
at 100 cm. The cutoff at 145 cm is to ensure the mode goes to zero at the plasma edge.
87
Figure 4.17: Radial mode eigenfunctions used in GYROXY calculations as a function of
flux (0=axis, 1=boundary).
figure 4.18). The interactions with the modes were the only source of heating present for
these calculations. The radial eigenfunction of the modes used in the heating calculation
are shown in figure 4.16 as a function of radius. They are plotted in figure 4.17 as a function of ψ. Near the magnetic axis, the modes are multiplied by a polynomial which goes
to zero at the magnetic axis. This is to avoid singularities that would arise in the poloidal
derivative if the mode amplitude does not go to zero.
In order avoid any heating due to a discontinuity caused by the polynomial, it is necessary to make both the function and its derivative continuous. In order to solve this, we test
the generalized case.
f (x) =


 g(x)
x≥a

 h(x)g(x) x < a
88
(4.58)
Figure 4.18: When the MHD modes are input into GYROXY, ion heating occurs. The black
line shows heating when modes are present. The red line shows that energy is conserved
when modes are absent. The initial difference between the red and black lines at t=0 is due
to the potentials from the perturbations.
89
At x = a, we set both sides equal to each other:
h(a)g(a) = g(a)
(4.59)
h(a) = 1
(4.60)
which makes
which we use to determine the relationship of the coefficients in h(x).
Taking the derivative of equation 4.58 gives
f 0 (x) =


 g 0 (x)
x≥a
(4.61)

 h0 (x)g(x) + h(x)g 0 (x) x < a
Setting the derivatives equal at x = a gives
g 0 (a) = h0 (a)g(a) + h(a)g 0 (a)
(4.62)
Since we know that h(a) = 1, this simplifies to
=
g 0 (a)
g 0 (a)
h0 (a)g(a) + (4.63)
0 = h0 (a)g(a)
(4.64)
Since g(a) 6= 0,
h0 (a) = 0
(4.65)
The function f (x) and its derivative can thus be matched subject to the two conditions
h(a) = 1
(4.66)
h0 (a) = 0
(4.67)
90
independent of the function g(x) and its derivative.
For the modes used in this heating calculations, a 4th order polynomial was used.
f (x) =


 g(x)
x≥a
(4.68)

 (C1 x4 + C2 x3 )g(x) x < a
From equations 4.66 and 4.67, we are able to solve C1 and C2 . Equation 4.66 gives
1 = C 1 a4 + C 2 a3
(4.69)
1
− C1 a
a3
(4.70)
C2 =
and equation 4.67 gives
4C1 x3 + 3C2 x2 = 0
(4.71)
1
−
C
a
a2 = 0
1
a3
3
4C1 a3 + − 3C1 a3 = 0
a
−3
C1 = 4
a
(4.72)
At x = a, we get
4C1 a3 + 3
(4.73)
(4.74)
which gives
C2 =
3
4
1
+
=
a3 a3
a3
(4.75)
4.5 Mode Heating Calculations
In order to compare the heating calculated from GYROXY with the heating calculated by
TRANSP, it is necessary to perform the calculations across the entire plasma. Figure 4.19
shows the heating per particle as a function of normalized ψ and initial particle energy,
91
Figure 4.19: The heating rate as calculated by GYROXY. Heating is calculated as a function
of energy and normalized ψ. The plasma temperature profile can then be used to determine
the amount of heating in the plasma.
where ψ is the flux function, defined as
∂ψ
= −RBr
∂z
∂ψ
= −RBz
∂R
(4.76)
(4.77)
and then normalized so that ψ = 0 at the axis and ψ = 1 at the plasma boundary.
It is important to note that this figure shows the heating per particle. To determine
the actual heating, it is necessary to multiply by the volume integrated density, using a
Maxwell-Boltzmann (also called Maxwellian) distribution. (see figure 4.20)
92
Figure 4.20: The plasma particles have energies distributed according to a MaxwellBoltzmann distribution. The higher temperature distribution peaks further out in energy
and contains more particles in the high energy tail.
93
4.6 Applying a Maxwellian Distribution to GYROXY Heating
The Maxwell-Boltzmann distribution gives the numbers of particles in an element of velocity space.
E
dn ∝ e− kT dvx dvy dvz
(4.78)
where E is the energy (≡ mv 2 /2), T is the temperature of the plasma (where E = 3kT /2),
k is Boltzmann’s constant, and the normalization constants have been dropped. Since the
particle density is only a function of energy, it is useful to convert the distribution from
velocity space into energy space. We start with the relation between energy and velocity
1
E = mv 2 .
2
v=
r
2E √
∝ E
m
(4.79)
(4.80)
Taking the derivative if each side of equation 4.79 gives
dE = mv dv.
(4.81)
Since the function is only a function of the magnitude of velocity and not the individual
components, we can write the volume element in terms of the scalar velocity v
dvx dvy dvz = 4πv 2 dv.
(4.82)
This makes equation 4.78
E
dn ∝ e− kT v 2 dv,
94
(4.83)
The normalization constants will be calculated numerically later. Substituting equations
4.80 and 4.81 into equation 4.83 gives
√
E
dn = N Ee− kT dE,
(4.84)
where N is the normalization factor. The Maxwellian distribution as a function of energy
is then given as
f (E) =
√
E
dn
= N Eee− kT .
dE
(4.85)
The normalization factor N is calculated by integrating over all energies and normalizing
it to 1.
It is useful to confirm that equation 4.85 gives a correct distribution. It can be confirmed
by calculating the average energy of a particle in the distribution and confirming that the
average energy is hEi = 3kT /2.
R
Making the substitution
hEi = R
E
E 3/2 e− kT dE
E
E 1/2 e− kT dE
S=
dS =
.
E
kT
1
dE
kT
(4.86)
(4.87)
(4.88)
(4.89)
gives
R
(kT S)3/2 e−S kT dS
hEi = R
(kT S)1/2 e−S kTdS
5/2
(kT ) Γ 52
=
(kT )3/2 Γ 23
95
(4.90)
(4.91)
3
= kT
2
(4.92)
which confirms that equation 4.85 is the correct distribution.
The normalization factor can be calculated numerically. If the heating rate is known
from GYROXY for discrete values of particle energy, the normalization N can be calculated from the sum
M
X
1
=
g(Ei , T )
N
i=1
(4.93)
where g(Ei , T ) is the normalized distribution function for a given plasma temperature and
M is the number of discreet energy points, chosen to be large enough that the high energy
tail contributes few particles.
If a volume of plasma, has a density, ne , the distribution in energy is
f (E, T ) = ne
1
g(E, T ).
N
(4.94)
In GYROXY, the particle heating was calculated for thermal particles at each 0.2 keV up
to 3 keV. (see figure 4.19) Using the temperature distribution from discharge, it is possible
to calculate the heating rate as a function of normalized ψ, by summing all of the terms in
equation 4.94 in that volume.
4.7 Comparing Measured and Calculated Heating
The heating profiles calculated from TRANSP and GYROXY are shown in figure 4.21.
The two profiles differ slightly, although both peak near the plasma edge, where theory
predicts the CAEs to peak. The difference may be related to the gyro-orbits and banana
orbits, as well as the fact that in GYROXY, particles on a flux surface were given random
gyro-phases, thus shifting the guiding center.
As shown in figure 3.10, the total excess heating power in the NSTX plasma as calcu96
Figure 4.21: Heating profile from GYROXY and excess heating TRANSP. Both profiles
are in terms of power (i.e. power density times the volume of each flux zone). The profiles
differ slightly in where they peak. This may be related to the gyro-orbits and banana orbits,
as well as the fact that in GYROXY, particles on a flux surface were given random gyrophases, this shifting the guiding center.
97
lated by TRANSP is approximately 5×105 Watts. Using the ion density, it is possible to
calculate the total heating that could occur from the modes in GYROXY. Summing over all
flux surfaces gives a total heating of 1.6 MW.
The total heating calculated by GYROXY is larger than the amount calculated by
TRANSP. It is possible that the estimate of the magnetic perturbation is too large, due to
neglected terms. It is also possible that the modes are not strictly compressional, although
it is not clear what effect this would have on the heating calculated in GYROXY.
Variation in heating as calculated by GYROXY is shown in figure 4.22. The heating
scales roughly as the square of the perturbation amplitude. This means that a factor of
two uncertainty in the CAE amplitude could be responsible for the discrepancy between
GYROXY and TRANSP excess heating calculations. Variation in the Fourier transform
amplitude for a single modes is shown in figure 4.23. The Fourier transform was performed
on the Mirnov coil data over 1 ms windows. Each data point represents a Fourier transform
over a 1 ms window. Also, as shown in table 4.2, the amplitude of the peak of the transform
depends on the window over which the transform was performed. The uncertainty in the
Fourier transform analysis leads to uncertainty in the mode amplitude.
Although the heating calculated by GYROXY exceeds the excess heating calculated in
TRANSP, uncertainties in the mode amplitudes cause uncertainty in the heating calculations which can explain this discrepancy. However, the full orbit calculations nonetheless
demonstrate the potential of the high frequency CAE modes to heat the plasma sufficiently
to explain the excess heating.
4.8 Excess heating in Future Magnetic Confinement Devices
TRANSP calculations show that excess ion heating is occurring in NSTX during neutral
beam-heated discharges. Full orbit calculations show that it is possible that compressional
98
Figure 4.22: The total heating calculated by GYROXY for various amplitudes on δB/B.
The heating scales roughly as the square of the amplitude (red curve). This corresponds
with the results of reference [17]. X=1 corresponds with full amplitude perturbations.
99
Figure 4.23: Amplitude of Fourier transform of 958 kHz mode over 1 ms intervals. The
mode amplitude can vary as much as 50%, depending on which 1 ms window the Fourier
transform is performed.
100
Alfvén eigenmodes (CAEs), which are driven by the presence of high energy beam ions,
are the source of this excess heating. The theory of the CAEs state that they are driven by
the presence of “super-Alfvénic” ions-that is, ions with velocities greater than the Alfvén
velocity [48]. The Alfvén velocity is given by
vA =
B
(4πni mi )1/2
(4.95)
In NSTX, full energy (90 keV) beam particles in the plasma can have a velocity several
times the Alfvén speed:
vf
= 3 − 6.
vA
(4.96)
However, it is generally expected that a tokamak power plant would require a much larger
magnetic field. As a result, neutral beam particles would not have a velocity greater than
the Alfvén speed.
However, the fusion α particles may provide the supply of super-Alfvénic particles. The
fusion αs in a D-T reaction have an energy of 3.5 MeV. Depending on the magnetic field
strength, vf /vA for the αs will be similar to what it is for the beam ions in NSTX. Thus,
they can provide the free energy necessary to drive the CAEs and directly heat the thermal
ions. It would be useful to design future tokamaks to take advantage of this heating.
4.8.1 Excess Heating and Neutron Production
The presence of high energy neutral beams on NSTX results in the production of neutrons
via beam-target collisions. The neutral beam particles are ionized through charge exchange
and collisions becoming fast ions or thermal particles. The fast ions are confined by the
magnetic field. These ions travel around the machine until they reenter the beam region. In
that region beam particles can impact the ions and undergo fusion. Since the neutral beams
injected in NSTX during this experiment were deuterium, the resulting fusion interaction
follows equations 1.2 and 1.3. Although fusion from the neutral beam injection does not
101
Figure 4.24: Neutron measurements and neutral beam injection during discharge 128820.
The neutron signal increases when the neutral beams turn on, saturating as the fast ion
population saturates. The normalized TRANSP calculations are similar to the measured
rate, which would not be the case if CAEs were causing significant perturbations to the fast
ion population.
provide enough energy to break even (i.e. energy production is less than the energy required
to continue it), this neutron production provides a way to study the fast ion population.
Figure 4.24 shows the neutron measurements and the neutral beam injected power. The
neutron signal generally follows the neutral beam power. When the beams turn on, the
neutron signal increases immediately as a result of the increase in beam neutrals for fusion.
As some of the beam ions slow down and fill in the fast ion distribution, the neutron level
saturates.
The CAEs absorb energy from the fast ions. However, the CAEs do not significantly
perturb the neutron production. The CAEs peak near the edge of the plasma and the
volume-integrated neutron production profile peaks near the core. (see figure 4.25) This
102
Figure 4.25: Volume-integrated neutron production and CAE profiles. The neutron production peaks near the magnetic axis, while the CAEs peak near the plasma edge. A change in
the edge fast particle population will not significantly affect the neutron production profile.
means that the CAEs are able to absorb energy from fast ions in the outer edge of the
plasma without significantly influencing the neutron production.
103
Chapter 5
Edge Neutral Density Diagnostic
5.1 Motivation
In calculating the ion power balance in NSTX beam heated plasma, it is necessary to know
the input power from the beams. Neutral beam power can be lost before entering the
plasma. Calculations using the TRANSP code also indicate that a significant amount of
beam power can be lost depending on the density of neutral hydrogen atoms in the edge of
the plasma. The neutral beam atoms interact with the neutrals by two methods. First, if the
neutral density in the beam duct is significant, it can lead to beam-neutral collisions; that is,
neutral particles inside the beam duct (outside the plasma) can collide with beam particles,
reducing the total beam density.
The beam loss from the beam-neutral collisions is calculated by
nloss =
Neutral
Species
H2
H
Z
σnn nb dx
σ (cm2 )
1.2x10−16
7x10−17
Table 5.1: Cross section for beam-neutral collisions
104
(5.1)
where nn is the neutral density of hydrogen in the beam duct, nb is the density of the
injected neutral beams, and the integral is through the beam duct, and σ is the collision
cross section, shown in table 5.1. Both nn and nb are functions of position. For the neutral
beams on NSTX, cryogenic panels in the large vacuum tank between the beam sources and
the duct pump the neutrals from the beam duct. A linear interpolation is made, assuming
that the vacuum vessel provides a source of neutral hydrogen at one end of the duct while all
neutrals are pumped from the vessel at the other end of the duct, and that the beam density
remains constant (i.e. losses are small and do not affect nbeam ). Also, it is assumed that
the vacuum vessel provides a source of hydrogen atoms. These particles might recombine
into molecules. However, recombinations would not affect these results, since the cross
section for hydrogen molecules is approximately twice as big as for hydrogen atoms, but
recombining into molecules would cut the density in half. These approximations simplify
equation (5.1) to
nloss = 0.5σ h dnn nb
(5.2)
where d is the length of the beam duct and the factor 0.5 comes from the linear approximation of the neutral density decay. The fraction of the beam power lost to these collisions
is
nloss
= 0.5σ h dnn
nb
(5.3)
For the NSTX beams, with a beam duct length of approximately 2.5 m, estimates can range
from less than 1% beam loss at densities below 1012 cm−3 to more than 10% beam loss at
a density of 1013 cm−3 . (see figure 5.1) Thus, the edge neutral density is important in
understanding the input beam power.
The second way that neutral particles can alter the power balance is by undergoing
charge-exchange with beam ions, that is, energetic, non-thermal particles. Beam ions can
orbit into the scrape-off layer, outside the last closed flux surface, where they can then
interact with neutral particles in the edge. Due to the low magnetic field in NSTX, beam
105
Figure 5.1: Estimates of the beam power lost to neutral-neutral collisions inside the NSTX
beam duct. If the edge neutral density approaches 1013 cm−3 , more than 10% of the beam is
lost. If the edge neutral density is less than 1012 cm−3 , a negligible amount of beam power
is lost.
106
ions have a large gyro-radius. In general, the gyro-radius is given by
ρ=
v⊥
ωci
(5.4)
where v⊥ is the ion velocity component perpendicular to the magnetic field and ω ci is the
ion cyclotron frequency.
v=
E 1/2
⊥
mi
ωci =
eB
mi c
(5.5)
(5.6)
For 90 keV deuterium beam ions in NSTX, the gyro-radius on axis is up to 9.6 cm on axis,
and increases to 12.3 cm near the edge. This means that a substantial fraction of the beam
ions have the potential to interact with the neutral particles in the edge, especially when
neoclassical effects including banana orbits are taken into account.
If charge exchange with edge neutral particles occurs, the beam ions become neutralized
and are no longer confined by the magnetic field and most are lost from the plasma before
they can deposit their energy onto the thermal particles, although a few re-ionize and are
confined. (see figure 5.2)
To calculate the impact of these potentially significant loss mechanisms, it is necessary
to obtain a measurement of the neutral density in the edge of the plasma.
5.2 Edge Neutral Density Diagnostic Design
The Edge Neutral Density Diagnostic (ENDD) is designed to measure the edge neutral
density profile on NSTX. It uses a 12-bit charge-coupled device (CCD) camera to measure
absolute Dβ (λ = 486 nm) emissions, then calculates the edge neutral density using a
collisional-radiative model along with electron density and temperature information from
the Thomson Scattering diagnostic.
107
Figure 5.2: TRANSP calculations showing power lost from the plasma. If the edge neutral
density is 1010 cm−3 , the total loss from charge exchange and direct loss is approximately
10% of the total input power. If the edge neutral density is 1013 cm−3 , the total loss increases
to 20% of the input power.
108
Figure 5.3: Schematic view of the Edge Neutral Density Diagnostic in NSTX. A mirror is
used to reflect emission from the outer midplane region into the entrance aperture of the
filter and CCD camera assembly. Once the image is numerically inverted, a radial profile
of the emission can be obtained for the field of view.
The camera for the ENDD looks through a window on the midplane of Bay I on NSTX.
The camera looks into a mirror, welded onto the flange of the port, which is angled to
provide a fan of tangential lines of sight through the edge of the plasma. (see Figure 5.3)
The camera focus is adjusted so that the tangential edge of the plasma is in focus. The
lens is covered with a filter that transmits in the Dβ wavelength. The filter used for this
diagnostic has a full width at half maximum transmission (FWHM) of 1.5nm.
The camera image is then Abel-inverted to provide a radial profile of intensity. Using
the collisional-radiative model, the neutral density is calculated. (See figure 5.4)
For a complete description of the Edge Neutral Density Diagnostic design and calibration, see Appendix B.
109
Figure 5.4: Schematic showing the ENDD analysis steps. The raw data is inverted and the
collisional-radiative model is used to calculate the neutral density profile.
110
Figure 5.5: Example of an image taken by the Edge Neutral Density Diagnostic. The
camera is angled so that the light goes to zero at the outboard edge of the frame. This
is necessary for a proper inversion. Variation in the z direction could be due to a higher
presence of neutrals from recycling off various diagnostics that stick into the plasma.
5.3 Edge Neutral Density Diagnostic Analysis
In order to calculate the edge neutral density profile, a collisional-radiative model is used.
The model assumes that all of the Dβ light emitted by the edge of the plasma is from neutral
atomic hydrogen. (See figure 5.5)
It is possible from this to calculate the local emission of the jth region of the plasma E j
as measured by a frame of the camera.
Ej =
X
i
4πIi L−1
ij ∗ f
(5.7)
where L−1
ij is the element of the “Abel” inversion matrix corresponding to the tangential
radius of the line of sight of the camera, f is the inverse of the exposure time, and I i is the
111
measured number of photons at pixel i. Ii has units of
has units of
photons
,
cm2 sr
while the plasma emission
photons
.
s cm3
The emission of the plasma in the Dβ wavelength can also be calculated by
(5.8)
Ej = N4 A24
where A24 is the Einstein coefficient for spontaneous emission corresponding to the Balmer
β series and N4 is the absolute density of atoms in the Nn=4 excited state.
In order to determine the number of atoms in the ground state, it is necessary to know
the ratio of atoms in the Nn=4 excited state to the Nn=1 ground state. This ratio, D41 , is
determined from a table calculated from first principles [50], and is a function of electron
density and temperature,
D41 = D41 (ne , Te ) =
N4
.
N1
(5.9)
It should be noted that this calculation neglects neglects excitations due to beam ions
which orbit outside the plasma and may excite neutral atoms.
A spline interpolation of calculated grid coefficients using the Thomson scattering density and temperature yields D41 . Solving equation (5.8) for N4 and substituting into equation (5.9) we obtain
N1 =
1 Ej
D41 A24
(5.10)
Ej is determined at each radius corresponding to the tangential point of the line of
sight. The electron temperature and density are also required at each point. However, the
Thomson Scattering diagnostic provides only a few data points in the region of interest
of the plasma. In order to obtain the electron density and temperature at each point, the
Thomson Scattering density and temperature data were fit with a hyperbolic tangent and
the fits were used to approximate the temperature and density at the required locations.
112
5.4 Results of the Edge Neutral Density Diagnostic
The Edge Neutral Density Diagnostic successfully measured the neutral density profile on
NSTX. The ENDD camera records the emission from the plasma as a 2-D image. The
image is multiplied by the pixel efficiency from the absolute calibration of the camera. The
image is then inverted using the onion layer inversion. The onion layer inversion is similar
to the Abel inversion, except the it uses discreet data points rather than an analytic function.
A collisional-radiative model is then applied to the inverted data. Using the electron density
and temperature, the neutral density profile is extracted.
As seen in figure 5.6 The neutral density was measured to be approximately 0.1 − 1 ×
1011 cm−3 . The neutral density is lowest near the separatrix, and increases moving outward
in major radius.
The neutral density as measured by the ENDD does not add a significant amount of
particle loss. The particle loss at 1011 cm−3 in the beam duct is approximately 1%. (see
figure 5.1) Additionally, the low neutral density means that the external charge exchange
loss is also low, amounting to approximately 0.15 MW. These two results indicate that
neutral density plays only a minor role in the ion power balance.
The ENDD has also been used to measure relative neutral density changes. A lithium
evaporator was recently installed on NSTX [51–54]. The lithium is coated onto the vacuum
vessel wall between discharges. This can affect the region near the separatrix by reducing
recycling [55, 56]. The ENDD has successfully been used to measure the relative density
changes due to the lithium deposition. [55, 57]
113
Figure 5.6: Edge neutral density profile from the ENDD diagnostic. The image is captured
from the ENDD camera and the neutral density profile is then calculated incorporating the
electron density and temperature.
114
5.5 Errors in the Edge Neutral Density Diagnostic Analysis
The neutral density profile in NSTX is necessary for a thorough understanding of power
balance. Errors in the ENDD analysis can potentially translate into errors in the overall
power balance.
There are two sources of error in the ENDD analysis. First, there are the errors due
to random fluctuations in measuring light. The light emitted from the plasma is a Poisson
√
distribution. The mean of a Poisson distribution is λ, and the standard deviation is σ = λ.
The fractional error is then computed as
√
λ
.
Error =
λ
(5.11)
(For more information about the Poisson distribution, see reference [58].) The pixel count
and error from discharge 128597 at t=485 ms is shown in figure 5.7. This error roughly
corresponds to error in the final analysis; i.e. if the pixel count is 100, the error in the final
measurement is approximately 10%.
Another source of error in the ENDD analysis is the Thomson scattering data. The
ENDD only measures the neutral density profile in the edge of the plasma. However, the
Thomson scattering diagnostic only provides a few data points in the region outside the
separatrix. The Thomson scattering diagnostic was reviewed in chapter 3. In order to
analyze the ENDD data, the Thomson scattering electron temperature and density data are
fit to hyperbolic tangent functions. This presents some difficulties, as the data points in the
plasma edge are prone to measurement errors due to the low electron density. (see figures
5.8 and 5.9) This limits the ability to automate the analysis of the ENDD data. Each MPTS
data time slice must be examined to ensure that it does not contain any erroneous data
points.
115
Figure 5.7: Pixel count and errors in discharge 128597. The pixel count does not correspond directly with radius, since it is line integrated. Errors due to random variations are
less than 10%.
Figure 5.8: Electron density and hyperbolic tangent fit for ENDD analysis. Error bars for
the Thomson scattering data are shown.
116
Figure 5.9: Electron temperature and hyperbolic tangent fit for ENDD analysis.
The neutral density in the ENDD model is not directly proportional to the electron
density and temperature. Instead, it is a complex function that has been calculated for
a variety of densities and temperatures. For the ENDD analysis, the measured electron
density and temperature are interpolated onto a table to obtain the neutral density. Figure
5.10 shows the error in the ENDD analysis due to errors in the Thomson scattering data.
In order to reduce the errors in the ENDD analysis, it is necessary to reduce errors
in the edge electron density and temperature data. The current capabilities of the MPTS
Thomson scattering diagnostic is limited to 30 fixed location channels, of which only 6
are in the region of interest for the ENDD analysis. If additional channels were added that
specifically targeted and optimized for measurements in the edge of the plasma, it would
improve the confidence in the ENDD analysis.
Another interesting proposal for obtaining edge temperature data was proposed by
Diem. [59] It was proposed to use the Electron Bernstein wave (EBW) antenna on NSTX as
a detector. By measuring the EBWs emitted by the plasma, it was possible to calculate an
approximate temperature, which showed good agreement with the Thomson scattering data
117
Figure 5.10: Errors in the ENDD due to temperature and density measurement errors.
The ENDD analysis was performed adjusting the Thomson scattering data within its errors
before the hyperbolic tangent fit was performed.
in the plasma edge, as shown in figure 5.11. If this capability were improved, it could provide temperature information for the ENDD analysis. If this was combined with a density
measurement, such as from a reflectometer, it could prove useful for the ENDD.
118
Figure 5.11: Measurements of the electron temperature using the EBW antenna. The EBW
antenna measures the Electron Bernstein wave emission from the plasma. Of particular interest is the large amount of data collected near the plasma edge. This image is reproduced
from Reference [59].
119
Chapter 6
Neutral Particle Analyzer and Ion
Power Balance
6.1 Introduction
The neutral particle analyzer (NPA) on the National Spherical Torus Experiment measures
the distribution function of fast particles deposited by the neutral beam sources. The neutral
beam particles are ionized by electron impact and charge-exchange in the plasma, creating
fast ions. These fast ions may then charge-exchange again and leave the plasma as fast
neutral particles, which can be detected by the NPA.
The NSTX NPA is an E k B spectrometer detector [60]. Neutral particles enter the
NPA aperture and are ionized in the stripping cell. The particles then enter a region with
parallel electric and magnetic fields. The E and B fields separate the particles by energy
and mass. The particles are detected by a multichannel electron multiplier plate. The NPA
typically has a detection energy range from 3 to 90 keV. For a complete explanation of the
NPA on NSTX, see reference [61].
The NPA detects neutral particles from a single collimated line of sight through the
plasma. The NPA can be scanned in toroidal and poloidal angle about a pivot point on
120
the outboard vacuum vessel at the midplane as seen in figure 6.1. The intersection of the
NPA sightline with the neutral beam trajectories through the plasma is important since the
neutral beam provides the majority of neutral particles for the fast ion charge exchange.
The evolution of a particle detected by the NPA is
Beam N eutral + thermal Ion → F ast Ion + thermal N eutral
Beam N eutral + e− → F ast Ion + 2e−
F ast Ion + Beam N eutral → F ast N eutral + F ast Ion
where the first two reactions show the processes which create fast ions, and the third reaction shows the neutralization of the fast ion. The fast neutral particle is then detected by
the NPA if it is directed into the NPA’s aperture.
6.2 Calculating NPA Viewing Pitch
Since the aperture of the NPA is small, only particles directed along the line of sight of
the NPA at the time of neutralization are detected. Since the charge-exchange region is
localized to the beam footprint, the particles that enter the NPA have a specific pitch relative
to the magnetic field, determined by the tangency radius of the NPA line of sight [61].
Figure 6.1 shows the viewing limits of the NPA The NPA lines of sight (LOS) with
tangency radii of 125 cm, 0 cm, and -75 cm are shown, where the negative sign indicates
that particles in this region must be moving in the direction counter to the toroidal magnetic
field to be detected. The NPA can be set to view any tangency radius between 125 and -75
cm.
The NPA detection difficulty can be visualized in figure 6.1. For example, in order to
see particles if the NPA has a tangency radius of 0 cm (pointed directly at the center stack),
the particles would have to have to have almost all of their energy in the perpendicular
121
Figure 6.1: Drawing showing the diagram of the NPA and the possible angles. The NPA
pivots between a tangency radius of 125 cm and -75 cm.
122
direction, and would therefore have to have a pitch (≡ vk /v) of nearly 0. However, as can
be seen from the diagram, the neutral beam particles initially deposited on that line of sight
are nearly tangential, and therefore have a pitch near 1. Thus, the fast ions would not be
seen unless they undergo significant pitch angle scattering on a time scale faster than their
slowing down time, which is typically around 30 ms.
If the NPA is pointed at a tangency radius of 125 cm, the diagram shows that the required pitch is near 1 for observation, while the pitch of deposited particles is less. Again,
few, if any, particles would be observed unless there was a significant perturbation to the
fast ion distribution.
In order to know the actual pitch of a particle observed by the NPA, it is necessary
to follow particles backward from the observation point to determine where they were
originally born.
It is possible to determine the location of the NPA/Beam intersection point based on
the NPA and beam geometry. The GYROXY code is an important tool for calculating the
pitch of an observed particle. (See Appendix A for an explanation of the GYROXY code)
Since the GYROXY code is toroidally symmetric, we can arrange its axes to line up
with the NPA pivot point. From measurements, the distance from the vessel center to
the NPA pivot point is known. Thus, the location of the NPA pivot point in Cartesian
coordinates is
N P A = (x = 0, y = RN )
(6.1)
where RN is the radius of the NPA pivot point. Also, it is important to note that this
analysis assumes the NPA is pointed in the midplane, and thus, all z values are implicitly
zero. When performing the full gyro-orbit calculation, the GYROXY code allows the z
value to be nonzero, but this analysis deals only with placing the particle, and thus it is
sufficient to assume that all z values are zero.
Since we also know the tangency radius of the NPA sightline, we can define an angle θ
(see figure 6.2) such that
123
Figure 6.2: A test particle is placed at the intersection of the NPA sightline with the neutral
beam path. We can determine this point by knowing the NPA pivot location, the NPA line of
sight tangency radius, and the beam injection radius. The beam in this diagram represents
a generalized beam, rather than the specific NSTX beam geometry.
sinθ =
rt
RN
(6.2)
cosθ =
l
,
RN
(6.3)
and
θ = sin−1 (
rt
),
RN
(6.4)
where rt is the tangency radius of the NPA line of sight and l is the distance to the tangency
point. Thus,
l=
q
2
RN
− rt2 .
124
(6.5)
The distance from the NPA pivot point to the beam intersection (l0 ) is determined from
engineering drawings. Since we know the distance to the intersection, we can determine
the (x, y) coordinates of the intersection, and thus, the starting point for the particle in the
simulation.
x0 = l0 sinθ = l0
rt
RN
l
= R N − l0
y0 = RN − l0 cosθ = RN − l0
RN
(6.6)
p
2
RN
− rt2
RN
(6.7)
6.2.1 Converting NPA to Cylindrical Coordinates
Although the calculations above are in Cartesian coordinates (x, y, z = 0), the GYROXY
code uses cylindrical coordinates (r, z, φ). Thus we must convert the coordinates of the
intersection of the NPA line of sight with the neutral beam (x0 , y0 , z = 0) into cylindrical
coordinates.
In cylindrical coordinates, the radius of the intersection is given by
q
R0 = x20 + y02
(6.8)
y0
)
x0
(6.9)
and the toroidal angle φ is given by
φ0 = tan−1 (
The tan−1 function returns a value in the range − π2 < φ0 ≤ π2 . The GYROXY code is
able to handle the possible negative toroidal angle, and it is necessary for determining the
velocity components, since the intersection will fall in the 1st or 4th quadrants. However,
this analysis applies only to the NSTX NPA geometry, and would have to be reworked for
other machines.
Since this simulation is designed to follow the particles that enter the NPA, the velocity
of these particles is pointed directly along the NPA sightline at the point of interest.
125
Overhead view of NSTX midplane
y
NPA sightline
Beam
/2
0
x
0
(x0,y0,z=0)
Figure 6.3: We are able to determine the velocity components knowing the intersection
position and the line of sight of the NPA
126
To determine the velocity in cylindrical coordinates, we define an angle α (see figure
6.3), which is given by
α=π−θ−(
π
π
− φ0 ) = − θ + φ 0
2
2
(6.10)
The radial velocity is then given by
vR = v0 cosα
(6.11)
vφ = −v0 sinα
(6.12)
and the toroidal velocity is given by
However, the GYROXY code does not use toroidal velocity, v φ . Instead it uses φ̇, which is
calculated by
φ̇ =
vφ
R
(6.13)
For particles entering the NPA, the z velocity is essentially zero.
(6.14)
vz = 0
6.3 Summary of Coordinate System Transformations
In terms of what is known,
R0 =
s
rt 2
) + (RN − l0
(l0
RN
φ0 = tan−1 (
RN − l0
√
l0 RrNt
127
p
2
RN
− rt2 2
)
RN
R2N −rt2
RN
)
(6.15)
(6.16)
(6.17)
z=0
rt
)
RN
(6.18)
π
− θ + φ0
2
(6.19)
θ = sin−1 (
α=
vR = v0 cosα
(6.20)
v0 sinα
R0
(6.21)
φ̇ =
vz = 0
(6.22)
6.4 NPA Measurements on NSTX
From figure 6.1, it is clear that as the tangency radius of the NPA line of sight is increased,
the pitch of the particles detected increases. However, the pitch of particles deposited at
the intersection decreases as the tangency of the NPA line of sight increases. Thus for
the majority of tangency radii, very little signal should be observed unless there is a large
perturbation to the fast ion population.
In order to determine the pitch observed by the NPA at a given tangency radius, a full
orbit calculation was performed.
Figure 6.4 shows the fast ion distribution as a function of energy and pitch, calculated
by TRANSP for NSTX neutral beam B for various NPA tangency radii (refer to figure 6.1
for beam identification). The NPA viewing pitch is shown as a black line in each frame.
The figure shows that as the NPA line-of-sight (LOS) shifts from a tangency radius of 5060 cm, the LOS intersection with the fast ion population shifts from one edge of the full
energy distribution to the other. When the NPA looks at a radius of 80 cm, the NPA cannot
observe the full energy particles.
If the high frequency CAE modes are driven by the fast ions, it is possible that the ions
are scattered in phase space so they are not visible by the NPA. If the modes are driven
128
Figure 6.4: Beam fast ion distribution calculated by TRANSP for source B injected at 90
keV of the and NPA line-of-sight intersection. The fast distribution is shown at various
tangency radii. Frame (a) shows the distribution at tangency radius of 50 cm. Frame (b)
shows the distribution at tangency radius of 60 cm. Frame (c) shows the distribution at 80
cm. The black lines indicate the pitch that the NPA can see at the beam intersection. Frame
(d) shows pitch visible for each beam at each tangency radius. The NPA line of sight only
intersects the full beam energy at a tangency radius of 50-60 cm.
129
Figure 6.5: NPA Spectrum from NSTX discharge 128600, with the NPA pointed at
Rtan=60 cm. The full energy particles are visible when the beam turns on at 420 ms.
However, no particles are visible at less than the full energy.
by the perpendicular energy of the fast ions, the distribution would shift to a higher pitch
(closer to
v||
v
= 1). The energy would also drop, and the particles would not be visible at
the full energy. If the NPA was pointed to see the full energy particles, the shift in pitch
would shift the particles out of the NPA’s acceptance angle, and particles less than the full
energy would not be visible. This is shown in figure 6.5.
If the NPA were able to view a whole range of pitch angles, the channels showing
the lower energies would fill up with particles as the high energy particles slowed through
collisions. The TRANSP code, which can simulate the NPA diagnostic, demonstrates this
effect. (See figure 6.6)
130
Figure 6.6: TRANSP simulations of the NPA diagnostic differ from NPA measurements.
The NPA only shows particles at the full energy, while the TRANSP simulation shows a
full slowing down spectrum. The lack of all energies in the NPA is likely due to the limited
pitch angle view, as calculated in this section. This demonstrates the inability of TRANSP
and the NPA to adequately analyze the fast ion distribution. The scale shows scaling for
each frame.
6.5 Conclusions from NPA Measurements
The NPA diagnostic on NSTX has limitations which make it unsuitable for analyzing power
balance in this experiment. First, with only one detection aperture, the line of sight of the
NPA is extremely limited. This prohibits sampling a large fraction of phase space at once.
Figure 6.7 shows an illustration of how the NPA aperture can limit the detection capability
of the NPA. If a particle decreases in energy, its new gyro-orbit may not direct it into the
NPA.
The acceptance angle of the NSTX NPA diagnostic is small due to the collimator. Thus,
if a particle’s gyro-orbit decreases by approximately the width of the NPA collimator slit,
the particle will no longer be detected. The gyro-radius of a fast particle is given by
ρ=
v⊥
ω
131
(6.23)
Figure 6.7: Fast ions could slow down out of the view of the NPA. As the particle slows
down, its gyro-radius shrinks, and the particle no longer orbits into the region viewed by
the NPA.
132
where v⊥ is the perpendicular energy
2
v =
2
v⊥
+
vk2
=
r
2E
m
(6.24)
and ω is the gyro-frequency. The magnetic field near the edge of the plasma is approximately 3.5 kG. The gyro-radii of beam ions in that region are
ρ90 ≈ 12.4 cm
ρ85 ≈ 12.0 cm
(6.25)
ρ80 ≈ 11.7 cm
where ρ90 , ρ85 and ρ80 , are the gyro-radii of deuterium ions with energies of 90, 85, and 80
keV respectively.
On the NSTX NPA, both collimator slits are approximately 4 mm wide and 1 mm thick.
If the NPA is set to view the full energy particles at a given location, particles at 85 keV
(the next NPA detection energy channel) might not be visible and 80 keV particles at the
same location would not be visible at all since the gyro-radius would be 7 mm smaller, and
the particle would not enter the acceptance angle of the NPA. The problem is even more
pronounced because the magnetic field pitch angle means that a change in gyro-radius also
changes the vertical position. This means that if the NPA is set to view a particle with a
given pitch and energy, even a small reduction in energy would likely cause the particle to
no longer orbit in the acceptance region.
The second limitation is that the TRANSP code is unable to adequately simulate the
NPA. The TRANSP code, which averages the distribution toroidally, does not include geometric considerations of the NPA line of sight and the neutral beam deposition. Since this
geometry is important in the NPA measurement, the TRANSP simulation fails to match the
measurement. Although this is not strictly a limitation of the NPA, it makes the analysis of
the NPA and the fast ion distribution more difficult.
133
Third, the NPA design limits the area of phase space that can be viewed. The position of
the NPA pivot point is fixed relative to the neutral beams. Since these beams are responsible
for both the fast ions and the charge exchange neutrals, the geometry between the beam and
the NPA dictate which particles will be visible. Since the NPA cannot be moved around the
machine, it cannot view all of the fast ion distribution.
6.6 Overcoming NPA Limitations
The NPA has a limited viewing angle. As a result it is incapable of simultaneously measuring the entire fast ion distribution. This is particularly true since the fast ions detected by
the NPA are neutralized only in the region of the beam intersection.
One possible way to overcome this problem is using solid state detectors. Solid state
detectors are small and relatively inexpensive. However, they have some limitations. First,
the energy resolution of an SSNPA is not as good as the energy resolution of the NPA.
Also, solid state detectors do not discriminate in pitch angle.
Modifications to the SSNPA could be made to make it discriminate in pitch angle.
If collimators were attached to the NPA, it might be possible to measure pitch. Also,
since they are less than 10% of the cost of the NPA, it is possible to construct an array
of SSNPA, which would simultaneously measure a much larger area of phase space, and
could presumably provide a complete view of the fast ion distribution, which would greatly
help the power balance study.
134
Chapter 7
Summary and Future Work
7.1 Summary
Ion power balance is a topic of critical importance in fusion plasmas. The rate coefficients
for thermal fusion reactions are functions of the ion temperature only, while power loss is
a function of ion and electron temperatures. For ignition, it would be advantageous to have
a mechanism that would preferentially heat the ions. Such a mechanism would reduce the
total power required for ignition.
TRANSP calculations show that heating is occurring above the amount normally predicted by classical calculations in some beam heated discharges in NSTX. This excess ion
heating occurred over a series of similar plasma discharges. The heating coincides with
the turn on of the third neutral beam source. The heating occurred in the outer edge of the
plasma, peaking at R=134 cm. The total excess heating to the plasma sometimes exceeded
500 kW, or nearly 10% of the total injected neutral beam power.
Previous studies have looked at the possibility of heating by high frequency CAE
modes. [17, 62]. In NSTX, these modes increase in strength when the third beam turns
on. The radial profile of the mode amplitude calculated from the eigenmode equation and
fit to figure 4.9 peaks near 140 cm. The high frequency modes increase in strength when
135
the third beam turns on, which coincides with the appearance of excess heating.
The CAEs coincide temporally with the requirement for excess heating. They also
have a similar shape and peak near the peak of calculated heating. In order to confirm
that the CAEs are responsible for the calculated excess heating, full gyro-orbit simulations
were performed. When the modes were excluded the particles did not heat. When the
modes were included, the particle energy increased. This indicates that the CAEs could
potentially be the source of the excess heating required to satisfy power balance.
Using a mode spectrum obtained by taking a Fourier transform of the Mirnov coil
signal on NSTX, the total heating integrated over a Maxwellian distribution of thermal
ions is approximately 1.6 MW. This exceeds the excess heating calculated by TRANSP.
However, this discrepancy can by explained by uncertainties in determining the amplitude
of the perturbation. Thus, even though the magnitude of the possible heating exceeds the
value calculated by TRANSP, it represents a clear demonstration that CAE modes could be
responsible for the excess heating.
The Edge Neutral Density Diagnostic was designed and implemented on NSTX. The
edge neutral density population was measured to be ≤ 1 × 10 11 cm−3 . The neutral density
was highest near the outer wall of the vacuum chamber, decreasing toward the separatrix.
This information is useful for analyzing the neutral beam input power and charge exchange
losses. Calculations including the neutral density measured by the ENDD showed very
little loss of beam particles due to neutral collisions.
7.2 Future Work
In order to more accurately quantify the heating by the modes, it is necessary to get a more
accurate estimate of the mode profile and amplitude. This is important for calculating the
correct amount of heating, as the mode amplitude represents the largest uncertainty in the
heating calculations above. An ideal method for doing this would be an array of optical
136
interferometers. Using two sets of interferometers, one arranged with poloidal lines-ofsight, and the other arranged toroidally around the plasma, it would be possible to obtain
both the toroidal and poloidal mode profiles to confirm the accuracy of the CAE theory. It
would also be possible to obtain a direct measurement of the mode amplitudes, eliminating
a significant source of uncertainty in the heating calculation.
With direct measurements of the CAE amplitude and mode numbers, the full orbit
calculation could then be performed to investigate whether the modes are responsible for
the heating calculated by TRANSP.
It would also be worthwhile to examine the fast ion population in more detail. The
Neutral Particle Analyzer (NPA) on NSTX is not capable of examining the entire phase
space of fast ions. An alternative to using the NPA would be to use a solid-state NPA
(SSNPA). These detectors are already in use on NSTX. However, they do not provide as
good energy resolution. More importantly, they do not provide pitch angle information.
One possible way to use the SSNPA to obtain pitch information is to use collimators to
only permit a certain pitch to enter the SSNPA. (see figure 7.1)
This design has several advantages. The collimators could be designed to block out
the majority of unwanted radiation such as x-rays and energetic ions. The cost of the
SSNPAs is low compared to the cost of the NPA. Also, by using several SSNPA‘s in an
array looking at different pitches, it would be possible to simultaneously see more of the
fast ion distribution, which would, in turn, provide a more complete picture of the power
balance.
137
Figure 7.1: Schematic of how collimators could be used with the SSNPA to measure pitch.
An array of these SSNPAs could simultaneously provide information at a variety of pitches.
138
Appendix A
Magnetic Ripple in NSTX
A.1 Overview of Magnetic Ripple
Perturbations and variations in the magnetic field that destroy perfect toroidal magnetic
symmetry can lead to particle losses and contribute to the Ploss term in equation (2.28).
One common magnetic asymmetry in magnetic confinement devices is magnetic ripple. [5]
Magnetic ripple arises from the discrete number of coils used to produce the toroidal field.
To understand magnetic ripple, it is instructive to consider the field arising from the
coils of a solenoid. Assuming an infinite number of coils, the magnetic field through the
solenoid will be straight and uniform. (see Figure A.1)
If some coils are removed, the magnetic field is stronger near the coils and weaker farther
away, resulting in a nonuniform field (see Figure A.2)
A similar effect occurs in toroidal geometry. Rather than being toroidally uniform, the
magnetic field lines have maxima at their closest points to a coil, and minima in between
coils. The result is the creation small magnetic wells that could potentially trap particles.
Since ions always drift in the same direction due to the ∇B drift (the direction of the
current in the center rod of the toroidal field coil), trapped particles that are caught in these
139
Figure A.1: If a solenoid has infinite, closely spaced coils, the magnetic field will be straight
and uniform.
Figure A.2: If some of the coils of the solenoid are removed, the field is weaker farther
from the coils and stronger near the coils, creating a nonuniform magnetic field.
140
magnetic wells at the lower end of their banana orbits will rapidly drift out of the closed
flux region. These rapid loss orbits are called “super banana” orbits. (see Figures A.3,
A.4) Compounding the problem, all particles in that region of phase space will eventually
be lost to the super banana orbits unless they undergo collisions to scatter them out of the
loss cone. These super banana orbits result in an increase to the diffusion coefficient at low
collisionality. (see Figure A.5)
Even if collisions prevent super banana direct losses, ripple can still increase diffusion
by stochastically altering the bounce point of banana-trapped particles. As the particles
reach their bounce points, they will randomly bounce in areas of slightly higher or lower
field based on the toroidal location. (see Figure A.6) Since the flux surface is defined by
the bounce points, a particle that bounces at a maximum will shift slightly out in flux space.
This results in a ”random walk” of the nominal flux surface. For a complete analysis, see
reference [5].
A.2 Ripple Loss in NSTX
The spherical torus shape of NSTX complicates the analysis of ripple loss. There are two
reasons to believe that magnetic ripple may represent a significant loss mechanism for fast
ions in NSTX. First, NSTX has only 12 toroidal field (TF) coils. This is less than some
tokamaks such as the Joint European Torus (JET), which has 32 toroidal field coils. Second,
the low aspect ratio design of the spherical torus is unique in that the center stack of the
device is very small. Thus, the distance between coils is significantly larger on the low field
side than on the high field side.
However, the design of NSTX means that the outer return conductors of the TF coils
are far from the plasma, which serves to mitigate the ripple. In order to determine the ripple
loss effect, the vacuum magnetic field ripple was calculated based on the full structure of
141
Figure A.3: Ripples in the magnetic field can trap particles. If an energetic particle is born
inside a magnetic well, it will leave the machine almost instantly. Reproduced from [5].
142
Figure A.4: Super banana orbits cause high energy particles to leave the machine almost
instantly. Since particles always drift in the same direction due to the ∇B drift, the particle
drifts rapidly out of the machine. Reproduced from [5].
143
Figure A.5: Super banana orbits can alter the diffusion coefficient. At high collision frequency, particles are pitch angle scattered out of the well before they can diffuse significantly. Reproduced from [5].
144
Figure A.6: Particles can shift out in flux space by shifting bounce location due to ripple
effects. Frame (a) shows a particles ordinary banana orbit and nominal flux surface (dotted
line). Since the bounce point is determined only by |B|, a particle that encounters a strong
ripple field will bounce early, shown in frame (b). The new bounce point defines a new flux
surface for the particle (frame (c)). The particle’s new banana orbit is farther out in radius
(frame (d)).
145
the coils. The coils were approximated as a series of short, linear segments and a BiotSavart integration was performed to calculate the magnetic field ripple. The ripple was
approximated by calculating the numerical integration of the Biot-Savart relation at a point
in the plane of an outer return leg and at a point that was halfway between two legs. The
difference was taken as the peak-to-peak amplitude of the magnetic ripple. This ripple is
approximated as varying sinusoidally since the major Fourier component of the ripple is
used which has the dominant effect on particle motion.
When determining ripple loss from a toroidal confinement device, the calculations are
generally performed using the guiding center approximation. In the guiding center formalism, the Hamiltonian, and hence the guiding center equations of motion, depend only on
~ The Lagrangian is given by ( [5])
|B|, rather than B.
~ + ρk B)
~ · ~v + µξ˙ − H
L = (A
(A.1)
~ is the magnetic vector potential, ρk (= vk /B) is the normalized parallel velocity,
where A
µ(≡ mv⊥ 2 /2B) is the magnetic moment, ξ is the gyro-phase, and H, the Hamiltonian, is
given by
H = ρk 2 B 2 + µB + Φ
(A.2)
~ Continuing the analysis in reference [5]
where B in equation A.2 is |B|, rather than B.
leads to equations of motion for ψp (the poloidal flux function of the guiding center) and
ρk , the particle position and velocity respectively:
∂B
I
∂B
I ∂Φ
g ∂Φ
g
+ (µ + ρ2k B)
+
−
ψ˙p = − (µ + ρ2k B)
D
∂θ
D
∂ζ
D ∂ζ
D ∂θ
(1 − ρk g 0 )(µ + ρ2k B) ∂B
(1 − ρk g 0 ) ∂Φ
D
∂θ
D
∂θ
0
2
(q + ρk I 0 ) ∂Φ (q + ρk I )(µ + ρk B) ∂B
−
−
D
∂ζ
D
∂ζ
ρ˙k = −
146
(A.3)
−
(A.4)
where g, D, and I are defined by White [5], and are functions only of ψ. For the purposes
of this analysis, it is sufficient that equations A.3 and A.4 are functions of |B| rather than
~
B.
Since equations A.3 and A.4 are only functions of |B|, any perturbations in B can be
modeled only as perturbations to the field amplitude |δB|.
A.3 Implementing Ripple in ORBIT
A.3.1 Analytic Approximation to |δB|
The ORBIT code is a guiding center code written by R.B. White [49]. Rather than follow
the particles through the full gyro-orbits, the position of the particles are approximated by
the guiding center location. Particles are progressed in time using equations A.4 and A.3
above.
In ORBIT, an analytic form is used to perform ripple calculations. The ripple in the
machine is calculated then fit to the equation
δB
= d0 e
B0
√
(x−x0 )2 +bz 2
w
(A.5)
where x, z are the cylindrical coordinates, and d0 , x0 , b, and w are free parameters used to
optimize the fit. Each free parameter is given an initial guess and a range. The parameters
are then randomly generated and compared with the best guess. This process is repeated
until optimal parameters are obtained which give the minimum error. It is important to
check that the values all fall within the parameter range rather than at the edge of the range.
A value at the edge of the range would indicate that the optimal solution might be beyond
the range specified and the calculation should be performed again using a larger range.
147
A.3.2 Fortran PsuedoCode
The following is Pseudo FORTRAN code for optimizing d0 , x0 , b, and w.
dmin=1e6
re = calculated δB
!
Initial guesses d1, x1, b1, w1
!
Ranges dr, xr, br, wr
Do 10 n=1, ntry
del=0
d0=d1+2*dr*(ranx() - 0.5)
x0=x1+2*xr*(ranx() - 0.5)
b =b1+2*br*(ranx() - 0.5)
w =w1+2*wr*(ranx() - 0.5)
do 20 k=1, km
do 20 j=1, jm
rt=d0*exp(sqrt((x-x0)**2+b*z**2)/w)
del=del+(re(k,j)-rt)**2
20 continue
!
Now del is the total error
if del.lt.dmin then save del, d0, x0, b,w
10 continue
!The final d0,x0,b, and w are the solution
A.4 Results of Guiding Center Calculations
Magnetic ripple in NSTX is small. The calculations from section A.3.2 give values for d 0 ,
x0 , b, and w as
d0=1.8 × 10−5
148
Figure A.7: Magnetic ripple magnitude as a function of radius for guiding center calculations. ORBIT only requires the ripple magnitude since the guiding center representation
only relies on magnetic field magnitude rather than individual components.
x0=37.0 cm
b =0.0374
w =18.5 cm
Figure A.7 shows the magnetic ripple magnitude as a function of radius. Even at the edge
of the plasma, the magnetic ripple is only 0.5% of the field at the axis. The magnitude of
magnetic ripple is added to the total field, and the derivatives with respect to the various
directions are calculated using a spline. The particles were then deposited on a flux surface
and allowed to move forward in time. Guiding center calculations using 5000 particles at
each energy and flux surface show that including magnetic ripple does not siginificantly
increase the number of lost particles. Figures A.8 and A.9 show the ratio of lost particles
from simulations with and without magnetic ripple. Simulations with magnetic ripple did
not have significantly greater loss than simulations without ripple.
149
Figure A.8: Ratio of ripple to non-ripple lost particles as a function of energy for guiding
center calculations. The number of lost particles does not change significantly when ripple
is added. (A ratio of 1 would represent exactly the same amount of loss)
150
Figure A.9: Ratio of ripple to non-ripple simulations as a function of normalized minor
radius. The particles were deposited on various flux surfaces with and without magnetic
ripple. In each simulation, the particles were given a uniform energy up to 2 keV, and the
results of the various energies were summed.
151
A.5 Guiding Center vs. Full Lorentz
For NSTX, it was not originally certain that using the guiding center formalism is permissible in calculating ripple loss. Even when the magnitude of the magnetic ripple is known,
the size of the orbits, particularly for high energy beam particles, is large enough that the
particle whose guiding center would normally be ripple trapped might escape the trapping
by sampling a field larger than the ripple well. Alternatively, it could be argued that the
larger sampling of field makes it more likely that a trapped particle would undergo the
random walk effect of the bounce point adjustments.
To investigate this, simulations were also run using a full Lorentz calculation of particle
positions. Whereas the ORBIT code uses the guiding center formalism to calculate particle
motion, the GYROXY code is a full Lorentz calculation, where each particle individually
satisfies the Lorentz force law
~ + ~v × B).
~
F~ = m~a = q(E
(A.6)
For these calculations, we assume that the electric field is zero, leaving
~
m~a = q(~v × B)
(A.7)
Since the full Lorentz treatment requires the vector magnetic field, care must be taken to
~ = 0. This is necessary to ensure energy conservation. If ∇ · B
~ 6= 0, then
ensure that ∇ · B
the magnetic field might do work on the particles, leading to erroneous results.
In principle, the Biot-Savart integration would provide a divergenceless magnetic field.
However, numerical errors as well as approximations such as spline interpolations could
cause the divergence to be non-zero at a particle location. To avoid this difficulty, it is
necessary to solve for two components of the magnetic field and then calculate the third
term to maintain the divergenceless property. To calculate the magnetic field perturbation,
152
we let
~ T ot = B
~ 0 + δB
~
B
(A.8)
~ 0 is the equilibrium field, and δ B
~ is the perturbation caused by magnetic ripple.
where B
Taking the divergence leads to
~ T ot = ∇ · (B
~ 0 + δ B)
~ =0
∇·B
(A.9)
~ T ot = ∇ · B
~ 0 + ∇ · δB
~ =0
∇·B
(A.10)
The equilibrium field is divergenceless by construction, so we are left with
~ =0
∇ · δB
(A.11)
In cylindrical coordinates, (r, z, φ) we can make use of the sinusoidal variation in the per~ as
turbed field and write δ B
~ = [δB1x (x, z)x̂ + δB1z (x, z)ẑ + δB1φ (x, z)φ̂] sin (N φ) +
δB
[δB2x (x, z)x̂ + δB2z (x, z)ẑ + δB2φ (x, z)φ̂] cos (N φ)
(A.12)
where N is the number of toroidal field coils (NSTX=12). Taking the divergence gives
~ = 1 ∂ (xδB1x ) sin (N φ) + 1 ∂ (xδB2x ) cos (N φ) +
∇ · δB
x ∂x
x ∂x
1 ∂
(δB1φ sin (N φ) + δB2φ cos (N φ)) +
x ∂φ
∂
∂
δB1z sin (N φ) + δB2z cos (N φ)
∂z
∂z
(A.13)
Setting this equal to zero and making use of the fact that the coefficients are functions
only of (x, z), we can collect the sine and cosine terms into separate equations, since these
153
equations must hold regardless of toroidal angle φ. For the sine terms, we get
1 ∂
N
∂δB1z
(xδB1x ) − δB2φ +
x ∂x
x
∂z
(A.14)
∂δB1x N
∂δB1z
1
δB1x +
− δB2φ +
x
∂x
x
∂z
(A.15)
0=
which simplifies to
0=
Similarly, for the cosine terms, we get
0=
∂δB2x N
∂δB2z
1
δB2x +
+ δB1φ +
x
∂x
x
∂z
(A.16)
Since φ = 0 is an arbitrary location in the calculation, we only need either equation A.15
or equation A.16. In reality, the two equations are equivalent, with the sign change coming
as a result of the phase shift between sine and cosine. Arbitrarily, we keep equation A.15
and set the coefficients from the other equation to zero.
Solving for δBφ gives
δBφ =
x 1
∂δBx ∂δBz δBx +
+
,
N x
∂x
∂z
(A.17)
where the subscripts 1 and 2 have been dropped. Setting the coefficients of equation A.16
to zero also simplifies equation A.12 to
~ = [δBx x̂ + δBz ẑ] sin (N φ) + δBφ φ̂ cos (N φ).
δB
(A.18)
δBx and δBz are determined by the calculation of the ripple and δBφ is calculated from
equation A.17.
154
Figure A.10: Magnetic ripple calculations based on Biot-Savart integration. The n=12
mode is clearly visible in the Br and Bφ fields, but not in the Bz plot, since these calculations were done at z=0. The n=1 mode (visible in the Bz plot), is the result of an asymmetry
in the final link connecting to the power supply.
A.6 Implementing Ripple in GYROXY
In order to implement magnetic ripple into the GYROXY code, the ripple is first calculated
by the Biot-Savart integrator in the x,z plane for various toroidal positions. The magnitude
of the ripple is obtained by taking the difference between the maximum and minimum
ripple as the peak-to-peak amplitude of the sinusoidal variation. This amplitude is then
normalized to the magnetic field on axis. This step is necessary since different magnetic
fields in different discharges would have different ripple magnitudes, but the normalized
value would not change. In other words, a field twice as strong would have twice the ripple,
but it would still only represent a certain fraction of the field, which would not change from
shot to shot.
The amplitude of the ripple is calculated at each point. The derivatives are calculated
in the code using a spline. The δBφ term is then calculated from equation A.17. The full
155
Figure A.11: Ratio of ripple to non-ripple particle loss versus energy for full orbit calculations. The number of lost particles does not increase when ripple is added, indicating that
ripple is not a significant factor even in the full orbit calculations.
ripple is calculated from equation A.18.
A.7 Results of the Ripple Calculation in GYROXY
As with the guiding center calculations, adding ripple to the full orbit calculations did not
significantly increase the number of particles lost. (See figures A.11 and A.12) This is
true across all energies and all radii. The low ripple is insufficient to induce loss through
stochastic means, and the energy of the particles and the gyro-radii are too large to be
caught in the magnetic well of a “super banana” orbit. While the particles would eventually
bounce out from stochastic means, the required time would exceed the time required for
these energetic particles to deposit their energy onto the thermal particles.
156
Figure A.12: Ratio of ripple to non-ripple particle loss versus normalized minor radius for
full orbit calculations. Even at the edge, the ripple does not induce particle loss.
157
A.8 Conclusions
Magnetic ripple does not significantly alter the fast ion population in NSTX. It causes very
little fast particle loss. This is true in both guiding center and full orbit calculations. In
both cases, the number of lost particles did not increase significantly when the ripple was
included in the calculations.
158
Appendix B
Edge Neutral Density Diagnostic Design
and Calibration
B.1 ENDD Setup
The Edge Neutral Density Diagnostic (ENDD) uses a CCD camera to measure D β emission, using a collisional-radiative model along with electron density and temperature information from the Thomson Scattering diagnostic to infer the neutral particle density profile
on the National Spherical Torus Experiment (NSTX).
The ENDD consists of a CCD camera looking through a window on the midplane of
Bay I on NSTX. The camera looks into a mirror, welded onto the flange of the port. The
mirror is angled to provide a tangential line of sight to the plasma. (see Figures B.1 and
B.2) The camera focus is adjusted so that the tangential edge of the plasma is in focus. The
lens is covered with a filter that transmits in the deuterium Balmer-series Dβ wavelength
(486 nm). The filter used for this diagnostic has a full width at half maximum (FWHM) of
1.5nm. A movable shutter is attached to the flange and to a pneumatic motor. When the
shutter is closed, it protects the mirror from lithium coating. (see Figure B.3) The flange
was attached to the vacuum vessel. The shutter is controlled automatically and closed to
159
Figure B.1: Overhead schematic showing the position of the camera looking into the
plasma. The camera looks through a mirror tangentially into the plasma. Once the image is Abel-inverted, a radial profile of the emission can be obtained for the field of view.
protect the mirror and port window from coatings.
The mirror allows the camera to look through a notch in the beam armor. Figure B.4
shows a photograph of the mirror and shutter as seen from the inside of the vessel.
B.2 Calibrating the Camera
An individual pixel on the camera receives light emitted by the plasma. The counts that the
camera registers depend on several factors. The counts registered by pixel p ij is
pij = E
Ω
ηθ ij τ
4π
160
(B.1)
Figure B.2: Photograph showing the flange with the mirror welded to stick into the vacuum
vessel. Also shown is the shutter, driven by a pneumatic motor.
Figure B.3: Photograph showing the shutter in the closed position. The pneumatic motor
is shown in the lower part of the photograph. The valve is automatically controlled and is
shut during lithium coatings.
161
Figure B.4: Photograph of the in-vessel view of the mirror mount. The shutter is visible in
the photograph. When the shutter is open, it does not block the view of the plasma. The
camera can be seen behind the shutter.
where E represents the emissivity of the plasma (or calibration source),
Ω
4π
represents the
solid angle subtended by the detector aperture, η represents the geometric effects including
the camera lens, θij represents the sensitivity of the pixel, τ represents the filter effects
such as attenuation, and the ij refer to the x,y pixel of the CCD chip. The CCD chip had a
resolution of 128×127 pixels. It is worth noting that the emissivity is line integrated along
the sightline through the plasma.
Since η represents the geometric effects of the port and mirror, the camera must be
calibrated in situ. Such a calibration is valid only while the camera and mirror setup has
not changed. An example of a change that would invalidate the calibration is the coating
of the window with lithium or other coating during plasma operation, which would reduce
the transmittance of the window. Also, while, η, τ , and θij could be measured separately, it
is not necessary, since only the ratio of terms is required. When calibrating the camera, η
and τ cancel, leaving only the desired term.
To calibrate the camera, a “white plate” is used. The white plate is illuminated by two
250 Watt halogen bulbs, and is assumed to emit uniformly intense light at all angles.
162
The absolute emissivity of a data frame is given by
Eij =
pijdata
pijwhiteplate
∗ C ∗ E abs ∗ ∆λ
(B.2)
where C represents the relative calibration of the white plate, E abs is the absolute emission
of the calibrated source in
photons
sec cm2 sr Å
, and ∆λ is the equivalent width of the filter in Å. C
is given by
C=
Nabs
(B.3)
Nwhiteplate
where Nabs is the average pixel count of a region of interest from the calibrated source, and
Nwhiteplate is the average pixel count of the white plate over that same region of interest.
Substituting equation (B.3) into equation (B.2), we get
Eij =
pijdata
pijwhiteplate
∗
Nabs
Nwhiteplate
∗ E abs ∗ ∆λ
(B.4)
For simplicity, it is useful to define a quantity Mij such that
Mij =
1
pijwhiteplate
∗
Nabs
Nwhiteplate
∗ E abs ∗ ∆λ
(B.5)
This simplifies equation (B.4) to
(B.6)
Eij = pijdata ∗ Mij
This is useful because the matrix M , which has units of
Emissivity
,
count
contains all of the
calibration information and does not change during operation. This matrix can be stored in
a database, rather than recalculated for each pixel in each frame.
163
B.3 Data Inversion
In order to perform the analysis, the image from the camera must be inverted to obtain a
profile. Each pixel on the camera measures the line integrated emission from the plasma–
that is, each pixel receives emission from the whole line of sight through the plasma.
Since only the midplane of the plasma is used in this analysis, the toroidal geometry of
NSTX is approximated as a cylindrical system, where R is the major radius of the machine,
z is the vertical axis of the machine, and φ is measured toroidally around the machine.
B.3.1 Abel Inversion
The Abel inversion (also called an inverse Abel transform) was derived by N.H. Abel as
a tool for analysis in spherically or cylindrically symmetric systems. [36] The inversion
analysis assumes that the relevant parameter is solely a function of radius.
f = f (r)
(B.7)
The line integrated signal F (y) (e.g. emission from the plasma) at a line of sight with
tangential radius y is given by
F (y) = 2
Z
∞
y
f (r)r dr
p
.
r2 − y 2
(B.8)
To determine the function f (r), one can invert equation (B.8) to obtain
1
f (r) = −
π
Z
∞
r
dy
dF
p
.
dy
y2 − r2
164
(B.9)
Figure B.5: The Abel inversion assumes that the parameter to calculate is a function of
radius only. The function can be inverted assuming that the line of sight extends out to
where the function is zero.
165
B.3.2 Onion Layer Inversion
In practice, it is not usually possible to perform the integration required for a complete
inversion, since F (y) is rarely known analytically. For the example of plasma emission, a
detector would require infinite spatial resolution. Instead, it is useful to divide the plasma
into regions defined by the resolution of the detector. For example, if the detector is a camera, then the resolution is limited by the dimensions of the pixels. Each pixel corresponds
to a line of sight with a certain tangential radius. This creates a series of concentric regions
or layers that give rise to the inversion’s name.
For the analysis, the plasma is divided into concentric annular regions. (See figure
B.6) Each ring of plasma is assumed to have a fixed emission intensity. The integrals in
equations (B.8) and (B.9) then turn into sums over the products of the emission intensity
times the length of the line of sight through that zone. For the example in figure (B.6), the
amount of emission recorded by the plasma in the pixel covering only the outermost zone
(with central radius R) is
r
a
F (1) = 2E1 (R + )2 − R2
2
(B.10)
where a is the radial thickness of a given zone, and E1 is the emissivity of the outermost
zone. Using the same reasoning, the emission at the second pixel would be
F (2) = 2E1
p
(R +
a 2
)
2
− (R −
a)2
p
+2E2 (R − a2 )2 − (R − a)2
166
p
− (R − a2 )2 − (R − a)2
(B.11)
and the emission reaching the nth pixel would be given by
p
(R + a2 )2 − (R − (n − 1)a)2
p
a 2
2
− (R − 2 ) − (R − (n − 1)a)
q
(2n−3)a 2
2
(R − 2 ) − (R − (n − 1)a)
+ . . . + 2En
F (n) = 2E1
(B.12)
If there are N such zones, then equation (B.8) becomes
F (N ) = 2
N
X
f (R(i))
i=1
p
R(i)2 − R(i − 1)2
(B.13)
where R(i) is the tangential radius at position i, and i = 1 corresponds to the outermost
radius.
Equation B.13 can be rewritten as a matrix equation:
y = Ax,
(B.14)
where A is a sub-diagonal matrix containing the radial elements of the sum in equation
B.13 (the terms under the square root), y is an array containing the measured signal from the
camera F (N ), and x is an array containing the emissivity of each zone f (R(i)). The inverse
of the matrix (A−1 ) can then be numerically calculated. The array x is then calculated as
x = A−1 y,
giving the radial profile of the emissivity.
167
(B.15)
Figure B.6: For a practical Onion Layer inversion, the function is divided into distinct
regions. The signal at a pixel on the camera is then the sum of each line of sight times the
emission of each zone.
168
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