PFC/RR-85-20
DOE/ET-51013-160
UC20
AN EXPERIMENTAL STUDY OF VISIBLE AND ULTRAVIOLET
IMPURITY EMISSION FROM THE ALCATOR C TOKAMAK
Juan Carlos Moreno
October, 1985
Plasma Fusion Center
Massachusetts Institute of Technology
02139
Cambridge, MA
This work was supported by the U.S. Department of Energy Contract
No. DE-AC02-78ET51013. Reproduction, translation, publication, use
and disposal, in whole or in part by or for the United States government is permitted.
AN EXPERIMENTAL STUDY OF VISIBLE
AND ULTRAVIOLET IMPURITY EMISSION FROM
THE ALCATOR C TOKAMAK
by
Juan Carlos Moreno
B.S., University of Maryland
(1978)
SUBMITTED TO THE DEPARTMENT OF
PHYSICS IN PARTIAL
FULFILLMENT OF THE
REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY IN PHYSICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 1985
©Massachusetts Institute of Technology, 1985
Signature of Author
Department of Physics
August 26, 1985
Certified by
Dr. Earl S. Marmar
Thesis Supervisor
Accepted by
Professor George F. Koster
Chairman. Graduate Committee
- 1-
AN EXPERIMENTAL STUDY OF VISIBLE
AND ULTRAVIOLET IMPURITY EMISSION FROM
THE ALCATOR C TOKAMAK
by
Juan Carlos Moreno
Submitted to the Department of Physics
on August 26, 1985 in partial fulfillment of the
requirements for the Degree of Doctor of Philosophy in
Physics
Abstract
Densities of carbon, oxygen, and silicon in Alcator C tokamak plasmas have
been computed from spectroscopic measurements of the absolute brightnesses of
visible and ultraviolet emission lines in combination with a one dimensional numerical calculation which models the charge state and emissivity profiles. Profiles of all
the charge states of a particular impurity were calculated by utilizing independent
measurements of plasma density and temperature and solving the coupled system
of transport and rate equations connecting the ionization states. These profiles
were then used to calculate emissivity and brightness profiles by solving the matrix
equation relating the level populations through atomic processes such as electron
impact excitation, de-excitation, spontaneous emission and cascades from upper
levels. Good agreement was found between predicted impurity line brightnesses
and experimentally measured brightnesses of different charge states.
Three different types of limiter materials, molybdenum, graphite and SiC
coated graphite have been used on Alcator C. It was determined that the principal impurities in the plasma, under most conditions, depends upon the type of
limiter being used. However, the sources of the impurities are both the wall and the
limiters, since it was observed that the wall becomes coated with limiter material
due to plasma discharges.
A significant influx of impurities directly from the limiters, causing an increase
in the effective ion charge, was often seen during the application of lower hybrid
RF power to the plasma. This RF induced influx of impurities exhibited a toroidal
asymmetry for the low ionization states. Numerical simulations of toroidal transport
of impurities were consistent with these observations.
Thesis supervisor: Dr. Earl S. Marmar
Title: Principal Research Scientist
- 2 -
To my parents
Manuel A. Moreno
Gladys Martinez de Moreno
- 3-
Acknowledgments
The experimental and computational work for this thesis would not have been
possible without the support of the entire Alcator group. There are several people
I would like to. thank in particular.
Alcator C was kept running reliably and well through the efforts of Frank Silva,
Charlie Park, Dave Gwinn and the technicians and machine operitors who worked
at all hours of the day and night. I would like to thank Mark Iverson and the
other members of the machine shop who did most of the machining and welding for
my experiments. Bob Childs, Harold Shriber and Ed Sudenfield assisted on all the
vacuum work that was required. Laurie Pfeifer and the secretarial staff were always
available to help me with my graduate and thesis work.
I would particularly like to thank my advisor Earl Marmar for continually
encouraging me and helping me with my research. I would also like to thank the
other physicists in the plasma-wall interactions group, John Rice, Jim Terry and
Bruce Lipschultz, for supporting my work and being there to answer any questions
I might have. A final thanks to the other graduate students from Alcator, past
and present, whose friendship and assistance enabled me to complete my graduate
studies.
- 4 -
TABLE OF CONTENTS
Chapter I
Introduction..................................................
1) M otivation for W ork ...................................................
7
8
2) Alcator C Tokam ak ...................................................
13
3) Thesis O utline ........................................................
15
Chapter II
Experimental Set-up .......................................
17
1) Visible - UV Spectrograph ............................................
17
2) UV M onochromator...................................................
20
3) Edge Plasma Diagnostics..............................................
25
4) Additional Diagnostics................................................
29
Chapter III
Impurity Modeling .........................................
33
1) Plasm a Conditions....................................................
33
2) Models that Describe Impurity Distributions ..........................
36
3) Neoclassical Impurity Transport.......................................
39
4) 1-D Radial Transport M odel ..........................................
41
5) Population of Energy Levels...........................................
47
Chapter IV
Im purity Studies ...........................................
58
1) Impurity Density Calculations.........................................
58
2) Zeg M easurem ent .....................................................
63
3) Lim iter-Plasma Interaction............................................
66
4) Impurity Influx during Ohmic Discharges..............................
71
5) R adiated Power.......................................................
78
6) Impurity Injection during Lower Hybrid Experiments ..................
87
- 5 -
7) Error Analysis ........................................................
Chapter V
Toroidal Transport of Impurities .........................
99
103
1) Measurements of Toroidal Asymmetries ..............................
103
2) 1-D Toroidal Transport Model .......................................
104
3) 2-D Radial and Toroidal Transport Model............................
109
4) Comparison between Theory and Experiment.........................
115
Chapter VI
Conclusion ..................................................
125
1) Sum m ary of Results .................................................
125
2) Suggestions for Future Work .........................................
127
R eferences ...............................................................
129
Appendix A
Solving Partial Differential Equations .................
134
Appendix B
Explicit M ethod ..........................................
135
Appendix C
Im plicit M ethod ..........................................
135
Appendix D
Steady State Solutions ..................................
136
-6-
Chapter I
INTRODUCTION
The density and spatial distribution of impurities in a tokamak discharge can
greatly influence the characteristics of the plasma. Their importance is especially
evident in experiments connected with controlled thermonuclear fusion, since even a
modest amount of impurities can affect energy confinement, particularly the heavier
impurities which do not become fully stripped and radiate mostly from the central
region of the plasma. Too large a concentration of impurities can cause disruptions
or result in considerable radiative power loss [1]. For these reasons it is essential
that plasmas in reactor regimes be kept relatively clean of impurities. However,
small amounts of light impurities can be tolerated, and may in fact be beneficial,
since they radiate from the edge region of the plasma and should therefore help
keep the temperature at the edge low.
This in turn can reduce sputtering from
edge structures [2).
Spectroscopic instruments, operating from the visible to the x-ray region of
the spectrum, are typically used to study impurities in tokamak plasmas [3-5]. This
thesis will deal mainly with low Z impurities such as C, N, 0 and Si which radiate
primarily in the visible and vacuum ultraviolet. The principal instrument used in
the experiment is a spectrograph covering the wavelength region from 2000
8000
A. Data from
a vacuum UV monochromator (300
A to
1400
A to
A) and Langmuir
probes were also used extensively.
Experimental measurements that were made for this study consisted initially
of a survey of impurity emission lines using the visible - UV spectrograph with a
film attachment. The principal measurements, comprising the bulk of the thesis,
- 7 -
-
were time histories of absolute brightnesses of impurity lines. Various aspects of
the plasma and its interaction with material surfaces were studied. Some of these
aspects which influenced impurity concentrations and transport include: the effect
of different limiter materials; main plasma parameters; edge plasma conditions; the
use of auxiliary heating. Impurity line brightness measurements were also combined with computer analysis to calculate impurity densities, effective ion charge,
line radiated power and impurity influx for typical operating conditions of Alcator
plasmas. Both radial and toroidal transport of impurities were examined. During
the injection of lower hybrid RF power, there is a local source of impurities at the
limiter which allows a study of toroidal transport of impurities. Toroidal asymmetries were observed and compared to numerical models of impurity behavior.
I.1
Motivation for Work
The ultimate goal of plasma fusion experiments is the development of controlled
thermonuclear fusion. In order to achieve the breakeven point for nuclear fusion,
the plasma must reach a sufficiently high temperature (> 10" eV). Additionally,
the product of the density and the energy confinement time must be greater than
a theoretical value.
N,7E > .6 x 1014
cm
(1.2)
This is usually called the Lawson criterion [6]. There are many experiments underway which are pushing the plasmas ever closer to the breakeven point. Alcator C
has recently achieved the distinction of being the first machine to exceed the NrE
value. This was accomplished by injecting frozen hydrogen pellets into the plasma
and raising the central density [71. The principal method of heating tokamak plasmas is ohmicallv with the plasma current. Auxiliary heating such as lower hybrid
RF and ICRF have been tried on Alcator with moderate success so far [8.
-8 -
Table 1.1
BT............. Toroidal magnetic field
r . ............. Impurity flux of charge state z
D..............Impurity diffusion coefficient
I,..............Plasma
mb,
... .. ... ..
current
.. Mass of background gas
N .............
Electron density
n .............
Principal quantum number of level p
qo..............Safety
factor on axis
qL.............. Safety factor at limiter radius
ri ..............
Minor radius
Ro ...
Major radius
.. ... ... ..
T .............. Electron temperature
Ti..............Ion temperature
v...............Impurity convection velocity
Zbg ............
Charge of background ion
Zeff ........... Effective ion charge
Because of the importance of impurities in the development of fusion plasmas,
most of the major tokamak experiments have undertaken studies of impurity behavior. A thorough review article covering techniques and experimental observations
of impurities in tokamaks can be found in ref [9]. Several experiments pertaining
to this thesis will briefly be reviewed here. To facilitate the reading of the thesis
commonly used notation for machine and plasma parameters are listed in Table 1.1.
On TFR in 1975 101, measurements were made of visible and UV impurity
emission.
Zff ::: 6 was measured and approximately 40% of the ohmic input
-9-
power was lost through impurities radiating from the edge. The electron densities
for these discharges were typically, N, = 3 x 1013 cm- 3 , while Ip = 140 kA and
BT
= 30 kG. It was concluded that the plasma was composed of two separate and
independent regions: a central hot core with properties depending on I, and BT,
and a peripheral shell which was dominated by the interaction with the walls and
limiter. More recently (II], TFR has managed to produce cleaner plasmas with Zeff
on the order of 2 - 3 , partly due to the higher plasma densities, N, > 1 x 1014 cm- 3 .
In contrast, Alcator A was able to achieve very clean plasmas. Impurity studies on Alcator A [12,131 determined that the plasmas had an effective ion charge
of Zff a 1. This value for Zff was confirmed by measurements of plasma resistance, electron temperature and soft x-ray bremsstrahlung emission. Total oxygen,
nitrogen and carbon concentration was .3% during the high density plateau portion
of the shot. Power loss due to line radiation from the lower ionization states of
impurities was found to be 7-20% of the ohmic input power.
Impurity injection experiments on Alcator A and C allowed the study of impurity transport as a function of various plasma parameters 114.
Trace amounts
of non-intrinsic impurities were injected into the plasma using the laser blow-off
technique r151. It was observed that the impurities penetrated to the plasma center
and then left gradually. The impurity confinement time was determined by monitoring the decay time of central impurity emission. An empirical expression for the
confinement time r, was formulated.
.075riMg R;.75 Zf fsec
q1 0 Zbg
(1.1)
Comparing these results to a computer code showed that impurity transport could
be accurately described by an anomalous diffusion term or diffusion plus a small
- 10
-
amount of inward convection.
Neoclassical theory is in disagreement with these
results.
Impurity injection experiments have been performed on other machines. One
of the first experiments of this type was performed on ATC [16,17]. Trace amounts
of Al, Au, Cu, Fe, Mo and Si were injected into the plasma. The time behaviors
of various charge states of the injected impurity were monitored and compared
to an impurity transport code similar to the code used for Alcator. Comparison
of the results with theory indicated that the impurity transport was consistent
with neoclassical transport to within a factor of two. Another impurity injection
experiment, on the Impurity Studies Experiment (ISX) tokamak (18], gave impurity
confinement times well described by inward convection predicted by neoclassical
theory, in contrast to the Alcator results. The differences in the machines could
be attributed to Alcator being relatively clean (Ze11 ~ 1), and operating at much
higher plasma densities. Similar results to Alcator were also obtained on TFR [191,
FT-1 [20], PLT [21] and PDX f221.
Transport models have been used extensively to study impurity behavior and
calculate radial profiles of impurity densities and radiated power. Hawryluk et.al
[23] performed an analysis of low Z impurity transport models and applied a transport model to spectroscopic data from ATC and PLT. The experimental results
indicated the need for an anomalous transport term. It was also concluded that the
density and power radiated by the low ionization states are sensitive to uncertainties
in rate coefficients and plasma conditions at the edge. Thus, several impurity lines
should be studied for an accurate understanding of impurity behavior. However, by
monitoring the 0 VI and C IV r.esonance lines it was possible to estimate the total
power loss from these impurities to within 50%.
-
11 -
Poloidal variations in impurity densities have been observed on several tokamaks. Poloidally asymmetric impurity emission has been observed from high density
Alcator A discharges [24]. This asymmetry was attributed to the B x VB drift of
impurity ions. Strong support for this explanation came from reversing the toroidal
magnetic field and observing a change in the sense of the asymmetry.
Vertical poloidal asymmetries of hydrogen isotopes and low Z impurity radiation was seen in PDX plasmas [25]. A qualitative difference was observed between
impurities in the Pfirsch-Schliiter regime (C II to C IV) and those in the plateau
regime (C V). Impurities in the shadow of the limiter were strongly affected by
edge processes, such as recycling, while C V which was located inside the limiter
boundary varied less with edge conditions.
The limiter can produce toroidal asymmetries in impurity emission. Allen et
al. measured EUV emissions from light impurities on the Alcator A tokamak at two
toroidal locations (one of them a limiter port) simultaneously [261. The emission
exhibited a toroidal asymmetry in addition to differences in the time histories and
spatial profiles at the two ports. Measurements of toroidal asymmetries on Alcator
C will be discussed in detail in chapter V of this thesis.
Impurity control has been studied on ASDEX
27! and PDX [28] through op-
eration with poloidal divertors. Both these machines use internal coils enclosed by
divertor chambers to form separatrices. Field lines, outside the main plasma, pass
through target plates located in the chambers. In ASDEX the amount of oxygen
did not depend on whether a divertor or stainless steel limiter was used. Gettering
reduced the level of oxygen by almost an order of magnitude.
Although much research has been undertaken on the subject of impurities in
tokamak plasmas, it is still not well understood. There are several aspects of Alcator C that make the study of impurities on it interesting, such as the very high
- 12
-
plasma densities, its cleanliness, the RF auxiliary heating that is often utilized and
the different limiter types and configurations that have been tried. The emphasis
of this thesis is an experimental study of intrinsic impurity emission in the Alcator
C tokamak in order to gain an understanding of how impurities enter and interact
with a plasma. Sources of light impurities are examined, in addition to the plasma
conditions which influence the influx of impurities. A method for computing impurity densities from spectroscopic measurements is outlined and utilized to calculate
low Z impurity densities. These impurity concentrations are then used to calculate
Zff
and power radiated. Finally, the transport (radial, poloidal and toroidal) of
impurities is examined. All of these issues are relevant to the future construction
of fusion reactors. Of particular impact on impurities are the walls and limiters of
a reactor which must be chosen and designed so as to minimize the introduction of
impurities into the plasma.
1.2
Alcator C Tokamak
The Alcator C tokamak j29] is a high magnetic field, high density plasma
confinement device with a toroidal geometry (fig 1-1).
A tokamak consists of a
toroidal vacuum chamber with a transformer situated at the opening in the center
of the torus. This transformer generates an electric field along the toroidal direction
which results in a plasma current when gas is fed into the torus. Plasma stability
is achieved with the aid of a toroidal magnetic field generated by coils which wrap
around the torus. Equilibrium is maintained by the poloidal magnetic field resulting
from the toroidal current plus vertical and horizontal fields generated by additional
external coils.
- 13 -
MA"
ORC
ConhAinmwU Stitcute
a
OMIC MEATNG
EF
coa
TOODAL FEUD MAGNET
TurboqoigcugwW
-PLASOA
VACtA)M PtjMp
4p
Dwagnostc ACCESS PM
E
V
F o
E F cc
ALCATOR C
I
A
I-,
Figure 1-1
The Alcator C tokamak
- 14 -
CPHAjM
Table 1.2
Alcator C
Parameters
Standard Value
R,
64 cm
rt
16.5 cm
Range of Values
57
-
71 cm
10 - 17 cm
BT
80 kG
30
IN
450 kA
100 - 700 kA
Ne
2 x 1014 cm-3
T"
1500 eV
1000 - 3000 eV
Ti
1100 eV
500 - 2000 eV
-
130 kG
.1 - 20.x 1014 cm-3
The parameters that describe Alcator C are listed in Table 1.2. An extensive
set of diagnostics are used on Alcator to measure plasma parameters. Diagnostics
that require a view of the plasma are placed at one of the 18 ports around the
machine. These ports are located at six toroidal locations, with three ports (top,
side and bottom) at each location.
1.3
Thesis Outline
In Chapter II, the experimental instruments used on Alcator are described in
more detail. Particular emphasis is placed on the spectroscopic instruments used
for most of the measurements.
These instruments measured the time history of
impurity emission lines and were calibrated for absolute brightness measurements.
Edge plasma diagnostics are also discussed. Chapter III contains an explanation of
various models that can be used to describe impurity distributions in a laboratory
plasma. A radial transport model which accurately describes impurity behavior in
Alcator plasmas and a method of calculating the population of individual energy
- 15 -
levels of impurity charge states are discussed. Examples of numerical results, including simulated spectra, will be shown. Chapter IV consists of the basic experimental
results obtained with the visible and UV spectrographs and the edge diagnostics.
These results include calculations of impurity densities, influxes, radiated power
and Z,ff. An investigation of the influence of the limiter on low Z impurity concentrations is reported. Impurity influx and radiated power during lower hybrid and
ICRF experiments is detailed and analyzed. Finally, a discussion is given of the
uncertainties in both the experimental measurements and the numerical modeling.
Chapter V contains an analysis of toroidal impurity transport. This was performed
by studying impurity emission from different toroidal ports during the injection of
lower hybrid RF power into the plasma. Computer simulations of toroidal transport
are compared to the experimental observations.
- 16
-
Chapter II
EXPERIMENTAL SET-UP
Spectroscopy of line emission from impurity ions was the principal technique
employed here to investigate impurity behavior in Alcator C plasmas. The basic
design of a spectroscopic instrument generally consists of a diffraction grating which
disperses the light in wavelength and a photomultiplier type detector which counts
photons. Instruments used on Alcator C covered the spectral range from the extreme ultraviolet (300
A)
up to the visible (8000
A).
In addition to spectroscopy,
measurements of the plasma edge parameters are also important since conditions
near the plasma-wall boundary can greatly influence impurity influx and transport.
Edge plasma diagnostics on Alcator C consisted of Langmuir probes, thermocouple
probes and thermocouples placed on limiter blocks.
11.1
Visible - UV Spectrograph
Most of the measurements reported here have been obtained using a 1.5 meter
grating spectrograph with a Wadsworth mounting, covering the wavelength range
4000
A
to 8000
A
in first order and 2000
A
to 4000
A
in second order.
The
instrument has a reciprocal linear dispersion of nominally 10.8 A/mm in first order
using a concave holographic grating with 600 G/mm. One of the advantages of
this instrument's wavelength range is that no vacuum is needed and the absolute
calibration, using a tungsten lamp, is straightforward and relatively accurate. Some
of the disadvantages of this spectral range are that it is more difficult theoretically
to obtain impurity densities from the characteristic lines because the emission is
mostly from the edge region of the plasma where there are more transient effects
- 17 -
and the plasma parameters are less well known. In addition, as will be discussed
later, poloidal and toroidal asymmetries are in some cases significant.
There are two modes of operation for this spectrograph. The first mode of
operation consists of using a film attachment that takes film spectra of the plasma.
The film holder allows film to be positioned along the entire exit plane of the
spectrograph so that the full spectral range can be covered.
Film spectra were
recorded every few months in order to obtain a survey of the emission lines and verify
which impurities were present in the plasma. Lines were identified by comparing
observed wavelengths to published tables of wavelengths of impurity ions 130,31]. A
portion of a densitometer trace of one such spectrum is shown in fig. 2-1. The main
emission lines which are observed are the hydrogen Balmer series and low charge
states of light impurities such as C, N and 0. Other impurities such as Si and Mo
can also be observed in this spectral region depending on the type of limiter and
the plasma conditions. Many of the lines were observed in second order. In order
to separate out which lines were first and second order, spectra were taken with
and without a borosilicate glass slide placed in the light path. The glass absorbs
radiation in the ultraviolet (below 3800
A),
which is the spectral region giving
second order lines.
In the other, more common mode of operation, the absolute brightnesses of
emission lines were measured as a function of time. Up to 10 emission lines were
recorded simultaneously by sending the light through quartz optical fibers to PM
tubes as shown in fig.
2-2.
The optical arrangement consists first of a mirror
located directly underneath the port. By adjusting the angle of the mirror, radial
scans can be undertaken on a shot to shot basis with a typical spatial resolution
of 1 cm. A convex lense is located between the entrance slit and the mirror to
focus emission from the plasma onto the entrance slit. In the spectrograph housing,
- 18
-
I
'CC
0
"a
I ::F
o
io
If4
N
C
'U
C.,
---i
N
IZ
C..,
V
0
z
Figure 2-1 Film spectrum for H 2 plasma showing H Balmer series, oxygen and
carbon line emission.
- 19 -
the diffraction pattern created by the grating is focused onto the exit plane. Fiber
arrays are connected at one end to a brass plate which attaches along the focal
plane of the spectrograph. Slots had been milled into the plate at the locations
where chosen emission lines were focused on the exit plane. The fibers transfer the
light passing through the slots to the photomultiplier tubes, whose output signals
are then digitized and stored on a computer.
The visible-UV spectrograph was calibrated with a tungsten lamp so that absolute brightnesses could be determined. Calibration of the lamp for spectral radiance
was performed through direct comparison to NBS standards of spectral radiance.
The uncertainty in the calibration of the lamp is estimated to be less than 5%
over the wavelength region from 2500
A
to 7500
A. To
calibrate the spectrograph,
the same optical arrangement was used as when the spectrograph was viewing the
plasma, except that in this case the lamp replaced the plasma as the source and no
mirror was used. Using this method, the complete spectrograph system, including
the fiber optics and the PM tubes, could be calibrated as one unit. The total error
in the calibration is estimated to be less than 20%.
11.2
UV Monochromator
A UV monochromator covering the range 300
A to
1400
A was
also extensively
used on the Alcator C tokamak.
The method by which it was connected to a
tokamak port is indicated in fig.
2-3.
The monochromator itself has a Seya -
Namioka mounting and a focal length of 0.2 meters. The holographic grating is
coated with aluminum and MgF 2 and has 1200 G/mm resulting in a reciprocal linear
dispersion of 40 A/mm. A small stepping motor is connected to the grating holder
allowing the grating to be rotated by an external controller to scan in wavelength.
The vacuum system is separate from machine vacuum and is maintained by means
-
20
-
-PLASMA
MIRROR
LE N S
<:
>
DIFFRACTION
GRATING
1.5 METER
GRATING
QUARTZ
WADSWORTH
SPECTROGRAPH
OPTICAL
FIBERS
PHOTOMULTIPLIER
[6666
PREAMPLIFIER
Figure 2-2
mak.
TUBE
HIGH
VOLTAGE
CAMAC
Experimental set-up for connection of UV-visible spectrograph to toka-
- 21 -
of a turbo pump and an ion pump. In order to view the plasma during a discharge,
the gate valve is opened approximately 500 msec before the shot and is closed
immediately aftei the shot. Underneath the gate valve is a bellows which allows the
stand, on which the UV monochromator is mounted, to be rotated so that a radial
scan of the plasma can be performed. Next there is a ceramic break to electrically
isolate the system from the tokamak. Directly beneath the ceramic break is a preslit which determines the solid angle and thus the amount of plasma viewed by the
monochromator. The detector connected to the exit plane of the monochromator is
a continuous electron multiplier (i.e. channeltron). This channeltron was operated
in the pulse counting mode as shown in fig.
2-4. The current pulses from the
channeltron went first to a pulse amplifier discriminator and from there to a pulse
counter which converted the TTL pulses to an analog signal. One problem, which
was experienced when operating the UV monochromator underneath the tokamak,
was suppresion of electron flow in the channeltron due to the large (=z 1000 Gauss)
magnetic fields.
The magnetic field at the channeltron was reduced greatly by
shielding the detector with iron and mu metal. This eliminated the problem under
most plasma conditions.
The UV monochromator was calibrated at Johns Hopkins University using a
calibration system designed and built at Johns Hopkins (fig 2-5). The source for
the calibration was a small plasma discharge lamp which could be operated with
several different gases (typically H, He, Ne and Ar). Light from the lamp reflected
off a pre-monochromator to select an appropriate emission line from whichever gas
was being used. The radiation passed through a slot of known dimensions and then
reflected off a focusing mirror. This mirror could be tilted to send the light either
to a photodiode or to the UV monochromator. Comparing the signal from the UV
monochromator to the signal from the photodiode, which was absolutely calibrated
to NBS standards. allowed an accurate calibration of the monochromator at discrete
- 22 -
I
______________
AlAtnr C Ort
Gate Valve
Bellows
Ceramic Break
Ion Pump
Channeltron
Detector
Rotary Feedthru
for Preslit
Turbo Pump
Figure 2-3
tokamak.
UV Monochromator
Experimental set-up for connection of vacuum UV monochromator to
- 23 -
Channeltron Electron Multiplier
500 pF
Photon
1 MG
HV
Pulses in
Pulse Amplifier
Discriminator
Pulse
Voltage out
Counter
Pulses out
D
Pulses in
Figure 2-4 Schematic of channeltron circuit used to count photons in the vacuum
ultraviolet.
- 24 -
wavelengths ranging from 460
A to
1216
A. Fig.
2-6 shows a plot of the efficiency
of the detector system as a function of wavelength. The monochromator is most
sensitive to photons with wavelengths between 500
HI.3
A and
1000
A.
Edge Plasma Diagnostics
Much of the low Z impurity emission comes from the edge plasma. Analysis
of this impurity emission requires experimental measurements of the conditions of
the edge plasma. The types of edge plasma diagnostics used in these experiments
consisted of Langmuir probes to measure edge density and temperature, and thermocouple probes to measure heat flow at the edge.
Several Langmuir probes were constructed and then installed on Alcator C to
measure properties of the edge plasma at different poloidal and toroidal locations
[32). The basic design of a single electrostatic probe consists of a wire, which is
inserted in the plasma, connected to a voltage source. A drawing of such a probe
used in Alcator C is shown in fig. 2-7. The portion of the probe inserted into the
edge plasma consists of a molybdenum wire melted into a sphere at the tip, while
the rest of the wire is enclosed in a ceramic tube. The wire attaches to a stainless
steel rod which is connected, via a ceramic feedthru, to a bellows assembly so that
the probe position in the plasma can be adjusted on a shot to shot basis. Electrical
connection to the voltage source is through the steel rod. A resistor (several ohms)
is usually used in the circuit as a means of measuring the probe current.
The operating principle of a Langmuir probe is illustrated in figure 2-8 which is
a typical IV curve (i.e. probe characteristic) observed in a plasma with no magnetic
field i33]. At sufficiently large negative voltage (usually < - 30 V), the probe current
reaches what is referred to as ion saturation. This means that all ions reach the
probe at their flow velocity, while electrons are repelled.
-
25
-
The magnitude of the
Johns Hopkins Calibration
Lamp
Slit
Va :uum
Ch amber
Mirror
I
Diffraction
Grating
I
~'2
Grating
Cold Trap
Channeltron
Detector
Diffusion Pump
UV Monochromator
Figure 2-5 Experimental set-up for calibration of vacuum UV monochromator at
Johns Hopkins University.
- 26 -
Calibration
10-2
I
0
-
Co
0
-
U.
C)
U.
10-4
10-5t
4C 0
600
800
1000
Wavelengi h
Figure 2-6
1200
(A)
Sensitivity of UV monochromator vs wavelength.
- 27 -
1400
molybdenum
probe tip
-.
ceramic
insulators
stainless
steel rod
7:0
welded
vacuum
bellows
ceramic
feedthru
electrical
feedthru
radial
position
drive
Figure 2-7
Drawing of Langmuir probe used on Alcator C.
-
28
-
ion saturation current is proportional to the ion density. As the bias voltage goes
more positive, electrons begin to be drawn to the probe and the probe current
becomes more positive. This part of the curve can be used to measure the electron
temperature since it follows an exponential of the form eCv/kTe. For a larger positive
probe voltage the curve begins to turn over and eventually electron saturation is
reached.
In the presence of a strong magnetic field, the probe characteristic is
modified somewhat but it is still possible to use the ion saturation current to obtain
ion densities and the exponential part of the curve for T,. Data were obtained by
placing up to four separate probes in the plasma at different poloidal and toroidal
locations. Strong poloidal asymmetries were observed in the edge densities.
The other type of edge diagnostic which was employed consisted of thermocouples embedded in molybdenum blocks (fig. 2-9). The change in temperature of the
molybdenum blocks as measured by the thermocouples could be used to calculate
the energy deposited on the blocks. Thermocouple probes were moved in and out
of the plasma to determine the radial variation of energy depostion in the limiter
shadow region. Thermocouples were also installed on ten blocks of a molybdenum
limiter covering a full poloidal range (3600).
These thermocouples were used to
calculate the total energy deposited on the limiter.
11.4
Additional Diagnostics
Measurements from other diagnostics were used to supplement the spectro-
graph and probe data. Vacuum UV spectrographs used for other experiments supplied additional spectroscopic data. The first is a grazing incidence monochromator
covering the wavelength range 50
A to 500 A r341.
-
29
-
The second is a normal incidence
Probe Characteristic
80
A
A
A
A
A
A
A
A
I
A
I
A
A
A
A
A
A
A
A
I
I
A
A
A
A
A
I
A
A
A
..................
A
A
A
A
A
A
A
A
A
60
40
L----------------------A
A
A
A
A
A
A
A
A
I
A
A
I
A
A
A
A
I
A
A
A
A
A
A
A
A
A
I
A
A
A
I
A
A
I
A
A
A
A
A
I
A
A
A
A
A
I
A
A
A
A
I
A
A
I
A
A
A
A
I
I
A
I
A
A
A
I
A
.1
20
A'A
LA1flJIN~INI1kL
A
I
$
-
-
-
-
-
A
0
-20'
-3
0
-20
A
A
I
A
A
A~
I ON SJ~TURATIdN
--
-10
A
I
A
I
I
A
I
A
I
A
I
I
A
A
A
I
I
I
A
A
I
A
A
A
I
I
I
0
10
*
A
A
I
A
A
A
*
Voltage
Figure 2-8
A
A
A
A
A
A
A
A
A
A
A
Q)
I
A
Typical single probe characteristic.
--
30
--
A
A
A
j
A
A
A
I
I
I
A
I
A
A
A
-
I
30
40
molybdenum
blocks
..
9 CM
stainless
steel
spine
Figure 2-9
limiter.
Limiter with thermocouples used to measure
the energy deposition to
.
,
- 31 -
monochromator covering the range 1200
A to
2300
dence VUV spectrograph covering the range 15
A [351.
A to
1200
Finally, a grazing inci-
A was
used on Alcator
(36]. All of these instruments were absolutely calibrated.
There are many diagnostics which are routinely used on Alcator C to measure
basic plasma parameters. Chord averaged electron density is measured with a laser
interferometer
[371.
Electron temperature can be measured by several means: one
method uses Thomson scattering with a ruby laser, which measures local values
of T,; another method measures the second harmonic of ECE which can then be
converted to T, profiles [38]; finally soft x-ray emission can give T from the slope
of the soft x-ray intensity vs. energy [39]. Z f1 is determined by measuring visible bremsstrahlung emission [40]. Soft x-ray emission (photon energy > 1keV) is
typically measured on Alcator with photodiodes and using Be as a filter.
-
32 -
Chapter III
IMPURITY MODELING
This chapter deals with theory and numerical modeling of low Z impurity behavior in tokamak plasmas. Impurity density profiles can not be directly measured
in plasmas. They can be determined by making certain assumptions about plasma
conditions and relying on theoretical expressions. The quantities which must be
known are brightnesses of impurity emission lines, T,(r), Ne(r), impurity transport
parameters and relevant atomic rates. If all these quantities are accurately known
then a good model can be formulated to compute impurity densities. These calculated impurity density profiles can in turn be used to compute Zff(r), impurity
influx and radiated power. Only well monitored and well behaved plasmas were analyzed for these impurity calculations. The requirements for a 'good' plasma that
were used for this analysis are discussed in section 111.1. Subsequent sections discuss
different models of impurity behavior and how they depend on plasma conditions.
A detailed discussion is given of the numerical model used for Alcator plasmas.
111.1
Plasma Conditions
Impurity concentrations were computed for only a limited range of plasma
densities, 1 x 101
< N,
< 2 - 3 x 1014 cm- 3 , where the plasma was well behaved
and its .parameters could be accurately measured. Discharges were generally not
run at low densities, N, < 1 x 1014 cm- 3 , unless lower hybrid current drive was
attempted, in which case the plasma commonly exhibits non-thermal effects. In
addition, in this density range the dominant contribution to Z,ff is usually from
the heavier impurities which makes impurity calculations of light impurities more
- 33 -
complicated and less crucial. Therefore, not much data was taken at low densities.
For N. > 2 - 3 x 1014 cm- 3 the plasma can exhibit what are referred to as marfes
[41]. A marfe is probably caused by a thermal instability which results in a poloidally
asymmetric high density region of plasma near the limiter radius and on the inside
major radius edge of the plasma. Large variations are observed in the time behavior
of emission from the plasma edge. This is particularly evident in the brightnesses
of low charge states which can change by as much as a factor of ten in the course
of a few msec during a plasma discharge. High charge states are influenced less by
marfes and can be used more reliably for impurity density measurements.
A typical plasma discharge on Alcator C is displayed in fig. 3-1. The length
of a discharge is usually around 500 msec with the steady state portion of the
discharge beginning after about 100 msec and lasting for several hundred msec.
The top trace in fig. 3-1, which is the time history of the plasma current, shows a
gradual rise for about 100 msec until a flattop is reached. The second trace shows
the electron density as measured by laser interferometry. Each fringe corresponds
to a line average density of .55 x 1014 cm-3. Note that the variation in density
is small during the central portion of the discharge. The third trace is the soft
x-ray emission from the center of the plasma (photon energy > 1 keV). Sawtooth
behavior, seen during most of the discharges is due to internal m=1 disruptions
causing the current profile to repeatedly flatten after peaking to a q < 1 central
value. The last trace, visible continuum emission at 5360
A,
consists mainly of
bremsstrahlung radiation which is proportional to Zeff NTe-1/2.
Characteristic time histories of impurity lines are shown in fig.
3-2.
They
consist of an ionization spike at the beginning of the discharge when the plasma
is rapidly being heated and then after about 20 msec the ionization states reach a
- 34 -
I
I
I
-430
kA
IP
2 x 10 14Cm~
Sof t
X-ray
ViS.
I
100
200
300
400
500
time(m sec)
Figure 3-1 Typical Alcator C plasma shot showing the plasma current, central line
average electron density, soft x-ray emission and visible continuum emission.
- 35 -
near steady state equilibrium. It is for this steady state portion of the discharge
that the impurity densities have been calculated.
111.2
Models that Describe Impurity Distributions
Different models can be used to describe impurity distributions in a plasma,
depending upon the characteristics of the plasma. A brief review will be given of
some impurity models and their applicability.
At sufficiently high plasma densities, impurity behavior can be described by the
local thermal equilibrium (LTE) model. This model assumes that the population
densities of energy levels in the ions are determined by particle collision processes.
The population densities redistribute until equilibrium is achieved in which each
collision process is balanced by its inverse process. If the electron distribution is
maxwellian, then the population densities are given by the well known Boltzmann
distribution,
N,(p) = wZ(P)X.(p,q)/kT
N(q)
wz(q)
(3.1)
where N.(p) is the density in level p of charge state z, Xz(p, q) is the energy difference
between level p and q, and w is the statistical weight. The charge state densities
are given by the Saha equation,
N,+ 1 (g)
N,(g)
_wz+ 1 (g) 2 (27rrkT
h2
w,(g)
3/ 2
x (g) 'kT3
X
where x,(g) is the ionization energy from the ground state. A criterion can be
computed for when the LTE model is valid. The determining factors are the values
of the collision rates as compared to the radiative decay rates 331. It can be shown
that the density must satisfy the following condition for LTE.
Ne ;> 1.7 x 10" 4 T"
-
36
2X(p,q)
-
cm- 3
(3.3)
CIMi 4650A
CIY 2524 A
0
CY2271 A
Si IE[ 4552 %
SiXI 303X
Mo:
0
100
200
300
400
500
Time (msec)
Figure 3-2
Time histories of different impurity lines during an ohmic discharge.
- 37 -
The condition given by equation (3.3) is not satisfied in tokamak plasmas since
the densities are low enough that radiative decay rates are important. A more
accurate model for tokamaks is the coronal model which originally was developed
to explain spectroscopic measurements of the solar corona (42]. The processes that
create a balance in this case are
collisional ionization -
radiative and dielectronic recombination
collisional excitation t spontaneous decay
The steady state equations for the charge state distributions are given by
NN,(g)I(z,z + 1) = NNz+j(g)R(z + 1,z)
(3.4)
where I(z, z + 1) is the ionization rate from charge state z to z+1, R(z + 1, z) is the
recombination rate and g represents the ground state. This can be rewritten as
Nz+I(g)
N, (g)
_
I(z, z + 1)
R(z + 1, z)
which shows that the ratio of consecutive charge states depends only the ratio of
the ionization to the recombination rates. A detailed discussion of ionization and
recombination rates will be given later. The population densities of excited states
can similarly be related to the ground state density using the coronal model,
NeNz(g)X(g,p) = N(p)
A(p, s)
(3.6)
where X(g, p) is the excitation rate from level g to p and A(p, s) is the spontaneous
decay rate from level p to s. The above expression is useful for determining the
ground state density from the brightness of an emission line. The coronal model
can be applied to Alcator plasmas to calculate profiles of impurity charge states.
Simulated profiles of carbon charge states are shown in fig. 3-3. A more complete
- 38 -
analysis of impurity behavior which includes transport will be discussed in the
subsequent sections.
Impurity densities can be calculated with relatively few assumptions from measured brightness profiles of emission lines. For optically thin plasmas the brightness
at a radius r is the integral along a chord of the emissivity as given by the following
expression.
B(r) = 41r
E(r')
Vr'12 _ ,2
dr'
ph/s sr cm
2
(3.7)
.V2
The emissivity as a function of radius r for a transition from an excited state p to
a lower state q is
E(p, q; r) = N,(p; r)A(p, q)
ph/s cm
3
(3.8)
Abel inverting the brightness profile will result in the emissivity profile, which combined with equations (3.6) and (3.8) yields an expression for the ground state density
profile.
N,)
-
E(p,q) E A(p, s)
NX(g,p) A(p,q)
(39)
This expression is accurate if the dominant mechanism for populating the excited
state p is electron impact excitation from the ground state.
111.3
Neoclassical Transport
Classical theory predicts an inward transport of impurities in toroidal plasmas due to both temperature and density gradients. When neoclassical effects are
included in the Pfirsch-Schliter regime there is an enhancement of this inward diffusion. An accumulation of impurities in the center of the plasma would be expected if
- 39
-
Carbon
300
250
Ii
C VC
t
C ViC
2001
*
*
150
CIII
100
C VI
CI
50
0
0
3
6
9
radius
12
15
18
(cm)
Figure 3-3 Radial profiles of different charge states of carbon assuming coronal
equilibrium.
- 40
-
this was the dominant mechanism governing transport of impurities. The impurity
flux due to this effect can be written as [43J
I22(r) =
q2 2ii
ZOTi
i
NT 81
N
O(T + Ni
ar
NiZ, Or
aTi
T
8i
N
Or
OT)
Z, Or
(3.10)
where subscript I represents the impurity, q is the safety factor, vil is the ionimpurity collision frequency, pi is the ion gyroradius, Zz is the atomic number of
the impurity and a and # are given by
a =0.47 +
0.35
5
0.
(0.66 + NIZZ1Ni )
)3=0.3 +
0.41
(0.58 + NZ2/N)
Experiments measuring impurity transport on ATC [16,171 and ISX [181 found that
the transport was consistent with neoclassical theory. However on most plasma
devices neoclassical theory can not explain the observed impurity transport. It is
usually necessary to include additional self diffusion and convection terms.
III.4
1-D Radial Transport Model
Neoclassical theory does not agree with the observed impurity behavior in
Alcator C [141. The model chosen to simulate impurity profiles in Alcator has an
anomalous transport term in addition to the ionization and recombination terms.
A good review of different transport models can be found in Hawryluk et al. 23'.
In this model, charge state profiles as a function of time are computed by solving
- 41 -
a coupled system of transport and rate equations connecting the ionization states
[16,17,44),
aN (r,t)=
a
NNiI(1, 2) + NN 2 R(2,1) - V -r
at
2 (r,t)
at
NN
,= 1(1, 2) - NN
I(2,3) + R(2, 1) + N.N 3 R(3, 2)
2
-. V - r2
(3.11)
aN
at
=(r,t)
NN- i I(z - 1,z) - NeNzR(z,z - 1) -V r,
ri(r) = -DVNi - Niv
v(r) = var/ri
D(r) = Constant
where va is a constant and an initial profile is chosen for N, (r, t = 0). Methods
of solving such a system of partial differential equations are outlined in appendix
A [45]. The implicit method has generally been employed here since the solutions
will always be stable. There are essentially four parts to this impurity transport
model: ionization and recombination terms; electron density and temperature profiles; transport terms; the initial distribution of impurities.
The ionization rates used here are from Lotz '46],
I(z, z + 1) = 6.7 x 107
N
0
z(
where N is the number of subshells.
,7
exp(-x)
cm 3 /s
(3.12)
is the number of equivalent electrons in the
ith subshell and a, is a constant whose value is, in this case. 4.5 x
- 42 -
10-14.
Plots of the
ionization rate vs T, for all the ionized states of carbon are shown in fig. 3.4. The
lowest charge states have the fastest ionization rates and are only observed in the
low temperature region of the plasma. The expression used here for the radiative
recombination rate is from Beigman et al. [47),
R,(z,z - 1) = 1 x 10
1 2A
1
xi()2
T3/
1
xz(g)/Te + A 2
cm 3 /s
(3.13)
where A1 and A 2 are constants which depend upon the isoelectronic sequence. The
dielectronic recombination rate used here is (47,481,
Ao
Rd(z,z - 1) = 1 x 10-11
ezp
cm/s
(3.14)
P,9
where Ao depends on the transition and the isoelectronic sequence. Total recombination rates for the charge states of carbon are plotted in fig. 3-5. The peak seen
near the edge in several of the curves is due to dielectronic recombination.
The second part of the model uses independent measurements of electron temperature and density profiles (fig. 3-6). The electron temperature profile has been
studied extensively on Alcator C by measuring second harmonic electron cyclotron
emission [38] and under most conditions the profile is well approximated by a gaussian of width aT given by
(3qg
ar =
1/2
(2 qj
ri
cm
(3.15)
The electron density profile as measured by a multi-chord laser interferometer [37
can be described by the function, (1 - (r/r) 2 )", with m typically between .5 and 1.
The temperature and density in the shadow of the limiter, where the lowest charge
states exist, have been measured with Langmuir probes [321 and it was found that
T, is nearly constant in this region while Ne decreases exponentially in radius with
a scrape-off length of typically .3 cm.
- 43 -
Carbon
0-7
-
- -
- r
-
c-
--- - - -
- --- -
-
r r-- -- - - - - - - - - - - - - - r-------
108
I- -
T1
1
I
1
t. t. t I
7
1
II
J iJ
o ilj
ii
I
10-9
----
10
r
--
-
r -T -e1,-
- -
-
-
-
-
-
-
-
-
-
--
- - -- - - -- -rr ---
----- r
1
-
N
10-Il
0
10-12
-
---- -- ----
-
-
i
1
10-1
1
10
100
1000
Te (eV)
Figure 3-4
Ionization rate vs electron temperature for ionization states of carbon.
- 44 -
Carbon
1010
10-
-
* i
r
rr
-
-
11-
1
L
r
N
I.
l
I/
,I
rrI
-r
ir
I
:
\
0
-
.
t
r
S
1
T
1
.L-. ...
I1
a
I
. . .
I
T
. r
I
I
I
11
r
I
-
- -
-I
L,
7
7
7
7
1
L
-
----
1-----
-
---
1
I
I
I
-
Rcmiato
rat
vs
-
-
II
r
-
IL
I-
A I
L
-
-
-
-
-
---
-
-
-
r".'
I
-
I-
- -I
I-
~
-
----
I
I
10
10
Te
Fiue
- -
-L-L-
-
----
----
-
a
I
- -
I
.l
10
-14
- -.-
II
L
III
I.
I
I .o1
- ---
I -
:
electro
-
45
-
1000l
(eV).
temperatur
II
carbon.
II
fo
t
u
inzto
statesof
I
II
I
I
II
I
1500
250
200
-wis
1000
150
\\
4)
*
E
100
z
500
50
0
0
0
3
6
9
radius
12
(cm)
15
18
I
LIMITER
Figure 3-6 Typical experimentally measured electron temperature and density profiles used in numerical code.
- 46 -
Next we shall consider how the transport term is evaluated. An empirical relationship for the diffusion coefficient D, as a function of various plasma parameters,
has been determined by impurity injection experiments performed on Alcator A
and C.
D = 23Orlqcm
Ro 4 mbg Zeff
2 /sec
(3.16)
This formula for D, which assumes no inward convection, was initially used in the
computation. To investigate the effects of convection, an analytical expression for
the confinement time is also employed [49],
(77 + S2) (eS - S
(±=)
44S2
(56+S2)
1) ,2
D5S
sec
(3.17)
where S is the dimensionless convection parameter given by S = riva/2D. Convection can be included by choosing an appropriate value for va and adjusting D so
that the confinement time given by equation (3.17) remains the same.
Finally, the model requires an initial distribution of impurities.
The initial
condition usually chosen is a gaussian distribution for the singly ionized state peaked
near the limiter radius. The system is allowed to evolve in time, with each time
step dependent on how fast the profiles are changing. With this method, simulated
charge state profiles as a function of time can be computed.
However, for the
measUrements that have been made it is more important to determine steady state
profiles for all the charge states. The steady state solution of equation (3.11) is
computed by integrating the equations for several confinement times until essentially
all the particles are lost. The time integral density profiles are solutions for a steady
state source. This mathematical technique is explained in detail in appendix D.
111.5
Population of Energy Levels
- 47 -
Once the charge state profiles have been computed, the next step in the calculation of the absolute charge state density is to compute the emissivity and brightness
profiles and then finally to normalize all the profiles to the measured central chord
brightness of an emission line. The intensities of the emission lines observed here are
determined mainly by electron impact excitation and spontaneous decay between
the separate energy levels of the ion. Rate coefficients for each of the individual
transitions, from quantum number n = 1 to n = 3 or 4, are used in the calculations
(50-571.
Higher n levels have been included by using expressions for average
spontaneous decay rates and average excitation rates. The emissivity of an excited
state is directly related to its density through equation (3.8). Using the density of
this excited state and assuming steady state, a matrix representing a system of m
equations is solved to determine the density of all the individual energy levels up
to n = 4 and the average density for n > 4. The equations are
a.Nv(2;
a2 r)
r=0
=
N*(p)A(p,2) -
,(2)
A(2,1) + N,E
X(2,p)
Nz(p)X(p,2) + NNz-1Ic(2; z - 1,z)
+ N,E
P
aN,.(3; r)
at
(pNz)A(p, 3)- N(3)
P
=0 =
A(3,p) + N,
P
: X(3,p)
P
+ N" ENz(p)X(p,3) + NNz- 1 c(3; z - 1,z)
P
a
t
;r =,(0 = -N(m
A(M +41,P) + N, -
)
p
+ N,
X(m + 1,p)
p
N,(p)X(p,m - 1)
(3.18)
- 48 -
where N,(p; r) is the density of energy level p in charge state z at a radius r, X(p, q)
is the collisional excitation (or de-excitation) rate from level p to q and I, is the
collisional innershell ionization rate. Note that it is not necessary to include the
equation for the ground state density since the density of one of the excited states
is computed directly from the measured brightness of an emission line and we are
left with m equations and m unknowns.
Level populations for n < 4 are determined principally by spontaneous decay
and electron impact excitation among low n energy levels.
However, collisional
innershell ionization and cascades from high n levels also influence, to a lesser degree, the level populations and have therefore been included in the calculations.
Dielectronic recombination and charge exchange have been neglected here. These
processes could be important in determining the population of high n energy levels
but will have negligible effect on the intensities of the measured lines.
Fig. 3-7 shows level diagrams for C III and C V ions. The dashed lines indicate
allowed transitions between separate energy levels. Fine structure splitting caused
by the spin-orbit interaction has been neglected here but there is a distinction
between singlet and triplet states.
Three quantum numbers are important: the
principal quantum number n; the orbital quantum number L; and the spin quanLum
number S. Although intercombination lines (AS = 1) have weak spontaneous decay
rates, the electron impact excitation rates between these levels can be large if the
energy difference is small. This can lead to metastable levels such as the 2s 3 S state
of C III.
Electron impact excitation rate coefficients were determined by fitting a function to the collision strengths of each transition. The most commonly used functional form for the collision strength is
2(z)
=
CC - C 1 /z
C 2 /z
-
- 49
-
2
+ C.lnz
(3.19)
1s2
,/7
m..hu!.m4
\
7'
~
,
)~/
'~"
-%
'V
/
-
/,(
'9
/
\
V
/
~-
-4-3
/
I,
'S
'S
'S
C V Grotrian Diagram
(2 electrons)
lsnl Triplet
nd 'D
np P*
1,1
I,
He I sequence
Isnl Singlet
ns3
nd 3 D
3po
np
-/---3
/>,
------3
,7
/
'5
7
7
N' 7
7
-
7
.5~
'5
/
/
/
//
//
I,
-~-1
Figure 3-7
/
'I
1/
73
/
'I
\
\
\\
3
IN
/
//I,
ns
-..
~
/
I,
\~
/
~'%
,
/'
\
*
/
/
// /
-%N
'5-
/
A,
'.
I
if
/
/,
/
/
/
~
/
/
%.
~
$.~.
nd 3 D
np3P*
ns3S
'I
/
7,
''S
/
-.
nd 1 D
-. 7.-.. 4
/
,
1s2 2snl Triplet
2snl Singlet
np P*
ns
Be I sequence
(4 electrons)
C III Grotrian Diagram
Level diagrams for C III and C V.
- 50
-
7
7
7-
where x = E/x(p,q) and Co, CI, C 2 and C3 are constants [50]. Other functions
for the collision strengths were also used. The chosen function is integrated over all
energies, using a maxwellian distribution, to get the excitation rate coefficient as a
function of electron temperature.
00
X(pq; T
=
8.01 x 10- 8 X(pq)
(2S + 1)(2L + 1)Te/
2
n(X)e-X(Pq)z/T-d
(3.20)
Here S and L are the spin and angular momentum quantum numbers of the initial
state. Using equation (3.19) or some other reasonable function for fl(z), this integral
can be solved to obtain an analytical expression for the excitation rates.
Collisional innershell ionization is an atomic process where ionization occurs
through the loss of an innershell electron, leaving the ion in an excited state. This
rate is less than the total ionization rate and therefore has no measurable effect on
the charge state distribution. However, this term does act as a slight perturbation on
the population of excited energy levels and has therefore been included in equation
(3.18).
Excited states with high quantum numbers (n > 4) can influence the population
of low n states through cascades. This effect has been taken into account by using
average transition rates [58,59 between quantum levels p and q where n,
4. The
average spontaneous decay rate for levels with n, > 4 is given by
A(p,q) =
1.5
x 01Z4
x 100
n,(n
- n2)
(3.21)
sec~
An approximate expression for the electron impact excitation rate is also used for
these high n levels
1.58 x 10- 5 f(g)exp(-x(p.q)/Te)
X(q, p; T) =
-Cm
X ( p. q) Te '
- 51 -
3
/s
(3.22)
where (g) is an average gaunt factor and the average oscillator strength is
I(,
1
196/1
= nIn2)
-(3.23)
2
nfp/
fllpfq n
These expressions allow the computation of the total density of each energy level
with quantum number n > 4.
With these higher n excited levels the computed
densities of ionization states can change, particularly those with metastable levels
(e.g. C III, 0 V). This is due to the fact that these ions with metastable states have
a relatively large population in the high n energy levels and therefore cascades can
appreciably populate the low energy levels.
Equation (3.18) was represented by a matrix and solved every 0.1 cm along
the minor radius with the input initially being a radially constant emissivity profile
of the measured emission line. The solution at each radial location yielded the
densities of the energy levels, which were then added together to give the total
charge state density. This charge state density was made to agree with the steady
state charge state profile by adjusting the emissivity profile. Then by repeating this
procedure iteratively it was possible to obtain an emissivity profile which was selfconsistent with the charge state profile and also normalized to the measured central
chord brightness. Computed emissivity and brightness profiles for C III and C V
are shown in fig. 3-8. From this same procedure, the absolute densities of energy
levels in each charge state were computed as a function of radius. Fig. 3-9 shows
the level populations for the first few excited states of C III at a chosen radius. As
can be seen the triplet state 2s 3 S, which is metastable, has a larger population than
the ground state.
The brightnesses of all the emission lines included in the code were computed
using the calculated level densities. These calculated line brightnesses can be combined together to plot the spectrum of an impurity. A simulated spectrum for carbon. shown in fig. 3-10, includes emission lines from C III to C VI. The strongest
- 52 -
Carbon
5
CV
.
to
4
to
I
3o
-
3
C.
iCIII
IP1
co
I
' '
I'd-'
0
0
0
3
6
9
radius
Figure 3-8
12
15
(cm)
Emissivity and brightness profiles for C III and C V.
- 53 -
18
n2e=1x
C Ill Level Populations
Te- 25eV
ANI'.
iv
3P
2IS212s2p
108
10.
.*
02p2p
2p21,p
2*3pas
223
p2s3d
-
106
-
-
104
102
100
1
Figure 3-9
2
3
4
5
6
7
8
Level populations of the nine lowest energy levels of C III.
- 54 -
9
lines appear in the ultraviolet at around 1000
(1550
A) being
A,
with the resonance line of C IV
the brightest line in the spectrum. These predicted lines are cor-
pared to experimentally measured lines which are indicated in the figure with bars.
The agreement was in general within a factor of two.
Spectra were also calculated for oxygen and silicon using the same general
procedure (fig. 3-11). Emission from 0 IV to 0 VIII and from Si III to Si XIV are
included. A general trend which appears is that higher Z elements radiate more at
shorter wavelengths. This is due to the greater average energy difference between
levels. The Si spectrum is expected to be less accurate than the others since the
excitation rates and spontaneous decay rates for many of the transitions are not
well known.
-
55
-
Spectrum
Carbon
1017
10
I
I
15
101
4
101 3
*
-
I.
I
I
-
*
101 2
101
,I
I
jI
_-~
-
I
______
10
Wavelength
Figure 3-10
-
V
*
10 1
-
I
(A)
Simulated spectrum of carbon. Bars are experimental measurements.
- 56 -
Oxygen
Spectrum
1017
1016
1015
103
1012
,il
-
10 1
.:
10 2
10 3
10 4
Silicon Spectrum
1016
1015
1014
1013
102
1011
1010
100
101
102
Wavelength
Figure 3-11
Simulated spectra of oxygen and silicon.
- 57 -
10
(A)
104
Chapter IV
IMPURITY STUDIES
The numerical model and the absolute brightnesses of emission lines were used
to determine densities, flux and radiated power of impurities for a wide range of
plasma conditions and with three types of limiters: molybdenum; graphite; and SiC
coated graphite.
For plasma densities above N, ~ 1 x 1014 cm
3
, the dominant
impurities in ohmically heated Alcator C discharges were found to be carbon, oxygen, and silicon, with the relative amounts depending on the limiter being used.
In addition to the limiter material, impurity concentrations were also found to depend on basic plasma parameters and on external influences such as lower hybrid
and ICRF auxiliary heating. Values for Zff were obtained using both calculated
impurity densities and measurements of visible bremsstrahlung emission.
IV.1
Impurity Density Calculations
The main emission lines used to calculate densities were C III (977
C IV (312
(4552
A),
A,
1550
A),
Si XI (303
C V (2271
A)
A),
o V (630
and Si XII (500
A).
A, 2781 A),
A, 4650 A),
o VI (1032
A),
Si inI
Measured brightnesses of these lines
were reproducible from shot to shot if the plasma density and current were kept
constant and no major disruptions occurred. Time histories of the impurity lines
are consistent with the assumption of a steady state source. Figure 4-1 shows the
time evolution of C III (4650
A) and
C V (2271
A) on
separate days in which the
plasma parameters were similar to the plasma shot shown in fig. 3-1. Although
the time histories are somewhat different for the two days, the magnitude of the
brightnesses during the steady state portion of the shot are approximately the same.
- 58 -
A larger variation is usually observed in C III brightnesses, almost certainly due to
it being a low ionization state existing at the plasma edge where conditions are more
turbulent and asymmetric. C V exists in the bulk plasma and is therefore a more
consistent and accurate measure of C density. While C III emission measurements
are more uncertain than C V, they still exhibit the same general trends as C V and
can be used to calculate C densities. In cases where the edge conditions are not
well known, it may be necessary to calibrate C III brightnesses with C V, C VI or
visible continuum emission.
For the steady state plasma conditions of fig.
3-1, simulated charge state
profiles of carbon were calculated with the transport code (fig.
4-2). Transport
parameters for this case were v, = 0 cm/s and D = 3400 cm 2 /s. It can be seen
in fig. 4-2 that the different ionization states exist in 'shells' whose peaks depend
mainly on the ionization potential of the particular charge state and the temperature
profile of the plasma. The lowest charge states appear only in the low temperature
region of the plasma located at the edge. The higher charge states have higher
ionization potentials and therefore can be found farther in the plasma where the
temperature is higher. Fully stripped carbon exists over most of the plasma volume
and is the dominant contributor to the total carbon density. Including modest
inward convection causes the profile of fully ionized low Z impurities to be more
peaked, while partially ionized species are only slightly affected. Fig. 4-3 shows a
plot of the total C density for various values of the convection parameter; S = 0, S
= 1/2 and S = 1. Table 4.1 lists the convection velocities, diffusion coefficients and
confinement time for these different values of S. The amount of convection usually
chosen is S = 1/2, which yields a flat Zeff profile, consistent with profiles from
visible bremsstrahlung measurements.
-
59
-
I
shot # 30
Sep
17
9.5x 10 14
C v
1.5x1014
C III
0
100
200
300
400
shot # 22
Feb
8.8x10
500
2
14
C v
42
Feb
3210 14
C III
0
100
200
300
400
500
time(msec)
Figure 4-1 C III (4650 A) and C V (2271
similar plasma conditions.
-
60
-
A)
line emission on different days but
Carbon
2.0
W
c VII
* 1.5 -
1.0C VI
0.5
C V
C
V
C 111
0.0
0
3
6
9
radius
Figure 4-2
12
(cm)
Radial profiles of ionization states of carbon.
-
61 -
15
18
Total Carbon Density (cri 3 )
3x1012
~
~S=1-
s =1/2
1.5x 1012
's=0
0
3
6
9
12
15
18
r(cm)
Figure 4-3 Radial profiles of total carbon density for different values of the convection parameter S.
- 62
-
Table 4.1
S
va, (cm/s)
D (cm
2
/s)
r (msec)
0
0
3400
13.8
1/2
250
4100
13.8
1
590
4850
13.8
Charge state density profiles for 0 and Si have also been computed and normalized to central chord brightnesses. The same diffusion and convection terms that
were used for carbon, were also used for oxygen and silicon. More than one line
from each impurity is typically viewed to minimize the uncertainty in the density
measurement. For a central plasma density of N, = 2.3 x 1014 cm- 3 , the calculated
charge state profiles of 0 and Si are shown in fig. 4-4. In this particular case the
limiter was SiC coated graphite. Although the dominant impurity was C, significant
amounts of 0 and Si were also observed.
IV.2
Zeff Measurement
Zeff is a measure of the total amount of impurities in the plasma compared to
the plasma density. If the impurity density profiles are known then one can directly
calculate Zeff(r) from the following definition.
Zff (r) =
F Nj(r)Z2
=
_
E Ni(r)Zi
N(r) Z2
N,
V
(4.1)
N, (r)
Many different factors influence the amount and type of impurities in the plasma
and thus the value for Zeff.
Some of the contributing factors are: limiter and
wall materials; limiter configuration; vacuum vessel cleaning method and plasma
discharge conditions.
-
63
-
Oxygen
800
600
0 IX
400
C.)
0 VII
200
0 VIII
0
0 vi
0
0
3
6
9
12
15
18
Silicon
30
0)
*
Si XIV
Si XIII
20
15
Si XII
10
-i
s
siXI
XV
5
0
0
0
3
6
9
radius
Figure 4-4
12
15
(cm)
Radial profiles of ionization states of oxygen and silicon.
-
64 -
18
For the original limiter configuration in Alcator C, consisting of Mo limiters,
the calculated Zef f was typically 1.2 - 1.3 at a plasma density of N, = 2 x 104 cm- 3 ,
with 0 and C contributing equally. Mo does not contribute significantly to Zff
unless the plasma density is very low, N, < 1 x 1014 cm-3. Later when graphite and
then SiC coated graphite limiters were installed, C became the dominant contributor
to Zff. These calculated values for Zff indicate relatively clean plasmas. The
concentrations of the low Z impurities are sufficiently small that the bulk plasma
properties are not measurably affected.
The impurity density calculations were compared to Z~ff values obtained from
measurements of visible bremsstrahlung emission. The bremsstrahlung emissivity
is given by
2
dE(r)
0.95 x 10-13
1/2
-he
AA
f f N.2(r) Zf f (r)T,- (2r
)exp (A,4.)2) /cm
dA =A
gf~7ef7e~~zkAT,(r)
_
ph
3 )A
(4.2)
s
For typical density and temperature profiles on Alcator C one can determine a line
average
Zeff
from the central chord brightness [401.
f
dE(r)dr
2eff o
2
(
f
(4.3)
N,2(r)dr
Over the usual operating range, Zeff was measured to be between 1.1 and 1.5 with
an experimental uncertainty of ±20%.
Fig. 4-5 shows a comparison of a calculated profile for Zff (r) using equation
(4.1), to a profile measured experimentally from visible bremsstrahlung emission.
To distinguish between the different methods of determining Zeff the calculated
value will be referred to as Zeff(calc), while the visible bremsstrahlung value will
be called Zfff(brem).
Zff(calc) was made flat by adjusting the convection and
diffusion terms in the transport model in order to agree with the Zeff (brem) profile.
-
65 -
The experimentally measured flat Zeff(brem) profiles are another indication that
impurities do not accumulate in the center, but rather leave the plasma due to finite
confinement times. In the shadow of the limiter Zeff (calc) is high which means that
impurities can be a dominant constituent of the edge plasma.
Another method of determining Zff is from the plasma resistivity. For an
ohmic plasma the resistivity is the ratio of the electric field to the current density.
An expression for the resistivity has been calculated by Spitzer
[721,
with a correc-
tion.term for impurities containing Z~ff. A value for Zff can be obtained using
the Spitzer resistivity, the measured loop voltage and the measured plasma current.
Zff values calculated in this manner are consistent with the previous techniques
mentioned.
IV.3
Limiter-Plasma Interaction
Limiters are a very important component of tokamak devices.
They must
be able to withstand large thermal fluxes and protect the vacuum chamber. In
addition, since the limiter surface comes into contact with the plasma, they have a
major effect on plasma purity.
Experiments were conducted on Alcator C to investigate changes in impurity
concentrations during the use of different types of limiters. Table 4.2 shows measured central chord brightnesses, computed densities and Zff contributions of C,
o
and Si under similar plasma conditions for four periods of time when three dif-
ferent limiter sets were used. All these brightness measurements were made under
similar plasma conditions, so although the absolute values for impurity densities
may contain some error, the trends that were observed were very reproducible and
seen in all lines of each impurity.
-
66
-
4
Visible
Bremsstrahlung
3
1
-
I
-I
0
0
3
6
9
12
15
18
9
12
15
18
4
Numerical
Simulation
3
1
0
0
3
6
radius
Figure 4-5
profile.
(cm)
Comparison of experimentally measured profile of Zeff with simulated
- 67 -
Table 4.2
S, = 2.0 x
101 4
cm-3
T, = 1500 eV
Limiter
C V 2271
o
V 2781 Carbon
Brightness Brightness Density
4.8 x 1014
Moly.
(8/82 - 9/82)
1.9 x 1014
Graphite 1.8 x 1014 2.3 x 1013
(9/82 - 11/82)
Moly.
1.6 x 1015
1.6 x 1014
Oxygen
Density
7.1 x 10"
Silicon
Density
*
1.27
1.7 x 101 2 8.7 x 1010
*
1.29
1.5 x 1012
5.9 x 10"
*
1.40
1.5 x 1012
7.9 x 1010
4.5 x 10"
(2/83 - 3/83)
SiC coated 1.6 x 10'5 2.1
Graphite
X 1013
5.8 x 1010
1.31
(5/83 - 6/83)
*
Not observed
Limiters that were initially used on Alcator C for several years were constructed
of molybdenum. The limiter structure consists of molybdenum blocks attached with
pins to a stainless steel spine. This allows for the replacement of individual blocks if
damage occurs. Spectroscopic measurements made during the use of molybdenum
limiters revealed that the 0 and C density for a typical run were comparable, and
lesser amounts of N were also present. The 0 and C in the plasma comes from the
wall and the limiters. Although no quantitative study has been made here of Mo
density in Alcator C, it has been observed previously '601 that Mo becomes deposited
on the wall and will remain in the machine even after the Mo limiters are removed.
However, Mo only plays a significant role in discharges with N, < 1 x 10", while
using Mo limiters and also in discharges with pellet injection where Mo transport
-
68
-
has been affected [73].
Typical values of Zqf
(both calculated and from visible
1
bremsstrahlung) were found to be 1.3 ± .1.
When graphite limiters were installed in the device, the C density increased a
factor of four, while the 0 density immediately decreased a factor of ten. Zf1 (brem)
did not change from its value with Mo limiters within the accuracy of the measurements. A similar lack of 0 was observed in the TM-G experiment, where the entire
first wall was constructed from graphite 161]. There are several possible explanations for why 0 decreased so much with graphite limiters. One is that the source
of 0 was the molybdenum limiter. Mo-oxide may have formed on the limiters and
the wall. An oxide layer would not be expected to form on graphite. Also when
graphite limiters were installed the wall became coated with C which could prevent
O from leaving the wall.
Switching back to molybdenum limiters after several months of operation with
graphite limiters resulted in only a slight decrease in C density, indicating that the
walls were a significant source of C entering the plasma. This result was supported
by actually viewing the inside of the vacuum chamber with a boroscope and confirming that the walls were coated with carbon. A gradual increase of 0 density
occured over the period of several weeks with the Mo limiters. TFR showed a reduced influx of light impurities as the limiter to wall distance increased, indicating
that the wall was also in their case a principal source of impurities [35].
A SiC coated graphite limiter was next installed on Alcator C. It was found
that the dominant impurity was C, whose density was typically a factor of ten or
more larger than Si. The 0 density again decreased substantially compared to
the molybdenum limiter case.
The dominant contribution to Zeff was from C,
although as will be discussed later, Si as well as C can cause a change in Zff
during lower hybrid experiments. There is some uncertainty in the atomic rates of
-
69
-
Table 4.3
Non-limiter Port
Ion
C
C
C
C
III
V
V
VI
Wavelength
4650
2271
40.3
33.7
Brightness
1.2 x i0"
4.5 x 10"
3. x 10"
4. x 10"
C density
4.8 x 1011
4.5 x 10"
5.9 x 10"
5.7 x 10"
Limiter Port
Ion
Wavelength
Brightness
C III
C IV
977
312
2. x 1015
1.5 x 1014
C V
C density
3.3 x 10"
4.3 x 10"
40.3
C VI
33.7
6. x 1014
8. x 1014
12. x 10"
12. x 1011
Si and therefore uncertainty in the contribution of Si to
Zeff. A larger value for
the density of Si would be more consistent with Ze f(brem).
Toroidal asymmetries
of ionization states of Si is a better candidate for explaining
this discrepancy. This
will be discussed more in chapter V. Si, like C and Mo,
also deposited itself on
the walls and therefore appeared in plasma discharges
even after the SiC coated
graphite limiters had been replaced with different limiter
materials.
A comparison was made of C line emission at both limiter
and non-limiter ports
during ohmic discharges, to study further whether the wall is
the major source of C
in the plasma. Table 4.3 shows brightnesses of various C emission
lines along with
the corresponding total C density. There is not a significant
difference between
the calculated C densities at limiter and non-limiter ports,
although a larger C
density at the limiter port is perhaps indicated. This can be
explained by toroidal
asymmetries discussed in the next chapter. Zef(brem) and
Zff(calc) would not
be greatly affected by this asymmetry in C density.
-70-
These results all confirm that the wall is the main source of C during ohmically
heated discharges and that the distribution of C around the machine is toroidally
uniform except perhaps for small regions near the limiters.
IV.4
Impurity Influx during Ohmic Discharges
Plasma conditions at the edge largely determine the influx of impurities into
the main plasma. The mechanisms by which impurities enter the plasma are still
not fully understood, partly due to the edge plasma often being very non-uniform
and insufficiently monitored. Possible impurity influx mechanisms include sputtering, thermal desorption, particle induced desorption, evaporation and arcing.
Plasma surface interactions are reviewed in detail by McCracken [21. This section
shows a calculation of impurity fluxes and then has a discussion of experimental
measurements of the edge plasma in Alcator C and its relationship to impurity
behavior.
The flux of a given impurity charge state can be evaluated from equation (3.11)
using calculated charge state density profiles. The transport and rate equation for
an individual charge state can be expressed in the following form.
S=Nr,t)
NNi- 1 I(i - 1,i)
at
NeNI(i, i + 1) + NNi.,R(i + 1, i)
(4.4)
-NeNiR(i,
i - 1) - V-i
This equation can be rewritten in the form of an integral equation to solve for the
individual fluxes.
FI(r,t) =
r
[N/ - NeNi_. I(i - 1, i) + Ne NiI(i, i + 1) - Ne Ni+iR(i + 1, i)
0
at
+ N,N, R(i, i - 1) r'dr'
(4.5)
- 71 -
One can solve for the steady state case by setting
the expression IX(r) = -D
8N
= 0, or alternatively from
- v(r)Ni(r). Fig. 4-6 shows calculated fluxes for
individual charge states of C, along with the total flux of C, during the steady state
portion of the shot. The total flux (3.5 x 1015 cm- 2 s-') is relatively high as a result
of graphite limiters being used. The walls were coated with loosely bound carbon
which could easily enter the plasma. Possible mechanisms by which carbon enters
the plasma are thermal desorption and desorption induced by energetic ions and
neutrals at the edge.
Several methods are used to keep the plasma as clean as possible. The machine
is kept under high vacuum when not operating. In order to outgas impurities from
the machine, the walls and ports are heated at regular intervals to at least 1000
C. Discharge cleaning is another important method used to cleanse the machine of
impurities. On Alcator pulsed discharge cleaning is used with H 2 or D 2 gas and a
repetition rate of 2 - 4 Hz [741.
Basic parameters of the plasma such as temperature and density generally influence the influx and concentration of impurities. The ionization, recombination
and excitation rates are all strong functions of the electron temperature. The electron density and temperature determines the population of energy levels in a given
charge state and therefore the radiation emitted in decay from an excited level. A
dependence on plasma density was evident in a density scan of line emission from
several charge states of C and 0. Fig. 4-7 shows the variation in C III (4650
and C V (2271 )
A)
line emission with line average electron density for a sequence
of shots. The C V brightness shows a steady increase with N, up to the maximum
density obtained for that day. C III on the other hand shows a sharp rise in emission when the line average density reaches 2.3 x 1014 cm- 3 . This plasma density
corresponds to the onset of a marfe {41'. Since C III exists mainly at the edge, it
- 72
-
Carbon
4
I
I
I
I
I
I
II
I
I
total flux
2
ft
C VII
0
CC1I
-4
I
0
I
3
I
I
I
I
6
I
I
I
radius
Figure 4-6
12
9
(cm)
Radial profiles of carbon impurity flux.
- 73 -
15
18
is strongly affected by marfes while C V exists more in the interior of the plasma
and is therefore not greatly affected by marfes or other sources of asymmetries or
turbulence at the plasma edge. Although the brightnesses of impurity emission lines
increase as the electron density increases, Zeff(cac) remains nearly constant.
Measurements of plasma density and temperature at the edge were utilized
in the analysis of impurity behavior.
Langmuir probes, placed in Alcator C at
different toroidal and poloidal positions, were used to measure radial profiles of
the plasma density and temperature in the shadow of the limiter. Fig. 4-8 shows
radial density profiles at three different poloidal locations.
The density falls off
exponentially in this region with a scrape-off length of typically 4 mm. Comparison
of the separate probes shows a clear poloidal asymmetry of the density in the
shadow of the limiter. These observations of an inhomogenous edge region were
later confirmed and studied in more detail by an edge plasma diagnostic called
dense pack [62], which consisted of an array of closely spaced Langmuir probes
at a single port and covering nearly all poloidal positions. Temperature profiles
measured by Langmuir probes are relatively constant in the shadow of the limiter
and do not exhibit such clear asymmetries.
Thermocouples placed on limiter blocks allowed measurements of energy deposition on the limiter as a function of poloidal angle (fig. 4-9). Energy deposited
on the limiter blocks was calculated using the temperature rise measured by thermocouples and the masses and specific heats of the blocks. A poloidal asymmetry
was also observed here with peaks in energy deposition occurring on the inside and
outside major radius. These peaks corresponded to the positions where damage was
observed on the limiter blocks following high current ohmic operation.
Any asymmetry in density at the edge should also manifest itself in impurity
emission which originates near the plasma edge. For example emission from low
- 74 -
20
C III
C V
U,
' 10 -
5
El C3EOO
E
0 0c
0-0.5
I
I
1.0
1.5
Ne
Figure 4-7
]C
(1014
C III (4650 A) and C V (2271
density.
- 75 -
A)
2.0
cm- 3 )
brightness vs line average electron
1014
I
I
I
I
I
'*1'''I
I
I
I
I
I
I
I
I
I
I
i
I
I
I
A
0
A
C)
0
1013
A
A
0
C.)
C
0
O
A
0
0
C
0
1012
0
300
30*
A port
-C D port
A F port
O F port
101
I
16
.0
I
I
I
I 50*
210
I
16.5
M-i
I
I
_-
L
I
I
I
I
I
17.0
I -
I
I
17.5
I
I
I
I
18.0
radius (cm)
Figure 4-8 Ion density at different toroidal and poloidal locations in the shadow
of the limiter. The port and the poloidal angle are indicated. 360* is directly down
while 90' is at the inside major radius.
- 76
-
40
30
o 20
10
0
0
60
120
180
240
Poloidal Angle (degrees)
Figure 4-9
Energy to limiter vs poloidal angle.
- 77 -
300
360
ionization states of C, 0, Si etc. which exist primarily at the plasma edge would
be expected to be poloidally asymmetric to some degree. Single shot brightness
measurements by Terry (fig.
4-10) using a vibrating mirror indicated an in-out
asymmetry for the low charge states of C and 0, while the higher charge states
show little or no asymmetry [63].
Emission generally seems to be greater from
the inside major radius, especially for the low charge states. This phenomena may
be related to the marfes which are observed at higher densities. C III emission
was studied during several different types of discharges, both with and without
marfes. A multichannel light detector system viewed C III brightnesses at a side
port by positioning an interference filter (4650
A) in
the light path. This instrument
together with the visible-UV spectrometer on the bottom port allowed simultaneous
brightness measurements along many different chords. During a moderate density
plasma shot, C III brightness is fairly uniform as seen in fig. 4-11. However when
the density is raised, C III line brightnesses increase and are very non-uniform.
Fig. 4-12 shows a plasma shot where the marfe moved poloidally causing the C III
brightness to rise and fall accordingly.
IV.5
Radiated Power
Radiated power from impurities can be a major source of power loss for the
plasma. Principal sources of radiation due to impurities consist of line emission,
bremsstrahlung, radiative recombination and dielectronic recombination.
All of
these processes will be examined for the conditions in Alcator C plasmas.
Power emitted through line radiation is given by
Lh(p,hq)A(p, q)Nz(p)
Pi =
z
p
q
- 78 -
W/cm
3
(4.6)
15
10
CV"
5
C
IV
.0
18
-12
-6
0
radius
6
10)
18
(cm)
Figure 4-10 Experimentally measured radial profiles of carbon and oxygen line
brightnesses.
- 79
-
Carbon III
Ne =.8x1014
1 =480
10/5/82
CI)
CI,
I
0
100
I
200
300
I
400
500
time(msec)
Figure 4-11
Different chordal views of C III emission during non-marfe discharge.
- 80
-
Carbon III
Ne =2.3x10 14
10/5/82
I =480
I
I
I
e.........
..
(13
(13
flY
I
0
I
100
I
I
I
i
200
I I
I
300
400
I
I
WMMA
500
time(msec)
Figure 4-12
Different chordal views of C III emission during marfe discharge.
-
81 -
where the summation is over all the transitions and charge states. Only lines originating from low n (n < 4) energy levels were included in the calculation of radiated
power since these were the dominant lines. Electron impact excitation is the dominant process which populates these low n energy levels. The strongest line for a
given charge state is usually an electric dipole transition from the first excited state
to the ground state or a metastable state.
Bremsstrahlung emission results from free-free electron transitions in the field
of an ion. The power emitted due to the bremsstrahlung process is
Pb = 1.53
x
10-
32
NZ2 f T'l
N,
2
W /cm
3
(4.7)
z
where the gaunt factor is taken equal to one.
In radiative recombination an electron is captured in an excited level, then
cascades to the ground level and in the process emits radiation. If we take the
average energy of the electron before it is captured to be equal to the plasma
temperature then the power lost by radiative recombination is:
P, = 1.6
x
N- R , [Xz + TeI
10~'Ne E
W/cM 3
(4.8)
This power loss mechanism includes cascade line radiation from high n levels.
In dielectronic recombination the incident electron excites a bound electron
to a higher energy level while being captured in a very high n energy level with
excitation energy approximately equal to the ionization energy. The power density
is thus given by:
Pd = 1.6
x
10-'g NZ
NzRd [Xz - x (p, q)
W/cM 3
(4.9)
Radiated power due to line emission from the charge states of carbon is plotted
as a function of radius in fig. 4-13. The total line radiated power is the heavy solid
-
82 - -
curve. Over 50% of the line radiation is from the low charge states located near the
plasma edge. This characteristic of line radiation from low - medium Z impurities
is also evident in 0 and Si (fig. 4-14). These calculated profiles are consistent with
bolometer measurements made on Alcator C which found that the radiated power
is mostly from the edge region, for discharges in which the main impurities were low
to medium Z
[641.
Calculations of the total line radiated power indicate that it can
be a dominant power loss mechanism. For a plasma density of N, = 2 x 104CM~3
and a contribution to Zeff of 0.2 from C, 0 and Si, resulting in Z f1 = 1.6, we
have the following line radiated powers.
Pcarbon = 62
kW
Poxygen = 105 kW
P,iiico, = 20 kW
The contribution to radiated power from bremsstrahlung, radiative recombination and dielectronic recombination is relatively small compared to line radiation
for typical Alcator plasmas (< 2% ohmic input power). Fig. 4-15 shows the radial
distribution of the power densities due to these three processes. Radiative recombination is roughly a factor of ten greater power loss mechanism than the other two
atomic processes. However, if the temperature of the plasma is raised sufficiently
then bremsstrahlung radiation eventually becomes the dominant loss mechanism,
comparable to or greater than line radiation.
A typical plasma shot (X, = 2 x 10
4 cm-3)
has
-
1 MW ohmic input power
and a calculated line radiated power of 200 kW. Bolometric measurements of power
radiated are typically less than 500 kW. A major portion of the input power can
be attributed as being deposited on the limiters. Thermocouples placed on limiter
blocks were used to study total energy deposition onto the limiter. A density scan
of energy deposition to a limiter is shown in fig.
-83 - .
4-16. An approximately linear
Carbon
1.5
TOTAL
C IV
1.0
0
C H]
C V
C V
0.0
0
5
10
15
radius (cm)
Figure 4-13
Radial profiles of carbon line radiated power for Zeff = 1.25.
- 84 -
20
Oxygen
1.2
C.)
1 0
TOTAL
0.8
0.6
0
$M
0)
0.4
0.2
0.0
0
3
6
9
15
18
15
18
Silicon
50
*
40
C.)
TOTAL
30
0.)
0
20
10
01
0
3
6
9
12
radius (cm)
Figure 4-14
Radial profiles of oxygen and silicon line radiated power.
- 85 -
Carbon
50
Radiative
40
Recombin ation
CV)
30
0
Dielectronic
20
Recombination
0.
10
Bremsstrahlung
'II
U
0
3
6
9
radius
12
15
18
(cm)
Figure 4-15 Radial profiles of total radiated power due to bremsstrahlung, radiative recombination and dielectronic recombination.
-
86 -
increase was observed in energy to the limiter as the plasma density was raised. For
a plasma duration of 500 msec, these measurements show that on the average up to
300 kW are deposited on the limiter. The remaining power is believed to be lost to
other limiters, lost to the wall and also radiated away by heavy metallic impurities.
At higher plasma densities, the marfe region of the plasma can radiate away up to
300 kW of power [64]. Enhanced impurity emission near the limiter can contribute
to ' 10% of the radiated power.
IV.6
Impurity Injection during Lower Hybrid Experiments
RF power has been injected into Alcator C plasmas at the lower hybrid and
ion cyclotron range of frequencies to investigate both heating and current drive
effects. Unfortunately, an important effect of RF input power has been to introduce
substantial amounts of impurities into the plasma [65]. This has also been observed
on several other machines such as TFR [66], TCA [67 and PLT 68].
Limiters
appear to be a major source of these impurities, although other sources such as the
wall, antennas and Faraday shields can also be important.
Lower hybrid RF power at a frequency of 4.6 GHz has been used successfully
on Alcator C to heat plasmas and to drive plasma current '81. Fig. 4-17 shows a
drawing of the waveguide system used to launch waves into the plasma. RF power
injected into the plasma gradually increased over a period of months until over 1
MW was successfully launched into the plasma. With 850 kW of power injected
into a deuterium plasma at a density of N, ~ 1.3 x 1014 cmincreases of AT
3
, typical temperature
~ 0.7 keV and AT, ~ 1.0 keV were observed. The limiter material
in this case was SiC coated graphite.
As the RF input power was raised, the
impurity level was observed to increase in a non-linear fashion. AZ,ff(brem) was
- 87 -
200
150 0
0
00
3
100
0
500
0
*
a.
ra
.
0*e
*
0
.0
.00
50
0+
0
0
0
2
1
30
#
23
88 -
4
Ne (1014 cm-3 )
Figure 4-16
Energy to limniter vs line average electron density.
-
88
-
5
6
0
I
ICL,
Cf)
z
3::
C,
Wo
W~ -J-J
0
0
(f)40
Cf)
U
r h.
i-
_
J.,)
-J
,-
E
Of
W -J
-
(n <
Figure 4-17
Lower hybrid RF waveguide system.
- 89
-
measured during a sequence of shots in which the RF power was varied (fig. 418). Below approximately 500 kW of RF power there is little effect on Zff(brem),
but once the RF power begins to approach 1 MW there are large increases in
Zff(brem). The increases in Zeff are consistent with the increases in Si and C
in the plasma as inferred from the change in brightness of the higher ionization
states. The percentage increase of Si line brightnesses was measured as a function
of RF power (fig. 4-19). Roughly the same increases were observed in Si during
LHRF pulses as were observed in Zeff(brem), indicating that Si can be a major
contributor to the increase in Zeff. Radiated power during LHRF injection due to
Si line emission was calculated assuming that before RF the Si radiated power was
approximately 25 kW, which is a contribution to Zeff of 0.5. Fig. 4-20 shows the
calculated Si line radiated power as a function of RF power. It is evident that a
substantial amount of power can be radiated away by Si at large RF powers. These
values are an average of measurements from limiter and non-limiter ports. Toroidal
asymmetries that were observed for Si and experimental uncertainties lead to an
estimated error of 75%. Figures 4-21 and 4-22 show the increase in brightness and
computed radiated power from carbon ions as a function of RF power, assuming an
initial Zeff of 1.2. The same general behavior is observed and the total radiated
powers are comparable.
The source of impurities during the application of LHRF power is mainly the
limiter, as evidenced by the results obtained using different limiter materials. Considering first the case of molybdenum limiters, fig. 4-23 shows a typical plasma shot
with RF power of 950 kW. Before the RF power is applied, Zff(brem) = 1.5, with
carbon and oxygen being the main contributors to Zff. When the RF pulse is
applied, Mo increases dramatically while little or no measurable effect is observed
on C and 0 line emission. The increase in Zfff(brem) can be attributed solely to
the influx of Mo from the limiter into the plasma.
-
90
-
5
4
A
/
/
3
A
A
/
/
z
/
/
/
-
/
/
A'
/
L
/
1
0
0. 0
.
0.2
-
0
0.4
0.6
0.8
RF POWER (MW)
Figure 4-18
The change in Zeff as a function
of RF input power.
-
91 -
1.0
1400
L
I
I
I
0
1200
0l
1000
00
0/
CI)
800
a)
00
8
600
/
//
0
0-4
0
400
0
0
/0
&*
0
0
0
200
0
0
0.
0. U
' 9
I
u-d
0.4
0.6
RF POWER
Figure 4-19
0,
I
0.8
1.0
(MW)
Percentage increase in Si XI brightness as a function of
RF power.
-
92
-
400i
I
I
I
I
/
0
300
j
I
0'
0
0
0C
200
/
z0
Q
C
/
0
100
/
C/2
0
.0
I4t-
M
0
/"L4
0
RF POWER
Figure 4-20
Zeff = 1.5.
C C
C C
(MW)
Total silicon radiated power during an RF
pulse, assuming an initial
- 93 -
1000
I
I
I
I
K
I
I
I
-
I
800
/
I
Cn
/
4)
I
I
I
I
600
I
I
/
I
Co)
/
400
I
200
0
0.
I -
U
-
II
I
0.4
RF POWER
Figure 4-21
I
0.6
I
0.8
1.0
(MW)
Percentage increase in C V brightness as a function of
RF power.
- 94 -
500
5
400
mI
300
0o
--
C1
4j
/
i
,'
200
i
.I
-
~
M
uI
--
I
Q
100
0.
U
U.
0.4
0.6
RF POWER
Figure 4-22
Zef f = 1.2.
0.8
1.
0
(MW)
Total carbon radiated power during an RF pulse, assuming
an initial
- 95
-
4/22/83
IP
e
Soft
X-rays
Mo
Vis.
Cont.
LHRF
I
0
I
I
I
I
100
200
300
400
500
time(m sec)
Figure 4-23
Plasma with molybdenum limiters and injection of LHRF power.
-
96
-
The molybdenum limiters were replaced with limiters made out of SiC coated
graphite. Again during the LHRF pulse, the level of impurities increased as can
be seen in fig. 4-24. However, in this case the cause for the increase of Zeff is the
influx of Si and C. Mo levels are a factor of 20 to 30 lower than with Mo limiters and
only small increases (< 30%) in Mo are observed during LHRF [65]. The increase
in Zff is generally larger with SiC coated graphite limiters than with Mo limiters
at similar power levels. In addition with SiC coated graphite limiters much larger
increases in T, and Ti during LHRF heating are observed. Some of the ion heating
can be explained by better coupling with electrons and the possible transition into
the 'Z mode' [69]. Reduced Mo radiation could also contribute to the improved
heating.
A study of the sources of Si and C during LHRF experiments was done on
Alcator [70]. Sputtering by ions appears to be the main cause of impurity influx
below 550 kW of RF power. For RF power > 550 kW, the impurity influx increases
very rapidly with RF power. Evaporation would be the most likely cause of impurity
influx at these higher RF powers.
In the ion cyclotron radio frequency (ICRF) heating experiments on Alcator
C [8], up to 500 kW of RF power at 180 MHz was injected into the plasma. An
antenna, consisting of two half turns, was used to couple power into the plasma
at either the first or second harmonic of the ion cyclotron frequency. Good results
were obtained in the hydrogen minority regime. However, as in the case of LHRF,
when ICRF power was sent into the plasma, it also resulted in significant impurity
influx. Possible sources of impurity influx in this case are the antenna and Faraday
shield (constructed of stainless steel), the limiters (constructed of graphite) and
the wall. Constituents of stainless steel appeared to be the dominant impurities
for most ICRF discharges, although the brightnesses of all impurity lines increased
- 97 -
In
1.7 x 10
CM-3
e
Soft
X-rays
800 kW
Visible
Contiruin
I
I
0
100
I
I
300
200
I
400
time(msec)
Figure 4-24
Plasma with SiC coated graphite limiters and LHRF power.
-
98 -
500
171). Analysis of impurity behavior is complicated here by the fact that the edge
temperature and density, as measured by probes, also increased dramatically during ICRF. Figure 4-25 shows a plasma discharge in which an ICRF pulse is fired.
Observed increases in soft x-ray emission and visible bremsstrahlung emission are
due mainly to impurity injection into the plasma. Time behaviors of carbon and
oxygen lines, seen in fig. 4-26, show a large increase during an ICRF pulse. Part
of this increase may be due to the increases in edge temperature and density. A
systematic study of impurity influx during ICRF was not undertaken here.
IV.7
Error Analysis
This section contains an analysis of the various uncertainties in the experimental measurements. The possible sources of error are listed in Table 4.4. Calibration
of the spectrometers was not a significant source of error compared to the other
uncertainties. The error in the transport model was due to uncertainty in choosing
the correct combination of diffusion and convection in the code. Electron density
and temperature profiles can be accurately measured in the central region of the
plasma, however near the edge their values are not well measured. In addition, the
electron temperature is not measured during every plasma shot. All of these factors
account for the relatively large error that is listed for T, and N,. The ionization,
recombination and excitation rates that were used are theoretical values. The error in these rates was estimated by comparing different published sources. Finally,
poloidal and toroidal asymmetries in lines emission could be a source of error for
the low charge states.
By changing the input values in the numerical simulation by amounts equivalent
to the estimated errors listed above, it was determined that the total uncertainty
in the computed impurity density was approximately a factor of two. This was
-
99
-
I
I
I
I
I
I
I
I
I
I'-'
IP
Re
J
4
jl
Soft
x-ray
Vi S.
I
Cont.
LAAW
ICRF
I
0
I
10
200
300
time(msec)
Figure 4-25
Plasma with injection of ICRF power.
- 100
-
400
500
600
0
IV
0
V
cn
C III
0
100
200
300
400
time(msec)
Figure 4-26
Time histories of impurity lines for plasma with ICRF.
- 101 -
500
Table 4.4
Source
Error
(a) Calibration
± 20%
(b) Transport model
± 50%
(c) T, and N. profiles
± 50%
(d) Atomic rates
± 75%
(e) Asymmetries
± 50%
consistent with a comparison of the measured brightnesses of different emission
lines of the same impurity. A final check of the calculated values for the impurity
densities is how well they agree with visible continuum measurements of Zeff. In
general the agreement has been good over the range of plasma densities where the
light elements are the main impurities.
- 102
-
.
Chapter V
TOROIDAL TRANSPORT OF IMPURITIES
This chapter contains a study of toroidal transport of low Z impurities. Experimental observations of impurity injection during LHRF were used to study impurity
transport parallel to magnetic field lines. The source of impurities during lower hybrid experiments was localized at the limiter location. Spectroscopic measurements
at different toroidal locations showed that the impurity emission was non-uniform
along the toroidal direction during a LHRF pulse. The low charge states exhibited clear toroidal asymmetries. These results were then compared to a theoretical
model for toroidal transport of impurities. Good agreement was found between
numerical simulations and experimental results.
V.1
Measurements of Toroidal Asymmetries
Injection of lower hybrid RF power into the plasma can cause an increase in
impurity emission.
There is evidence that this is due to an influx of impurities
mainly from the limiter. The relative increase in brightness is not the same for
all charge states. At a non-limiter port it is generally observed that the higher
charge states show a substantially greater increase during an RF pulse than the
lower charge states. Fig. 5-1 shows time histories of various. charge states of C
and Si during a plasma shot with LHRF heating. The higher ionization states, in
this case C V and Si XI, show a noticeable increase in brightness during the RF
pulse, while the lower charge states show little or no increase. This variation in
emissivity can not be accounted for by the increase in temperature during lower
hybrid RF. It is however consistent with assuming that the limiter is the main
- 103 - .
source of impurities during a LHRF pulse. Then by using this assumption and
a simple transport model one can show that a toroidal asymmetry will occur for
the low ionization states of an impurity coming off the limiter. This effect is more
evident by comparing impurity emission from limiter and non-limiter ports. Table
5.1 lists relative increases of various charge states during LHRF, from limiter and
non-limiter ports. When viewing from a limiter port it is observed that emission
from all the ionization states increases roughly the same percentage, in contrast to
a non-limiter port where only the higher charge states show an appreciable increase.
The parallel transport model, explained in more detail in the next section,
predicts that the low charge states will be localized near the limiter if the main
source of impurities is the limiter. This is due to the ionization times of the low
charge states being comparable to the collisional diffusion times along the field lines,
while the ionization times of higher charge states are much longer than the diffusion
times. Therefore the low charge states become further ionized before they leave the
vicinity of the limiter.
Studying toroidal transport of impurities through the lower hybrid experiment
has several advantages over other techniques. One is that the impurity line emission
can be monitored before the RF pulse since there is a steady source from the wall.
This in turn allows a calculation of the percentage increase in brightness of various
charge states. With no background source of impurities the absolute brightnesses
of the impurity lines would have to be measured making analysis of the toroidal
distribution more difficult. Finally, the LHRF pulse is typically on for more than
50 msec allowing a steady state approximation to be used in the analysis.
V.2
1-D Toroidal Transport Model
- 104 - .
CIII 4650A
Ci 2524
A
CY 2271
A
0
SiH[ 4552 %
Sif
X303%
0
RF Pulse
200
100
300
400
500
Time (msec)
Figure 5-1 Charge state emission from carbon and silicon during plasma shot with
LHRF heating. Illustrates dependence of brightness increase on charge state.
- 105
-
Table 5.1
Limiter Port
Ion
Percentage Increase
C III
120%
C IV
100%
0 IV
0 V
0 VI
80%
120%
90%
Non-Limiter Port
Ion
C III
C IV
Percentage Increase
10%
10%
CV
60%
Si III
Si XI
10%
200%
A transport model has been developed to describe impurity behavior along
a field line. The following simultaneous equations which relate all the ionization
states of an impurity species are solved using an implicit finite difference method.
aN(Xt)
aNatt
0N
8N 2 (z,t))
at
NeN(1,2) + NN
-
=X
+1,
2 R(2,1)
V I'i
2'(,2)
(2, 3 )
+ R(2 1) +
,V R( ,2
N,Ni I(1, 2) - NN2 I(2,3) +R(2,1)] + NN 3R(3,2)
(5.1)
aNt
at
= NeNziI(Z
-
1,z)
zR - NNzR(ZZ - 1)- V(
- 106 -
Both the temperature and the density are assumed constant along a field line. The
flux in this case is assumed to be purely diffusive,
ri(z,t)
= -DzVNi(x,t), where
the diffusion coefficient is taken to be classical.
D,(r)
The thermal velocity is
Vth =
cm 2 /s
Ve2I
2
(5.2)
VT,/m, and the ion collision time is given by,
rNiZ(r)
2.09 x 107T3/2 I./
NZ 2 lnA
2
sec
(5.3)
where pi is the ratio of the ion mass to the mass of hydrogen and InA is the
coulomb logarithm. Combining the two previous equations yields an expression
for the parallel diffusion coefficient in terms of temperature, density and type of
impurity,
D.(r) =
/ 12
5.0 x 1012
N Z 1
cm 2 /s
where yil is the ratio of the impurity ion mass to the mass of hydrogen.
(5.4)
Since
equation (5.1) is one dimensional, it can only by solved at specific radial locations.
Time evolving profiles and steady state profiles of all the charge states were
calculated. Fig. 5-2 shows the time evolution of C III along the toroidal direction
from 3 to 30 ms after the initial injection. No constant source term is assumed
here. By 6 ms C III has diffused outward along the entire toroidal extent, but in
the process the total number of C III ions has decreased due to ionization. Note
that this is at a radial location in the limiter shadow, which is why the density is
forced to zero at the limiter location.
The steady state profiles of several charge states of carbon are shown in fig.
5-3. The radial location again corresponds to the limiter shadow region. C III and
C IV show a strong asymmetry, while C V and C VI are distributed more uniformly
along the toroidal direction. These profiles are for a steady state source, which is
- 107
-
.
CIII
Te = 10
r =
17
10
I
N
x 1014
1.5 x
---
/
-
1.0
I
i
/
8
C*
cm
time = 3 to 30 ms
105
I WANW-Wwuw
3 MS11
6
/
-
4
/
/
-
6 ms
2
-------
...
..
.0
0
30
60
toroidal angle
I
(degrees)
I
PORT
LIMITER
Figure 5-2
90
Numerical simulation of time evolution of C III toroidal density profiles.
- 108 -
not quite the situation during a lower hybrid pulse. However, a finite time source
does not change the results measurably unless the length of time is under 20 msec.
Lower hybrid pulse lengths were typically greater than 50 msec.
The next figure (5-4), also shows steady state profiles of carbon, however in
this case the radial position is inside the limiter radius and therefore the density is
not forced to zero at the limiter location. These profiles exhibit the same general
behavior as in the previous case, with the main difference being that the higher
charge states (C V and C VI) are now essentially completely uniform along the
toroidal direction.
V.3
2-D Radial and Toroidal Transport Model
A more accurate model of impurity behavior, which includes both radial and
toroidal transport, has been investigated here. The equations are of the same form
as before, except now both radial and toroidal effects are included together.
N (r,xt)
N, NiI(1, 2) + NN
at
aN 2 (rxt)
-
NNiI(1,2) - NN
at
2
2 R(2,
1) - V . r
[I(2,3) + R(2,1)1 + NN
3 R(3,2)
(5.5)
NVz(r.z,t)
t
at
-,Nz_
izI(z - 1,.z) - NNzR(z,z - 1) - V .-
,
The flux is now given by
f (r, z, t) = -D.(r)V.N,(r,z,t)i - D,
- 109
-
7 ,.N(rz, t)r
(5.6)
Carbon
Te
.0 x
=
r = 17 cm
250
K
200
=
= 10
1014
1.5 x 105
/I
C IV
(0
*
*
150
C~
III
100
50
0
0
30
60
toroidal angle
I
90
(degrees)
IPT
LIMITER
PORT
Figure 5-3 Simulation of steady state toroidal profiles of carbon charge states in
shadow of limiter.
- 110-
Carbon
r=
12
N
=
Te = 40
1.0 x 1014
=5.0 x 106
16 cm
10
C V
C VI
8
.&
6
-
SC
III
4
-C
IV
2
0
0
30
I
LIMITER
Figure 5-4
60
90
toroidal angle (degrees)
I
PORT
Simulation of steady state toroidal profiles of carbon charge states
inside the limiter radius.
-
111
-
Once again we assume an initial profile for the singly ionized species and then let
the system evolve in time. The initial profile was chosen to be a gaussian both
toroidally and radially, sharply peaked just outside the limiter position.
By using 2-D equations rather than 1-D equations, we are now including the
influence of cross field transport and ionization. This results in an asymmetry of
the toroidal density profiles of the higher charge states, such as C V and C VI, in
addition to the asymmetry already noted in the low charge states. Fig. 5-5 shows
three dimensional surface plots of steady state density profiles of C III and C V
assuming the only source is the limiter. The radial axis is from the plasma center
to the plasma edge while the toroidal axis begins at the limiter location and extends
to a toroidal position halfway between limiters. Whereas in the 1-D model C V is
toroidally uniform, in the 2-D simulation we see a non-uniform toroidal distribution
of C V. There is approximately a factor of six difference in the density of C V from
its maximum at the limiter position to its minimum halfway between limiters. In
fig.
5-6 we see that C VI has a density variation of only a factor of two while
fully ionized C VII has little toroidal variation in density. The main result from
these 2-D numerical simulations is that all the ionization states of carbon, except
for fully stripped, may have some degree of toroidal asymmetry if the principal
source of carbon is the limiter. Other impurities such as oxygen and silicon have
not been analyzed with the 2-D model because of the excessive computer time that
is required.
A comparison of the ionization time to the diffusion time for both C III and C
V illustrates why C III has a greater degree of toroidal asymmetry than C V. The
ionization time is defined to be
=rin
--
N, I
- 112 -
(5.7)
A
C III
limiter
r
"
IIr'o-
CV
A
i...
\
~'A.*K\
1.~N
*
I
1~.
ANt/
~
N1
Figure 5-5
3-D plots of C III and C V radial and toroidal profiles.
- 113 -
z
N
-N
I,,7
42.4 0-
Fiur
56
-DpltsofC
I
ndC
IIraia
- 114 -
ad
oridl
roils
where I is the ionization rate. The parallel diffusion time is taken to be
Tdiff =
"mfp
) 2r
1i
(5.8)
where Amfp is the mean free path length. Fig. 5-7 shows a plot of ionization time,
diffusion time and density for C III and C V. In the case of C III we see that
near the peak in density the ionization time is comparable to the diffusion time
which means that C III will become further ionized before it can distribute itself
uniformly toroidally. However for C V the diffusion time is generally much less than
the ionization time over a good portion of the plasma where there is C V emission.
This allows C V ions to become distributed more evenly toroidally before ionizing
further.
V.4
Comparison between Theory and Experiment
The 1-D model for toroidal transport of impurities shows good qualitative
agreement with spectroscopic measurements of impurity emission. Numerical modeling results shown in figures 5-3 and 5-4 predict that the increase in emission from
all the charge states should be large at a limiter port during an RF pulse. This
is consistent with observations from limiter ports. The model also predicts that
the change in emission during an RF pulse at a port 60* away toroidally will be
small for low ionization states such as C III and C IV compared to higher ionization
states. This also agrees with measurements at a non-limiter port where for most
shots virtually no increase was observed in C III and C IV emission during an RF
pulse. The 2-D model predicts nearly the same results with the difference being
that some asymmetry is also predicted for the higher ionization states.
A toroidal asymmetry has been observed in the high charge states of silicon
(Si XI, Si XII) by measuring absolute brightnesses at limiter and non-limiter ports
- 115 -
100
1011
C III
10-1
LU
1010
0-2
DIFFUSION TIME
-
108
/.
C.,,
1085
LU
10 -41
TI/I E
-IONIZATION
10-5
10
12
14
18
16
10 12
101
17
-
C
100
10
10 - 1
DIFFUSION TIME
1011
C-)
LU
C.,,
LU
ZT
10-2
109
IONIZATION TIME
10~4
10
12
14
16
18
radius (cm)
Figure 5-7 Comparison of diffusion time to ionization time for C III and C V. The
dashed line is the charge state density.
- 116
-
Table 5.2
Si XI (303
A)
Si XII (500
Brightness
Brightness
Limiter port (SiC)
.5 - 3 x 1016
1 - 3 x 1016
Non-limiter port (SiC)
2.5 - 6 x 10"
2 - 6 x 1015
Limiter port (Mo)
1 - 2 x 1015
A)
during ohmic discharges. Table 5.2 shows a comparison of the range of brightnesses
of two Si lines at limiter and non-limiter ports. Although there is a large variation in
the brightnesses of Si lines, the general trend is greater emission at the limiter port
when the limiter material is SiC coated graphite. This is consistent with the limiter
being the primary source of Si for ohmic discharges and most of the ionization
states of Si (up to Si XII) having a toroidal asymmetry. When the SiC limiters
were replaced with Mo, the brightness of Si XI at the limiter decreased about a
factor of ten. The amount of Si in the plasma in this case was a factor of ten
lower than with SiC coated graphite limiters. Calculations of Zeff in cases where
Si contributes significantly to Zeff could be affected by toroidal asymmetries.
These results can be compared to a study done on Alcator A in which two
monochromators were used to view impurity emission at two ports simultaneously
26I. It was observed that emission from a limiter port was usually a factor of two
to three higher for different charge states of oxygen and nitrogen. The explanation
given was that either recycling or a local source at the limiter caused this asymmetry.
A simple model predicting a scale length for the toroidal distribution of impurities
was in qualitative agreement with the results. The results from Alcator A are also
in agreement with the l-D and the 2-D transport models that were used here.
A comparison was made of the 2-D simulation results to experimental measurements of the relative increases of carbon line brightnesses during a LHRF pulse.
- 117 -
Experimentally measured time histories of C III, C IV and C V are shown in fig.
= 2.0 x 1014 cm- 3 . The percentage increases during an RF pulse are
5-8 for I,
compared to a 2-D numerical simulation in fig. 5-9. A uniform background was
added to the simulated profiles to take into account the source from the wall. The
level of the background could not be measured directly and therefore had to be
adjusted to get the best agreement with experiment. In this case the background
level is relatively large before the RF pulse. Fig. 5-10 shows measurements of C
lines for which there are very large increases during the RF pulse. A comparison
to the 2-D numerical model using similar plasma parameters is shown in fig. 511. Here the background level must be much lower than in the previous case to
get good agreement with experiment. This plasma density in this case was low
(Re = .7
x
104).
Non-uniformity in the toroidal density profiles of impurity charge states can
be important when comparing Zqf calculated from the impurity density model to
Zff determined through visible bremsstrahlung measurements. This is illustrated
in table 5.3 which shows typical calculations of carbon and silicon densities both
before and during a LHRF pulse, using measurements at limiter and non-limiter
ports. The carbon density was calculated from the brightness of a C V line while
the silicon density was obtained from a Si XI line. The 1-D radial transport model
was used here to relate the brightnesses to absolute densities. The density values
were corrected to take into account toroidal asymmetries of C V and Si XI. Fig.
5-12 shows a comparison of the 1-D model to the 2-D model for C V. In the 2-D
simulation the ratio of the C V peak density to the total density varies depending
on the toroidal angle.
The ratio for the 1-D model is indicated on the figure.
Correction factors for C V and Si XI were around a factor of two.
Zff(calc)
was then determined assuming that carbon and silicon are the only impurities and
Zeff(brem) is the value from visible bremsstrahlung emission. There is still some
- 118 -
6/1/83
C III
CI)
c Iv
I
0
I
100
I I9 m
200
300
I
400
500
ti me(m s ec)
Brightness measurements of different charge states of carbon for plasma
with LHRF injection.
Figure 5-8
- 119-
600
*-c
III
U-c iv
A-cv
500
cv
400 -
-
OC
IV
300 C III
C 200
100 -
01
0
30
60
90
toroidal angle (degrees)
Figure 5-9 Comparison of simulated profiles to experimentally measured increases
in brightness at a non-limiter port (with LHRF). The straight line at zero on the
right y-axis is the background level before RF. The solid triangles, squares and
circles are the experimental measurements.
- 120
-
5/17/83
C III
CO
4J
0
-
C IV
-
C
100
V-
200
300
400
500
time(msec)
Figure 5-10 Brightness measurements of different charge states of carbon for low
density plasma with LHRF injection. This discharge had a large impurity injection.
- 121 -
500
5-c III
U-c IV
A-c v
400
* 300
cv
M/
(U
C IV
200 -
100
C III
00
0
30
60
90
toroidal angle (degrees)
Figure 5-11 Comparison of simulated profiles to experimentally measured increases
in brightness at a non-limiter port (with LHRF). The straight line at zero on the
right y-axis is the background level before RF. The solid triangles, squares and
circles are the experimental measurements.
- 122 -
N, = 1.2 x 1014
cm- 3
Table 5.3
RF power = 800 kW
Zff (calc) = numerical model
Zq f (brem) = visible bremsstrahlung
Non-Limiter Port
C density
Si density
Zff (caic)
Z,ff (brem)
Before RF
1.5 x 1012
9.2 x 1010
1.48
1.5
During RF
Increase
3.0 x 1012
3.7 x 10"1
2.2
2.8
100%
300%
150%
260%
Limiter Port
C density
Si density
Zff (calc)
Zeff (brem)
7.8 x 1011
3.0 x 101
1.58
1.8
2.3 x 1012
2.1 x 1012
4.3
4.5
230%
600%
470%
340%
discrepancy between the two values of Zeff
for both cases but the numbers are
within the experimental uncertainties./medskip
- 123 -
C v
1.0
0.
8
41)
0. 6
0
E-
0. 4
l-D Model
0
0.0
0
30
60
90
toroidal angle (degrees)
t
t
port
limiter
Figure 5-12 Ratio of C V peak density to total C density as a function of toroidal
angle from 2-D numerical simulation. Assumes limiter is the only source. The ratio
for the 1-D radial model is indicated.
- 124 -
Chapter VI
CONCLUSION
VI.1
Summary of Results
An experimental study of impurities in Alcator C plasmas was undertaken us-
ing spectroscopic instruments operating from the visible to the extreme ultraviolet.
Carbon and oxygen were found to be the principal low Z impurities for ohmic discharges with Mo limiters in the density range, N, > 1 x 10"' cm- 3 . After SiC coated
graphite limiters were installed, significant amounts of silicon were also observed in
the discharges. The type of limiter employed in the tokamak was found to have a
major influence on the amounts and types of impurities which were observed. The
limiter material contained the principal low Z impurities observed in the plasma
when both graphite and SiC coated graphite limiters were used. When molybdenum limiters were used the level of oxygen in the plasma was much higher (a factor
of ten) than with the other limiters, leading to the conclusion that molybdenum
structures retain a great deal of oxygen. Plasma conditions also influenced impurity
emission. Impurity brightnesses were found to depend strongly on plasma density
and therefore large variations in edge density, such as occur during marfes, lead to
large changes in edge impurity emission.
An important experimental observation was that the wall becomes coated with
limiter material such as carbon and silicon. In the case of carbon, the wall is the
main source of impurity influx into the plasma. Therefore all ionization stages
of carbon are toroidally uniform around the torus during ohmic discharges. The
low states of silicon and oxygen on the other hand are observed to be toroidally
asymmetric during ohmic discharges with the limiter being the principal source.
- 125
-
Impurity densities were calculated with a numerical code consisting of two
parts, first a transport model that computes charge state profiles and second a
model determining the level populations of each charge state. These two models
were combined, along with absolute brightness measurements of impurity lines, to
determine impurity density profiles of all charge states. Impurity densities showed
consistency among brightnesses of lines from different charge states of the same
element, and also showed reasonable agreement with independent measurements of
Zff. Values for Zqff were generally found to be in the range 1.2 < Zeff < 1.5
for plasma densities of N, > 1
x
104 cm- 3 .
The numerical code also allowed
calculations of radiated power due to impurities, impurity influx and spectra of C,
0 and Si.
Lower hybrid heating experiments were performed on the Alcator C tokamak.
The injection of RF power into the plasma caused an increase in impurity influx. A
scan of LHRF power showed a rapid increase of AZeff with RF power, particularly
above 500 kW. The major source of impurities during LHRF was determined to be
the limiters. This localized source of impurities resulted in a substantial toroidal
asymmetry of the low ionization states of carbon and a smaller asymmetry for the
high ionization states of carbon. Silicon exhibited toroidal asymmetries up to Si XII.
The reason for these asymmetries is that the lower ionization states become rapidly
ionized before they distribute uniformly around the torus. A detailed transport
model was formulated and compared to the experimental measurements.
Good
agreement was found between the model and experimental observations. One of
the implications of these observations is that, in cases where the limiter is the
main source of an impurity, one must take into account toroidal asymmetries when
calculating total impurity densities and total line radiated power.
- 126 -
VI.2
Suggestions for Future Work
Plasma-surface interactions and impurity behavior are areas which still require
a great deal of research. One of the questions related to these topics which is yet
to be answered is which types of limiters and wall materials would work best in a
fusion reactor. The heat load to limiters will be quite large in fusion plasmas unless
substantial efforts are undertaken to control heat flow at the plasma edge. Accurate
plasma position control can reduce hot spots on the limiter. The trend seems to
be toward limiters composed of low Z materials such as graphite since they radiate
less power away. A possible problem with using graphite is that all surfaces inside
the machine can become coated with carbon. This coating of carbon could have
detrimental effects on auxiliary heating devices which require close proximity to the
plasma. Other possible limiter candidates are carbide (Si or Ti) coated graphite
and beryllium.
An alternative to limiters are magnetic divertors. Poloidal divertors have operated on several devices [27,28] and offer an attractive method of impurity control.
Although divertors do seem to reduce impurity densities, the effect on low Z impurities has been mixed and these impurities have not been well studied.
A good understanding of power balance in tokamak plasmas is another important goal of fusion research. Power lost through impurity radiation can lead to
lower energy confinement. The relationship between radiated power and power to
limiter structures is complex and needs further investigation.
Accurate calculations of impurity densities require simultaneous calibrated
measurements of several impurity lines, particularly from the higher ionization
states. However, these calculated values would be limited in accuracy by assumptions made about the profiles of fully stripped ions. A method has been developed of
- 127 -
measuring the profiles of fully stripped impurities directly. This technique consists
of using a diagnostic neutral beam and observing radiation due to charge-exchange
processes. These profile measurements in turn could be used to calculate any inward convection velocity of impurities. Another factor which impairs the accuracy
of impurity density calculations is the uncertainty in theoretical ionization rates,
recombination rates and excitation rates. There has been much theoretical and experimental work in this area in recent years which has helped improve the situation.
Asymmetries in impurity density can lead to enhanced local radiation, localized
damage to limiters and wall structures, instabilities and disruptions. More detailed
measurements of the poloidal and toroidal distribution of impurities can be achieved
by positioning several spectrometers around the machine and doing simultaneous
measurements of impurity line emission.
Injection of lower hybrid RF and ICRF power into a plasma often results in
large increases in impurity influx. In future large scale experiments, the power density from RF heating is expected to be less than the current Alcator C experiment.
This should result in reduced impurity influxes. However, research on this RF induced impurity influx should continue and methods of controlling this influx may
be necessary.
- 128 -
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- 133 -
Appendix A
Solving Partial Differential Equations
There are many techniques available for numerically solving differential equations. In general, these techniques involve solving the equations explicitly, implicitly
or a combination of the two. What is meant by explicit and implicit is explained
below.
Consider the one dimensional diffusion equation which is the most simple form
of equation (4.11) describing transport of impurities.
0
n=
0
) -
t
a2n(z, t)
82
(J
(A.)
This equation can be written as a finite-difference equation of the form
0= n(z,t + At) - n(z, t) _ Dn(z + Az,t) - 2n(z,t)
2 + n(z - Az,t)
(AX)
At
(A.2)
in which forward time differences are used. Alternatively, the finite-difference equation can be written using backward time differences.
0 = n(x, t + At) - n(z, t)
At(A3
D n(x + Ax,t + At) - 2n(x,t +2 At) -4-n(z - Az,t + At)
(AX)
(A.3)
If we define 62 (X, t) = !n(z+ Ax,t)-2n(z,t)+n(x- Ax,t)) then the finite-difference
equation can be written more generally in the form
n(x,t + At) - n(x,t)
At
_
D
2
(X, t + At) + (1
- O)6 2 (X, t)
2
(AX)
(A4)
where 9 is a real constant lying in the interval 0 < 9 K 1. When 9 = 0, then we
have equation (A.2) which is explicit. When 9 > 0, a set of simultaneous linear
equations must be solved to obtain the solution and the system is called implicit.
The stability condition for equation (A.4) can be easily derived and is given by
2DAt
(Az)
2
<
1
1- 2
.1
if 0 < 0 < 2
-
- 134 -
(A.5)
no restriction
if
1
- <
2
e<
1
This stability condition states that implicit equations which lie in the range
e<
j
<
1, are always stable while the explicit equation, 6 = 0, is only stable if the time
steps are chosen appropriately small. Therefore, it is usually preferable to use an
implicit method when numerically solving a differential equation.
Appendix B
Explicit Method
Using an explicit method is the most direct way of solving a partial differential
equation. Equation (A.2) can be rewritten in the form
n(z,t + At) = n(z,t) - (A
[n(z + Az,t) - 2n(x, t) + n(z - AZ,t)]
(B.1)
which expresses n(z, t) explicitly in terms of its value at an earlier time. An initial
profile must be chosen for n(z,t) and from there it is straightforward to calculate
n(z,t + At). The only restriction is that the time step At and the spatial step
Ax must be chosen carefully so that the solution does not become unstable as the
system evolves in time.
Appendix C
Implicit Method
A fully implicit representation of the diffusion equation can be obtained by
rewriting equation (A.3) in the form
n(x, t) =
DAt
2DAt
-DAt
n(z + Az, t + At) + (I )n(z,t -At) - (AX)2
2 n(z - Az,t + At)
(AX) 2
(AX)
(C.1)
which in turn can be expressed as
-- Ain
Bjnj - Cnj-= Dj
- 135 -
(C.2)
where
j
represents a spatial value. These equations are linear and can be solved
using standard methods for solving linear systems of equations. However, a special
technique can be used to solve these equations by using the fact that the matrix is
tri-diagonal. To use this method we let nj be given by the following expression.
nj = Ejnj+l + F
(C.3)
Now substitute for nj.. 1 into equation (C.2) using equation (C.3). The result is a
relation between nj and nj+1 .
n-+1
B, - CE-.... 1
+
CEj
+
CEj.1
(C.4)
By equating this expression to equation (C.3) we can find a recursion relation for
E and F.
E=
A,
B, - C;E;-1
F =;-E-I
By - CI;_j1
Using the boundary condition at
order of increasing
of decreasing
j
j.
j
j>0
j > 0
(C.5)
= 0, one can solve for all the E and Fj in
Then the nj can be computed using equation (C.3) in order
and using the boundary condition at j = j,.
This method is much more efficient than inverting the full matrix and then solving for nj. Extending this method to a more complex coupled system of equations,
such as equation (3.11), is straightforward .451.
Appendix D
Steady State Solutions
This appendix shows how the solution to equation (3.11) with a steady state
source is obtained by integrating the solution to the initial source problem over
time. Equation (3.11) can be written in the form
af(,t)
3f (z, t) = 0
at
- 136 -
(D.1)
where # is a time independent linear differential operator and f(z,t) is a function
with the boundary conditions, f(z,0) = So(z)r and f(x, oo) = 0. So(x) is a source
function and r is the time interval for the source. The equation of interest for a
steady state source is
Oh(z) = So(z)
(D.2)
The solution we seek is h(x). Integrating equation (D.1) over time with the limits
0 and oo yields
af
,t')dt' +
0
ff(x,t')dt' = 0
(D.3)
0
Since # does not depend on t', it can be taken outside the integral. In addition we
can utilize the boundary conditions at t = 0 and t = oo to write the equation as
00
0 f
(z't')dt'= So(z)r
(D.4)
0
and thus by inspection
00
h(x) =
f(x,t')dt'
0
This is the steady state solution of interest.
- 137 -
(D.5)