PFC/RR-82-15
DOE/ET-51013-45
THERMAL & NON-THERMAL
SUBMILLIMETRE EMISSION
FROM ALCATOR TOKAMAK
by
S. EMIL KISSEL
Department of Physics
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
June 1982
THERMAL & NON-THERMAL
SUBMILLIMETRE EMISSION
FROM ALCATOR TOKAMAK
by
S. EMIL KISSEL
S.B., M.I.T.
(1977)
SUBMITTED TO THE DEPARTMENT OF PHYSICS
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1982
Signature of Author
Department o0 PnysIcs
31 April 1982
Certified by
Prof.
Ronald R. Parker
Thesis Supervisor
Accepted by
Prot. beorge F. Koster
Chairman, Department Committee
Page 2
THERMAL & NON-THERMAL
SUBMILLIMETRE EMISSION
FROM ALCATOR TOKAMAK
by
S. EMIL KISSEL
Submitted to the Department of Physics on 31 April 1982
in partial fulfillment of the requirements for the Degree of
Doctor of Science in Physics
ABSTRACT
An experimental investigation
was
undertaken
of
submillimetre
radiation from The Alcator Tokamak.
Thermal
emission
at
the
cyclotron
frequencies
determine spatial electron temperature distributions.
temperature
electrical
profiles
corroborated
conductivity,
discharge behaviour.
and
classical
provided
was
to
Measurements of
models
graphic
used
of
plasma
displays of unusual
Temperature fluctuations, such as sawteeth, were
also investigated.
Non-thermal
frequency,
radiation
was
observed
the
and occurred in two distinct fashions;
feature, and a fluctuating emission of
fluctuating
plasmas.
at
emission
very
narrow
electron
plasma
a steady broadband
bandwidth.
The
has not been previously documented from Tokamak
Pertinent theoretical considerations are outlined.
Thesis Supervisor: Prof. R.R. Parker
Page 3
TABLE OF CONTENT
SECTION NULL
SEC
SEC
SEC
SEC
INTRODUCTION
0.1
0.2
0.3
0.4
1.1
1.2
1.3
1.4
1.5
1.6
SEC
SEC
SEC
SEC
SEC
2.1
2.2
2.3
2.4
2.5
SECTION THREE
SEC 3.1
SEC 3.2
SEC 3.3
SEC 4.1
SEC 4.2
SEC 4.3
SEC
SEC
SEC
SEC
NON-GAUSSIAN PROFILES
5.1
5.2
5.3
5.4
..........................
...........................
Current Rise
.....................
Impurity Radiation
...................
References & Figures
........................
15
17
19
23
25
27
35
.
....
. 35
36
40
42
43
51
51
53
56
60
60
............
Reflections & Optical Depth
......... 62
Profile Shape
... * ..... .........
63
..................
References & Figures
TEMPERATURE FLUCTUATIONS
SECTION FIVE
15
..................
Gaussian Profile
.....................
Comparison To Classical Resistivity
Profile Width
.........................
Profile Centre ..........................
...................
References & Figures
THIRD HARMONIC EMISSION
SECTION FOUR
...................
...............
Single Particle Emission
....
......
.......
Plasma Effects
Optical Depth
............................
............
Reflections & Polarisation
Temperature Profile ......................
...................
References & Figures
ELECTRON TEMPERATURE PROFILES
SECTION TWO
7
7
....................
The Alcator Tokamak
................. 8
Organisation Of Thesis
Introduction To Instrument ............. 10
12
List Of Symbols
.......................
REVIEW OF CYCLOTRON EMISSION
SECTION ONE
SEC
SEC
SEC
SEC
SEC
SEC
...................................
....................... 68
.............
Fabry-Perot Interferometer
...................
Sawtooth Instability
.........
Sympathetic Plasma Oscillation
References & Figures .....................
68
69
73
75
Page 4
6.1
6.2
6.3
6.4
SEC 6.5
SEC
SEC
SEC
SEC
SECTION SEVEN
SEC
SEC
SEC
SEC
7.1
7.2
7.3
7.4
SECTION EIGHT
SEC
SEC
SEC
SEC
SEC
8.1
8.2
8.3
8.4
8.5
9.1
9.2
9.3
9.4
9.5
9.6
9.7
SEC 9.8
SEC
SEC
SEC
SEC
SEC
SEC
SEC
Cerenkov Emission
.................
84
........... ...
................. ...
Ray Trapping
References & Figures
.......
Temporal Behaviour
86
89
96
FLUCTUATING PLASMA FREQUENCY EMISSION
.....................
97
.................... 99
Correlated Behaviour
...... 100
Comparison With Other Experiments
References & Figures
101
.................
.. 110
SPECTRAL MEASUREMENTS OF FLUCTUATING EMISSION
110
...............
Michelson Interferometer
.................
Fabry-Perot System
.. 112
.114
........... 116
Shifts In Emitted Frequency
................... 118
References & Figures
High-Resolution Spectra
..........
........
.....
....
126
126
Stimulated Emission ......................
...... 128
Related Experimental Observations
132
* ....
.................
Runaway Electrons
.134
Plasma Wave Modes .............. *.....
. 136
.....
Other Waves .................
*........137
Saturation Of Plasma Waves
139
.............
Linear Radiation Processes
141
...................
References & Figures
FLUCTUATING EMISSION: NON-LINEAR EFFECTS
SECTION TEN
81
81
....................
Spectral Character
......... 83
Review Of Previous Experiments
FLUCTUATING EMISSION: PLASMA WAVES
SECTION NINE
SEC
SEC
SEC
SEC
SEC
SEC
.............
DOMINANT PLASMA FREQUENCY EMISSION
SECTION SIX
*.....
144
10.1
10.2
10.3
10.4
10.5
10.6
.................
.144
Resonance Conditions
.......... .......... 147
Dispersion Relation
.................... 149
Non-linear Current
..
9..... ......... 151
Absorption
.153
...............
Relation To Experiment
.. .....
. ........
References & Figures
1
2
3
4
5
6
7
8
............ ........... .. 159
Optical Path
............. 162
Reflections & Polarisation
........... 163
The Michelson Interferometer
......... 165
Fourier Transform Spectroscopy
170
................................
Detector
9.......171
....... ....
Calibration
....174
Atmospheric Water Vapour Absorption
................... 175
References & Figures
99155
APPENDICES
APX
APX
APX
APX
APX
APX
APX
APX
T 0
T H E
H A P P Y
F E W
Page 5
ACKNOWLEDGEMENT
The Alcator experiment is a collective effort requiring the labours of
numerous
individuals.
I would like to acknowledge the contributions
by the support staff, technicians and scientists
which
allowed
this
work to be performed.
I am especially grateful for
occasionally
necessary
the
prodding
guidance,
provided
by
encouragement,
and
Dr. Ian Hutchinson.
While I might question
what
presented
is no doubt of the great personal gain which I
him,
there
pernicious
have received from our collaboration.
as
product
our
proximity
Of greatest importance
to
has
me,
a student, has been the inspiration Ian has provided while serving
as my role model of the true scientist and scholar.
I also
'seagulls'
thank
with
the
whom
eclectic
I have
community
lived
of
and shared.
graduate
Their indomitable
spirit surely makes them birds that have not lost faith
virtue of flying.
student
in
the
high
Page 6
Dedicated in memory of Therese Crossley
with rue my heart is laden
for golden friends I once had
for many a rose lipped maiden
and many a light foot lad
by brooks to broad for leaping
the light foot boys are laid
the rose lipped girls are sleeping
in fields where roses fade
Page 7
SEC 0.0
This
thesis
radiation
during
INTRODUCTION
made
the
presents
in
conjunction
period
requirements
1978
with
through
of
submillimetre
plasma
the Alcator tokamak experiment
1981.
As
by
stipulated
the
for the award of Advanced Degrees, the included material
represents novel results
research
measurements
documented
of
original
herein
is
investigation.
Most
of
related directly to the operation of
Tokamak devices and should be of interest to persons involved in
field,
particularly
the
area
of
of plasma physics.
this
submillimetre plasma diagnostics.
However, some of the items recorded are pertinent to the more
field
the
general
For this reason the reader is assumed to be
knowledgeable in the area of plasma physics, but not
specifically
au
fait in the arena of Tokamak research.
SEC 0.1
THE ALCATOR TOKAMAK
The Alcator C tokamak is shown schematically
plasma
is
geometry.
produced
in
fig.(1).
The
and confined within a magnetic field of toroidal
A large toroidal field is
externally
produced
by
magnet
coils, while a relatively smaller poloidal magnetic field is generated
by an induced toroidal current within the plasma.
is
necessary
both
to
current,
the
Because
plasma
of
generated
the
induced
This discharge time of several hundred
much
than
steady state
discharge.
is
characteristic
said
to
occur
nature
of
the
can only be maintained for a
finite time.
longer
current
provide a confining force, and as a source of
energy to heat the plasma.
plasma
The plasma
milliseconds. is
collision or transport times, so a
during
the
middle
portion
Typical parameters for steady state operation are:
of
a
Page 8
6 x10 14
Plasma Density:
.5 x101 4
Plasma Current:
200
-
600
Electron Temperature:
1000
-
2000
Electron Volts
Ion Temperature:
800
-
1500
Electron Volts
Toroidal Field:
50
Majority Species:
Hydrogen, Deuterium, Helium
Of course the temperature,
beginning
and
end
of
-
-
cm-3
kAmperes
100 KGauss
current,
and
a discharge,
density
so
are
nill
at
the
a rise and fall phase will
precede and follow the steady state portion.
The time evolution of
a
representative discharge is shown in fig.(2).
The hot plasma is confined entirely
chamber
so
that
measurements
of
within
plasma
a toroidal
properties,
conditions at the discharge centre, must be inferred
technique.
For
this
reason
by
information
to
be
so
some
benign
since
by
this
can be gathered without perturbing the discharge.
Observations of submillimetre
proven
particularly
the study of electromagnetic radiation
emitted by the plasma is well suited as a diagnostic
method
vacuum
useful
in
wave
this
radiation,
in
respect,
that
measurement systems of this kind have become
particular,
integral
in
have
recent years
components
of
most tokamak experiments.
SEC 0.2
most
ORGANISATION OF THESIS
Cyclotron radiation is the principle mechanism
responsible
submillimetre
A review of this
emission
from tokamak plasmas.
radiation process is outlined in sec (1).
emission
is
the
capability
for
An implication of cyclotron
of determining electron temperature, in
Page 9
particular the spatial dependence or
temperature
second
technique
harmonic
radiation.
This
profile,
has
become
recently, so there is nothing new in the presentation
profiles
per
se.
from
of
the
standard
temperature
The purpose of the first half of the thesis is to
apply the measured temperature profiles towards a better understanding
of the physics involved in the discharge.
In sec (2), (3) it is
consistent
with
the
confidence
the
measured
profiles
The importance of this is the justification
classical
toward
measurements.
that
are
a classical resistance model over the full range of
operating conditions.
applying
shown
the
Radiation
model
to
of
the plasma, and the generation of
accuracy
of
the
temperature
profile
at the third harmonic has been the topic of
recent discussion, chiefly as a means of determining electron density.
Sec (4) present
measurements
demonstrate that although
the
density
does
profiles,
it
of
method
self-consistent calibration of the
important
tokamak
practical
research.
fluctuations
the
third harmonic emission which
is
inadequate
represent
cyclotron
for
a useful
acquiring
technique
diagnostic.
This
for
has
consequences for anyone involved in this area of
Sec (5) presents
measurements
of
commonly observed during tokamak operation.
temperature
While these
results demonstrate the wide ranged applicability of the submillimetre
diagnostic, they also represent a worthwhile addition to the data base
regarding phenomena, such as sawteeth, which are
interest.
of
current
topical
Page 10
The remaining half of the
thesis
deals
with
Some kind of radiation of this sort has been observed by most
and previous experiments.
good
for
between
theory
and
some types of non-thermal emission.
the emission discussed in sec (6).
however
does
expand
The
upon
material
the
experiment
presented
existing
measurements
is
This is true of
data,
distinguish two similar types of non-thermal emission.
present
current
Theoretical explanations have been advanced
to the point where the agreement
section
a
origin, in particular emission near the plasma frequency.
non-thermal
quite
of
radiation
in
this
and
helps
Sec (7),
(8)
of a second type of emission which has not been
previously documented.
The measurements
in
these
sections
clearly
differentiate this new emission from the process discussed in sec (6).
Sec (9),
results
(10)
review
currently
the
pertinent
theoretical
and
experimental
available concerning this novel type of radiation.
The intent in these sections is not so much
to
processes
attention on the sorts of
involved,
as
it
is
to
focus
identify
the
mechanisms which can be reasonably expected to apply, and to
actual
indicate
those which cannot.
SEC 0.3
INTRODUCTION TO INSTRUMENT
Most of the submillimetre spectral measurements presented within
this
thesis
were
made
with a Michelson interferometer.
discussion of the operation of this instrument
appendix.
is
A detailed
presented
in
the
However, a brief introduction to the interferometer system
is appropriate prior to a presentation of measured spectra.
Page 11
Given some spectral intensity 1(w) of frequency w, the output of
an ideal interferometer is:
EQ (1)
J(x)
1(w) cos(wx/c) du
f
=
0
where x is the optical path difference of the
system.
In
operation
the optical path is varied so that J(x) is measured over a range of x.
The spectral intensity is related to the interferometer output by
the
inverse transformation:
1(w) =
EQ (2)
cos(Wx/c){J(x)-j} dx
-f
0
where T is the average value of J(x).
I(w)
can
be
constructed
provided
The
exact
that
J(x)
spectral
is
semi-infinite range of x. In practice this is of
Instead
intensity
measured
course
x is periodically varied over a limited range.
impossible.
The spectral
intensities deduced from eq (2) are thus based on incomplete
information,
and
therefore
will
spectra.
Since
it
sets
of
have limited frequency resolution.
The period with which x is varied sets
measured
over a
the
time
resolution
of
the
is necessary to assume a constant I(w)
during one measurement cycle, the spectra will
represent
an
average
over this cycle time.
Two important aspects of the interferometric measurements can be
summarised
by
the
application of some
output,
not
frequencies.
from
following:
form
of
The produced spectra result from the
eq (2) to
the
entire
interferometer
individual measurements of intensities at separate
Although there is a direct
relation
between
the
path
Page 12
difference
x
and
the
time
during which the measurement of J(x) is
made,
no such relation connects the intensity at a specific
with
a
specific time.
frequency
The intensity at a specific frequency results
from information gathered over the
entire
time
during
which
x is
varied.
SEC 0.4
LIST OF SYMBOLS
Tokamak research has
field.
Producing
produced
a lexicography
thankless chore, so the usage of
avoided.
a vocabulary
peculiar
to
the
of Tokamak terminology would be a
non-standard
terminology
has
been
The commonly used symbols are listed below as a convenience.
It should be noted that most of
these
quantities
are
functions
space or time or both.
a...... Temperature profile width
Y..
Relativistic factor
B...... Magnetic field (Toroidal)
t.....
Optical depth
n......
Plasma current
I(w)... Spectral intensity
Plasma frequency
np...
Gyrofrequency
n c-----
J(r)... Plasma current density
Resistivity
Plasma density
qn...... Refractive index
Subscripts, Superscripts
Rn...... Safety factor
Major radius of torus
0.
.
. . .
Ordinary
rlim--,. Minor radius, limiter radius
x
T ...... Temperature
0.... .
Central value
V ...... Toroidal loop voltage
e......
Electron
v ...... Particle velocity
i......
Ion
Zeff..--
Effective ion charge number
......
Extraordinary
of
Page 13
THE ALCATOR TOKAMAK
TOROIDAL FIELD
MAGNET
TOROIDAL
HORIZONTAL
ACCESS PORT
VACUUM CHAMBER
Scale 10 cm
Fig. (1)
T
EXAMPLE TOKAMAK DISCHARGE
PLASMA CURRENT (kamp)
-
400!
200
2.0
LINE AVERAGE DENSITY (1014 cm-)
-
1.0
____________________
-
I
I
I
ELECTRON TEMPERATURE (keV)
1.0
TIME (mwed)
---
100
200
300
400
Fig. (2)
Typical time evolution of Alcator plasma discharge. Current
is determined by measurement of magnetic flux, density by laser
interferometry, temperature by central second gyroharmonic radiation.
Fluctuations in temperature are caused by sawtooth instability.
Page 15
SEC 1.0
REVIEW OF CYCLOTRON EMISSION
Electron
cyclotron
acceleration
of
the
emission
charged
results
particle
in
from
the
centripetal
a static magnetic field.
'Synchrotron radiation' is a term connoting relativistic energies
is
and
misleading when applied to a plasma with non-relativistic particle
energies.
Cyclotron emission by the plasma can
superposition
be
regarded
as
the
of numerous independently radiating electrons, although
the effects of dispersion, absorption and other aspects of the Tokamak
environment must be considered.
The
submillimetre
discharges
results
emission
measured
from
typical
entirely from cyclotron emission.
Alcator
Because of the
larger magnetic fields encountered in Alcator, the emission occurs
higher
frequencies
at
but is otherwise similar to measured spectra from
other Tokamaks.
SEC 1.1
SINGLE PARTICLE EMISSION
The exercise of computing the emission by a single electron in a
magnetic
field can be found in numerous textbooksl, 2 , and consists of
determining the radiation field arising from the current
of a gyrating electron.
distribution
For the geometry of fig.(1) the emitted power
from a single particle is:
W
d2 ,v/c'(
dild w
EQ (1)
.27rc
n=1
=
x cote (cose - vI/c) Jn(*)
2
e2w2
ICJ
z (cose - vI/c) Jn(*)
nv1 /c [1-(v /c)cose]~1 sinE
w =. ne
[1-(v 1/c)cose- 1
n
MC
where the direction of the vector quantity being squared indicates the
Page 16
polarisation
of
the electric field.
radiation at harmonics of the
Two noteworthy features are the
gyrofrequency,
and
the
disparity
of
emission intensity into the two distinct polarisations.
The frequencies of
radiation
are
determined
by
the
fourier
harmonics of the periodic current distribution of the electron motion,
giving rise to integrals of the form:
EQ (2)
which
(sinwcet) exp(i .L sinwcet - wt) dt
C
f
will
be
non-vanishing
for
integer
multiples
of
the
gyrofrequency, thus accounting for the radiation at the gyroharmonics.
It is this type of integral which produces
eq (1).
the
Bessel
functions
in
The relative intensity of emission at subsequent harmonics is
approximately:
EQ (3)
=
[2n(*)I2n1()
In-1
occurs at the fundamental,
the
local
c
most
of
the
emitted
power
or first gyrofrequency.
The observed radiation
upon
-
4n
so that for a non-relativistic electron
depend
+
=
frequencies,
magnetic
as
field,
indicated
and
projected velocity of the emitting electron.
For
upon
in
the
eq (1),
energy and
a single
particle
these quantities are constants, so emission will occur at harmonics of
a narrow
broadening.
frequency,
The
the
width
relativistic
being
factor
determined
y
can
by
radiational
decrease the emission
frequency, but only for uncharacteristically energetic particles.
Page 17
The
two
modes
extraordinary,
and
of
the
called
and
ordinary
refer to the polarisation of the electric vector.
Differing amounts of
example,
are
propagation
radiation
ratio
of
are
emission
emitted
into
into
each
mode.
two
modes
of
the
For
nearly
perpendicular propagation at any harmonic frequency is approximately:
EQ (4)
n(z)
2
).2
v [nU'
In(x)
v, [j'()
so that a non-relativistic electron
r v-1
riv 2
is
expected
to
emit
cyclotron
radiation primarily into the extraordinary mode.
SEC 1.2
PLASMA EFFECTS
Electrons will radiate at
determined
by
harmonics
of
the
the local value of magnetic field.
gyrofrequency
as
The toroidal field
as a function of position is:
BR
B(r) = --0
EQ (5)
R+r
where r is the radial position along the midplane
geometric
axis.
broadening
eq (1).
For
broadening
produced
example,
at
by
the
the
This spread is
Doppler
the
second
much
larger
or relativistic terms of
gyroharmonic
the
various
processes acting upon a typical Alcator plasma (BT = 60 Kg
ne = 1014 cm- 3 , Te = 1 KeV) will have the following
collisional -
5x10- 8 ,
magnetic - 5x10- 1 . For
relativistic
from
A large spread in radiated frequencies will result
from the spatial variation of the field.
than
measured
relativistic - 6x10- 4,
angles
e within
10
effect: Aw/ 2 wce
=
Doppler - 5x10- 2 cose,
of
perpendicular
the
broadening term exceeds the Doppler width, however, both
Page 18
are small compared
variation.
Thus
to
the
width
by
produced
the
magnetic
field
the frequency extent of radiation at any particular
gyroharmonic will be determined
by
the
variation
of
the
toroidal
Because of limitations imposed by construction of
the
toroidal
magnetic field, this being the larger effect.
magnet,
optical
access
to
the
plasma
is restricted to directions
perpendicular to the magnetic field, hence the observed radiation must
be
travelling
in
this
direction. The two polarisation states
will
describe propagation perpendicular to the toroidal magnetic field. The
ordinary
mode
is
polarised
with electric vector oriented along the
field direction, while the extraordinary polarisation has
perpendicular
to
the field.
its
vector
The cold-plasma dispersion relation for
these two modes of perpendicular propagation is shown in fig.(2).
ordinary
mode
is
plasma frequency,
non-propagation
accessible
while
between
the
the
to
The
radiation with frequency above the
extraordinary
mode
upper-hybrid
and
has
a
region
right-hand
of
cutoff
frequencies.
Due to the spatial variation of the plasma density and
magnetic
field, the dispersion relations for the two modes will have positional
dependence.
each
In particular the cutoff
mode. will
change
for
and
resonance
each radial position.
frequencies
This is shown in
fig.(3), where the critical frequencies are plotted as a function
radius for a typical plasma condition.
first harmonic propagating outward
will
encounter
the
this reason no extraordinary polarisation first harmonic
This
of
Extraordinary radiation at the
upper-hybrid
layer before the plasma edge, and will be absorbed or reflected.
observed.
of
emission
For
is
situation would not be expected if, for example, the
i
Page 19
detection system was located at the inner
by
precluded
the
design
and
plasma
edge,
a situation
of the Alcator tokamak.
construction
Ordinary mode first harmonic emission can be observed,
provided
that
wpe/Oce < 1.
When harmonic cyclotron emission is considered in the context of
a magnetoactive
plasma
a necessary
radiation absorption by the plasma.
this
absorption
will
consideration
becomes that of
For a sufficiently
dense
plasma
alter the emission produced by a collection of
individual electrons to the extent that a single particle treatment is
inadequate.
SEC 1.3
OPTICAL DEPTH
In a plasma, radiation at any frequency must travel from a point
of emission to a point of observation along a path which may allow for
absorption and reemission of radiation at
equation
of
the
same
frequency.
The
radiation transfer describing this process of absorption
and emission along the ray path is3:
d
dI(w)
=
ds I ds 1
--
EQ (6)
.
dI(w)
-w
-
L
d92
where j. is the volume emissivity and a. is the spectral
governing
radiation
at
frequency
w.
Both of these quantities are
integrals of the total electron distribution function,
the combined
absorptivity
behaviour of individual plasma
electrons.
becomes:
=dI(w)
1- exp(- faw ds)
raypath
I
Q ()d
and
represent
The emission
Page 20
distribution
For a Maxwellian electron
law
Kirchhoff's
relates the
emissivity and absorptivity 4 :
EQ (8)
-.
aw
bb(w)
=
-1
N3
Ew
3
kT _]
87*3C2
with the result that:
EQ (9)
bb [ds
8l
raypath
where Ibb ( the temperature ) remains constant
across the resonance.
Since the radiation quantum energies are small compared to the thermal
plasma
energy,
hiw(kT, the blackbody formula can be expressed in the
Rayleigh-Jeans limit, Ibb
The optical depth
absorption
affects
=
is
T,
radiation
and the plasma
ignored
For
radiators.
w2kT/8w 3 2
T>>1
properties
to which
of the
transfer.
If Tr1, absorption can be
treated
as
a collection
of
independent
the radiation will approach an equilibrium with
the emitting electrons, so that its intensity
thermal
degree
a measure
of
the
plasma.
will
If the
characterise
optical depth is much
greater than unity for a particular frequency, the emission
at
that
in
thermal
intensity
approaches the blackbody level, provided that the
frequency
electron distribution is near equilibrium.
is
the
equilibrium,
it
cannot
Furthermore, if the plasma
radiate more than the local
blackbody level, IbbDetermination
absorptivity
along
of
the
T
by
eq (9) requires
raypath
for
integration
a particular
absorption is a local process provided that the scale
resonance
between
of
the
frequency.
The
length
the
of
wave and electrons is small compared to variations
Page 21
in electron density, or temperature, or magnetic field, and of course,
that no overlap of harmonics occur.
The scale length is determined by
the magnetic field variation together with the
appropriate
resonance
width:
On
L=
EQ (O)
R (2n)
=
where the relativistic width is used for
propagation.
of
of
as
the
above
scale
length
perpendicular
multiplied
by
the
local
The absorptivity can be found through the imaginary
refractive index 5 , or from
the
emissivity.
of
The optical depth is thus a local quantity, and can be
absorptivity a..
part
case
It is seen that L is much smaller than typical gradients
in the plasma.
thought
the
Kirchhoff's
law
and
the
The optical depth for the nth harmonic is6:
2
kT(r) n-i
R pe(r) n n
2 c wce(r) (n-1)! 2mc2
(r)
T(r)
0
=
0.15
2
kT(r)
kr2
T (r)
2
for the low density, wp<<wc, case
in
In the
a
wp
Caic
limit eq (11)
is
the
mildly relativistic limit.
fair approximation for harmonics
n > 1, but is not applicable to the first harmonic.
Page 22
Absorption at the first gyroharmonic requires special treatment,
for the extraordinary polarisation.
particularly
is that
while
extraordinary
the
ordinary
mode
has
mode
an
is
purely
electrostatic
The reason for this
electromagnetic,
component
elliptically
polarised perpendicular to the magnetic field.
It
that
component
with
left-hand
rising
density
polarisation,
associated
the
electrostatic
instead
of
the
with the extraordinary mode.
the electron gyrations and so reduces
the
has
been
right-hand
shown7
produces a
case
normally
This decouples the wave from
the
absorption.
For
Alcator
the theory 8 predicts an optically thick ordinary, and thin
conditions
extraordinary
completeness
mode
it
at
is
the
first
mentioned
harmonic.
For
the
measurements 9
that
sake
of
in other devices
indicate an optically thick extraordinary mode, this being
attributed
to a mode conversion of electrostatic (Bernstein) waves' 0 .
harmonic
the
and
3 rd
extraordinary polarisation, for perpendicular propagation in
properties"
The contribution of plasma dielectric
the regime wp<Wc.
to
2 nd
the
The concern in this thesis is primarily with
absorption
at
higher
harmonics
results
from relativistic
2
(vth/c) 2 and finite density (wp/Wc) effects. It has been argued1 that
a relativistic
treatment
is necessary for the
2 nd
harmonic, even at
low (1 KeV) temperature, but not for higher harmonics.
The finite density effect for E = w/2 has been calculated in the
non-relativistic limit for harmonic n13:
E
aE X =2,X(E.C.)
EQ (12)
S= 11
2
2 n2 -
1 n-3/2
On2
1
-
(1
+
- O)n1 2
n2
Page 23
x
where aw (E.C.)
refers to
eq (11),
the
Englemann,
Curatolo result.
For a situation of interest, wp/wc = .8, eq (12) yields the following:
EQ (13)
n=2:
a
=
1.13 . x(E.C.)
The absorption is enhance slightly at the
slightly
limit.
the
at
the
3 rd,
over
fact that T>>1.
that
T
be large.
.93 a x(E.C.
harmonic, and
decreased
2 nd
harmonic does not
change
It is noted here that nowhere in what
follows is an explicit value of
only
2 nd
=
the values predicted in the low density
The small correction to a at the
essential
a
n=3:
T
for the
second
harmonic
required,
This requirement is satisfied when the finite
density are included.
Since the blackbody level scales with the square of the radiated
frequency,
which
in turn is a linear function of the magnetic field,
high field devices such as
cyclotron
radiation.
Alcator
can
generate
higher
levels
of
However, the radiated power from Alcator due to
cyclotron emission is inconsequential compared
to
other
power
loss
mechanisms, and to the ohmic input power.
SEC 1.4
REFLECTIONS & POLARISATION
Fig.(4) shows representative submillimetre spectra
of
the
two
polarisations from a tokamak discharge, as measured with the Michelson
interferometer
corresponding
system.
to
the
Emission
first
three
harmonic feature being determined
variation across the plasma.
is
recorded
at
the
frequencies
gyroharmonics, the width of each
primarily
by
the
magnetic
field
As expected, emission intensities at the
optically thick frequencies are limited to the blackbody level,
while
Page 24
at
intensities
higher
frequencies
below this level.
are
harmonic extraordinary radiation is detected, in accordance
predicted
absorption
examination shows that
are
polarisations
description.
the
at
the
upper-hybrid
layer.
the
with
However, cursory
emission
of
intensities
No first
into
the
two
not in the ratios predicted by the single particle
This is particularly true at the
third
harmonic
where
the optical depth is less than unity for both modes, so that the ratio
of intensity should be nearly approximated by eq (4).
The presently accepted explanation for
observed
ordinary
mode
emission at the second and third gyroharmonics requires the reflection
of extraordinary mode radiation from the vacuum
reflected
wave
chamber
wherein
undergoes a change in polarisation, thus appearing to
the observer as if emitted with the ordinary polarisation.
irregular
internal
unreasonable.
shown
the
Given
the
construction of tokamaks such a hypothesis is not
The relative polarisation of the second harmonic,
5:2,
in fig.(4) is the largest value obtainable from Alcator, and is
consistent with
polarisation
measurements
from
other
model 15 ,16 .
scrambling
tokamaks 14
According
total intensity of every reflected ray
is
some
to
and
with
the
this model, the
fraction
r of
the
incident ray, where r < 1, and some portion p of the reflected ray has
changed polarisation state.
The polarisation ratio then becomes:
EQ (14)
10
P
Ix
1+p-r
Furthermore it follows that the extraordinary mode intensity is:
-T
EQ (15)
p = r-p 1-
x
1
-
peT
Page 25
For the stated polarisation ratio, and p = .5 as
measured
from
3 rd
harmonic emission [sec (4.1)] eqs (14),(15) are satisfied with p = .24
and r = .64.
r = .85.
For Alcator A17 this same treatment produced p = .15 and
The errors in the measurements are such that the two sets of
values are considered approximately
similar
internal
construction,
equal.
so
The
that
two
machines
have
comparable
results
are
disallow
ignoring
the
frequencies for optically thick frequencies.
The
expected.
The presence of
doppler
spread
reason being
absorbed
of
that
by,
reflections
any
reflected
does
not
radiation
must
recross,
and
be
the optically thick layer, and hence cannot be observed
from the outside.
For this reason the doppler
broadening
effect
is
ignorabl e.
SEC 1.5
TEMPERATURE PROFILE
For the optically thick second harmonic extraordinary mode,
spectral
width
is
governed by the magnetic field spatial variation,
with intensity at each frequency
blackbody level.
proportional
the
corresponding
frequency
can
be
The blackbody level at a
associated with the temperature at the
equivalent radial location within the plasma,
eq (9).
to
A conversion of frequency to spatial position can be
made by use of the magnetic field, eq (5)18.
particular
this
being
done
be
used.
treatment.
thick
From fig.(3) it is seen that the first harmonic
ordinary and second harmonic extraordinary
this
with
In this fashion spatial electron temperature profiles can be
mapped 19 . It is only required that radiation from an optically
frequency
the
mode
are
candidates
for
The second harmonic is preferred since it will have
Page 26
the
higher
thus
intensity,
Furthermore,
emission
at
sufficient plasma densities.
plasma
density
during
a better
the
first
Fig.(5)
signal
to
harmonic
can
shows
a discharge
the
causes
encroach upon the first harmonic, with the
noise
be
case
ratio.
cutoff
where
for
rising
the plasma frequency to
resulting
attenuation
of
first harmonic emission.
Electrons of
spectrum
through
relativistic
energy
can
broaden
the mass term and the doppler shift.
the
frequency
These effects
will spread the region of emission beyond the frequency domain set
the
magnetic
field variation.
The resulting spectra will eventually
become a continuum, with no distinguishable harmonic structure 20 .
example
of
plasma
An
broadband spectrum is shown in fig.(6) for extraordinary mode
emission.
band
by
The low frequency cutoff results from
the
extraordinary
frequency
continuous
is
spectrum
mode.
discussed
The
in
the
non-propagation
additional feature near the
sec(8).
in the vicinity of the
A depression
2 nd
of
the
harmonic, apparent in
fig.(6), results from absorption by the thermal electrons.
Electrons with y > 1 will emit radiation at the
as
thermal electrons in a lower magnetic field.
gradient this thermal layer
emission
will
always
and the observation point.
be
chamber
in
order
perpendicular to the field.
entire
plasma
cross
optically thick layers.
to
be
between
the
point
of
Doppler broadened radiation from
observed
being
back
since
After reflection
section,
frequency
Because of the field
relativistic electrons must be scattered from the
vacuum
same
subject
it
wall
of
the
it is not emitted
must
to
traverse
the
absorption by any
Cyclotron radiation by relativistic electrons
will be absorbed at the thermal layer, and the observed radiation will
Page 27
be characterised by the temperature of this layer.
the
2 nd
broadband
corresponding
cyclotron
suggested that
modify
the
that
to
the
thermal
to
a temperature
emission
by
level
of
with
the
850 eV.
relativistic
measured
It has been
electrons
will
intensity of the optically thick harmonic, and so perturb
the profile.
seen
fig.(6)
emission in the frequency range of the optically thick
harmonic is absorbed
radiation
in
Thus
However, from the previous argument and example,
such
it
is
emission will be absorbed to the appropriate thermal
level.
SEC 1.6
REFERENCES & FIGURES
1.)
Classical Electrodynamics
J. D. Jackson
Wiley & Sons, New York (1962)
2.)
W.K.H. Panofsky, M. Phillips Classical Electricity
& Magnetism
Addison Wesley, Reading Mass. (1962)
3.)
G. Bekefi Radiation Processes in Plasmas
Wiley & Sons, New York (1966)
4.)
The integra
1/n2 i.e. ki
similar term
angle of the
medium.
5.)
M. Bornatici, F. Engelmann, G.G. Lister
Phys. Fluids 22 1664 (1979)
6.)
F. Engelmann, M. Curatolo
Nuclear Fusion 13 497 (1973)
7.)
M. Bornatici, F. Engelmann
Comments Plasma Phys. Cont. Fusion 4 139
ovej k-sace will introduce a factor of
= w /n2 c . This will be canceled by a
which accounts for the.change in solid
ray as it passes through the refractive
(1979)
8.)
Y. Dnestrovskii, D.P. Kostomarov, N.V. Skrydlov
Zh. Tekh. Fiz. 33 922 (1962)
9.)
A.E. Costley, T.F.R. Group
Phys. Rev. Lett. 38 1477 (1977)
10.)
J. Hosea, V. Arunasalam, R. Cano
Phys. Rev. Lett. 39 408 (1977)
Page 28
11.)
12.)
13.)
14.)
C.M. Celata, D.A. Boyd
Nuclear Fusion 19 423
I.P. Shkarofsky
Phys. Fluids 9 561
K. Audenaerde
Plasma Physics
F. Stauffer
(1979)
(1966)
19 299
(1977)
Private Communication
15.)
A.E.Costley, R.J.Hastie, J.W.M. Paul, J. Chamberlain
Phys. Rev. Letters 33 758 (1974)
16.)
I.H. Hutchinson
Plasma Physics 21 1043 (1979)
17.)
I.H. Hutchinson, D.S. Komm
Nuclear Fusion 17 1077 (1977)
18.)
It is assumed that no overlap of harmonic frequencies
occurs, so that any point in space can be resonant
with only one harmonic at one particular frequency.
19.)
R. Cano et.al.
Nuclear Fusion 19 1415 (1979)
20.)
C.M. Celata, D.A. Boyd
Nuclear Fusion 17.735 (1977)
Page 29
GEOMETRY OF CYCLOTRON EMISSION
A
X
OBSERVER
B
Vii
B
0
A
z
A
y
A4
V
Sinl
t] - A[FVL
tj
y
COS
1
-e B
ELECTRON GYROFREQUENCY
o
A
t] + V,,t Z
'Ymc
Fig.(1)
Radiation is observed
perpendicular
to
the
magnetic field;
that is, along the x axis. The
radiation from a single particle will be symmetric
with respect to rotation about the z axis. y is
the relativistic factor.
y = 1 for
thermal
plasmas.
Page 30
DISPERSION RELATION FOR PERPENDICULAR PROPAGATION
3.0
k2c2
I
I
I
I
2.0
/
/
EXTRAORDINARY POLARISATION k2 c 2
2
2
r2
(W2
U) 2_ Wh 2
0 1
22
ORDINARY POLARISATION
k2 2
k2
/
1.0
1
-
2
-
)p
U-2---------------------------
/
o/
/
I
I
0.0
/
/
/
/
/
I
/
/
FREQUENCY --ph
P.
rC
1.0
2.0
Fig. (2)
Cold plasma dispersion for wpe/we
frequency branches are not shown.
3.0
.57
Low
Page 31
CRITICAL FREQUENCIES vs. RADIAL POSITION
1.5
C
h
-
1.0
Wp.
0,5
-10
0
RADIUS (cm)
.10
Fi g. (3)
For wpe/wce = .57 .Shaded
areas are regions of
non-propagation
for
the
extraordinary mode.
Ordinary mode does not propagate below the wpe
curve.
Radiation moves from left to right, where
it is observed on the outside (low field) side of
the Tokamak.
ro~
POLARISED CYCLOTRON EMISSION
I(W)
ARBITRARY UNITS
BLACKBODY LEVEL
EXTRAORDINARY POLARISATION
co
3 Cz
O
ORDINARY POLARISATION
200
400
600
FREQUENCY
800
GHz
Ordinary and extraordinary mode emission
Hydrogen plasma in 60 kGauss magnetic field.
spectra are plotted on same scale.
from
Both
3~
CUTOFF OF FIRST HARMONIC
Page 33
ORDINARY MODE EMISSION
I(CO)
WC
= 1.0
TIME = 170 msec
c
=.8
C
TIME = 100 msec
- =- .6
W6TIME
200
=30 msec
400
600
FREQUENCY
800
GHz
The situation Ope 'Oce is rarely encountered in
the
Alcator C because
plasma disrupts. This case for 60 kGauss. All
Fage
BROADBAND CYCLOTRON EMISSION
1(W)
BLACKBODY LEVEL FOR T
Wp
200
=
850 eV
2W
C
400
600
FREQUENCY
GHz
The shaded region is Fig.(6)
emission
level.
The
feature
at
800
wpe
above
is
the
thermal
discussed
in
sec (6).
The dip at 130 GHz corresponds
to
absorption at the hybrid layer, this being an
extraordinary mode spectrum for 60 kGauss field.
Page 35
SEC 2.0
ELECTRON TEMPERATURE PROFILES
The electron temperature profiles
mode
inferred
from extraordinary
emission at twice the electron cyclotron frequency have nearly a
Gaussian shape under normal circumstances.
thus
measured
is
found
classical resistivity
spectroscopic
Zeff.
Relative
as
equal
the
determined
central
predicted
by
the
temperature
value based upon
measured
voltage
and
The profile width is also in agreement with this
model, and along with
qO = .85.
to
The
the
central
changes
of
temperature
the
indicate
values
of
position of the profile centre
agree with changes in the outer flux surface position measured
during
plasma
of the
positioning
experiments,
while
the
absolute
profile centre provides an accurate measure of the
value
toroidal
magnetic
field.
SEC 2.1
The
GAUSSIAN PROFILE
optical
depth
of
the
second
harmonic
extraordinary
polarisation from the Alcator Tokamak is1 :
23.8 n(101 4 cm- 3) T(KeV)
EQ (1)
B(Tesia)
Under normal operating conditions of density
applies
over
and
temperature,
the
square
of
the
frequency
of
accuracy of the
eq (1.5)
which
temperature
After
dividing
according to eq (1.8), the second
harmonic spectral feature is converted to
application
> 1
most of the plasma cross section, so that emission from
any position will be at or near the blackbody level.
by
T
a spatial
temperature
converts frequency to position.
profiles
derived
in
this
fashion
by
The
is
Page 36
discussed in the appendix.
The measured temperature profile can be fit to a Gaussian:
EQ (2)
T(r) = To
exp [
where TO,rOa refer to the central temperature,
The
choice
of
a Gaussian is arbitrary;
expressions could
include
easy
be
used.
integration
Favourable
position
and
width.
a number of other analytic
features
of
the
Gaussian
and the direct correspondence of the width
parameter, a, with the perceived profile width.
Its major drawback is
its infinite extent, which fortunately introduces only small errors.
A temperature
fig.(1).
The
profile
three
and
corresponding
shown
The weighting favours
in
the
of peaked temperature and ignores the edges where the emitting
layers are not expected to be optically thick.
most
are
free parameters of the Gaussian are adjusted to
minimise a weighted mean square deviation.
region
fit
A common
feature
of
Alcator profiles is the slight increase towards the inside (high
frequency) edge which results from an overlap of the second and
harmonics.
third
For this reason too the edge points are not included in a
fit.
SEC 2.2
COMPARISON TO CLASSICAL RESISTIVITY
Classical resistivity results from
collisions
considered.
additional
between
two
Anomalous
processes
charged
resistivity
a treatment
where
coulomb
particles are the only interactions
then
is
any
deviation
due
to
such as non-collisional phenomena or collective
effects in the plasma, eg.
scattering
from
waves.
The
classical
Page 37
resistivity
calculated by Spitzer 2 for a pure hydrogen plasma will be
the basis for comparison, with any deviations contained in
/
ratio
With this definition the plasma resistivity is:
n = 5.22 x 10-3 lnA T-3 /2 (eV)[n/n*]
EQ (3)
(ohm-cm)
For an ohmic-heated plasma the local resistivity
by
the
the
ratio
of
is
determined
the local electric field and current density.
central current density is related to the toroidal magnetic
field
The
by
qO:
2 B
j 0
EQ (4)
The presence of sawtooth
determines
the
=
j_____
u0q=
R
oscillations
MKS
(MKS)
indicates
qO < 1 which
central current density 3 . The central electric field
is established by assuming the measured edge loop voltage is
across
the
assumptions
the
be
ignored.
With
resistive voltage and central temperature are
related:
EQ (5)
constant
plasma cross section, and by sampling data during periods
of constant current so that inductive effects can
these
then
V T(eV) 3 /2 = 522 lnA B-n-(MKS)
q0 r
Page 38
The
relation
resistive
voltage
range
the
of
between
the
measured
parameters.
and
The data spans the available
is shown in fig.(2).
two
temperature
central
Curves
eq (5)
representing
are
For
temperatures less than 2 KeV the data are consistent
with n/n* = 1.5.
However, above this temperature a marked discrepancy
indicated.
In fig.(3) the ratio of measured to predicted temperature is
occurs.
plotted versus density for the same discharges shown in fig.(2).
The
deviation from predictions based upon eq (5) occurs at lower densities
as well as higher temperatures.
production
of
Because these conditions
favour
the
runaway electrons, and because of known distortions of
the electron distribution which may
occur
under
conditions 4 ,
these
figs.(2) and (3) suggest the involvement of some related process.
For
example, the presence
of
the
cyclotron
creating
emission
non-thermal
the
electrons
false
could
impression
enhance
of
temperature, or it might be argued that a large anomalous
appears
at
lower
a higher
resistivity
densities as a result of runaway driven collective
modes 5 . However, the observed deviations
can
be
explained
by
the
enhanced impurity levels encountered during low density operation.
The presence of impurities
observed
resistivity
over
the
in
the
plasma
predicted
will
value
which
Zeff = 1. This increase can be absorbed into the factor
increase
the
is based on
n/n*,
which
is approximated by the analytic expression 6:
n
EQ (6)
.914 x Zeef
S
1.07
+=e
The Zeff is measured with continuum
visible
7
and
+
1.077 + Zeffaf
a prediction
of
.58 x Zef
bremsstrahlung
emission
in
n/n* is made according to eq (6).
the
A
Page 39
temperature
measured value of n/n* is obtained from the electron
and
A constant coulomb logarithm is used for all data, since the
eq (3).
variations which would be produced by changing density and temperature
are
than
smaller
the
inherent
inaccuracy
intrinsic
the
in
approximations used to generate the logarithm to begin with.
In fig.(4) the two values of n/n* are plotted versus one another
for
the
same
reasonable
data
shown
considering
The
parameters.
in
the
figs.(2)
inherent
measured
central
and
(3).
errors
The agreement is
in
the
temperature
and
component
voltage
consistent with the classical predictions provided that the effect
impurities
is
consisdered.
It
is
not
necessary
to
are
of
include any
anomalous effect in order to explain the relation between the measured
electron temperatures and voltage.
It would seem then that knowledge of Zeff and the
would
in
render
well
measurements,
introduced
Zeff
The loop
by
ohmic-heated
and
loop
voltage
plasmas.
Unfortunately
voltage,
are
measurement
is
not
these
easily
made with
susceptible
to
discounted
if
errors
measurements
are made during periods of relative steady state operation.
complication results from the measurement being
rather
two
changing currents in the plasma and in external coils.
These inductive effects can be partially
edge,
voltage
superfluous independent measure of electron temperature
behaved
accuracy.
loop
than
the
centre.
'The Zeff
made
at
A further
the
plasma
determined by continuum
bremsstrahlung will depend upon an absolute intensity calibration
furthermore
is a chord average measurement,
of Zeff will require an inversion
These
factors
contribute
of
and
so that the central value
several
chordal
measurements.
to an overall systematic uncertainty which
Page 40
does not effect the relative precision of
to
measurements
corresponding
subsequent discharges, for example, but may influence the absolute
accuracy of both by the same amount.
SEC 2.3
PROFILE WIDTH
If the voltage is constant across the plasma cross section
the
then
assumption of classical resistivity combined with the temperature
profile determines the current density:
EQ (7)
J(r) = JO exp
[r
-
r
2
Integration of this expression yields a quantity
which
should
equal
the total plasma current:
I = j0
EQ (8)
a2
03
If the central current density is fixed by sawtooth
oscillations
the
profile width can be written:
EQ (9)
r2
a2 = 3 qO
2 qlim lim
where rlim is the limiter radius and the value of q at the limiter
is
determined by the plasma current and the toroidal magnetic field:
EQ (10)
qlim =
2w B r{im
iR
(MKS)
u01 R
Because the Gaussian profile does not become
zero
near
the
limiter
radius as a real profile must, this treatment will be in error by :
Page 41
E Q (11)E(1)exp
-32ir im2%
2 at
asa2
Fig.(5) is a plot of the profile
calculated from eq (10).
of
the
limiter
This same value
was used in conjunction with the previous discussion of
temperature
scaling
with
classical resistivity.
that the qO inferred here is not a measure
density.
of
central
It should be noted
the
central
current
It is a number which works best within the framework of the
applied models.
current
q
A least square fit of a power law to
agrees with eq (9) and indicates qo = .85.
data
9O
versus
The data are an average of a large number of
discharges, all having sawteeth.
the
width
To date no
distribution
has
direct
been
measurement
made,
of
qO,
or
of
the
so that a comparison with the
inferred qO is impossible.
A good bit of scatter is present in the data
because
of
averaging
is not apparent.
of
fig.(5)
which
If the data are divided into
extremes of temperature most of this scatter vanishes.
This is
shown
in fig.(6), where the data of fig.(5) is replotted to include only the
temperature extremes.
if
the
data
are
this difference
sawtooth
The obvious separation of data does
resulted
the
discharge,
which
difference
results
is
from
an
variation
not
from
the
an
unequal
would
case.
actual
be
It
sawteeth,
sec (5),
constant, so it seems unlikely that
current
density on axis.
averaging
noticed
is
effect
manifestation of the fourier transform that
of
occur
sorted by density, current, or magnetic field.
periods
Measurements
not
several
during a single
unclear
whether
the
in the plasma or is the
generates
indicate
some
over
If
the
profiles.
that qO is relatively
discharges
have
a higher
The difference may be explained by allowing
Page 42
an accumulation of impurities on axis, thus producing a local increase
in
Zeff.
The
increased
resistivity
on
axis would then produce a
higher temperature for the same current density.
This explanation can
neither be confirmed or disproved with the existing data.
SEC 2.4
PROFILE CENTRE
The central position inferred from the Gaussian fit depends upon
the
toroidal
magnetic
field value used.
This value is not measured
directly, but is arrived at indirectly through an
measurement
of
the magnet current.
from this technique.
field
such
Other
inherently
Substantial errors have resulted
possible
influences
upon
the
magnetic
as the ripple produced by the toroidal asymmetry produced
by diagnostic ports and current penetration times in the
been
inexact
magnet
have
ignored since they are unmeasured quantities and are expected to
be small.
There is nothing in the
data
which
suggests
that
these
effects should be included.
The cyclotron emission spectrum provides
direct
measure
of
magnetic field.
for a series of discharges,
reflect
actual
plasma
measurements made
techniques
with
reflect
changes
motion.
the
only
independent
Given that the field is constant
relative
magnetic
the
When
pickup
in
profile
compared
loops 8,
with
flux
position
fig.(7),
same relative change in position.
magnetic loops are sensitive to the outer
positions
surfaces,
the
two
Since the
while
the
profile fit is governed by the plasma centre, the agreement in fig.(7)
indicates motion of the entire plasma cross
various
applied
vertical
control
agreement of the
relative
change
fields.
in
section
Fig.(7)
position, and
in
response
indicates
should
not
to
the
be
Page 43
interpretted as measuring an absolute shift of the centre with respect
to the outer flux surface.
outward,
Although such a shift, approximately
1 cm
is expected due to toroidal effects, the observed 'shift' is
just as likely due to a systematic
overestimation
of
the
magnetic
field.
If, based upon evidence from the pickup loops or the soft
array,
the
temperature
profile
magnetic field is
determined
harmonic
has
profile
by
a maximum.
is assumed to be centered, then the
the
frequency
at
which
REFERENCES & FIGURES
1.)
Engelmann & Curatolo
2.)
L. Spitzer Physics of Fully Ionized Gases
Wiley & Sons, New York (1962)
3.)
S. vonGoeler, W. Stodiek, N. Sauthoff
Phys. Rev. Letters 33 1201 (1974)
4.)
J. Rice
5.)
0. Buneman
Phys. Rev. 115 503 (1959)
6.)
W.M. Stacey Jr. Fusion Plasma Analysis
Wiley & Sons, New York (1981) p. 264
7.)
E.S. Marmar
To be published
8.)
MIT (1979)
Rev. Sci. Instrument
P. Pribyl S.M. Thesis
also PFC/RR-81-21
2 nd
a recalibration
of the current will bring the two techniques into agreement.
Ph.D. Thesis
the
If this value disagrees with the
field determined by the magnet current measurements,
SEC 2.5
x-ray
MIT (1981)
Page 44
Te(KeV)
GAUSSIAN TEMPERATURE PROFILE FIT
*.
1.0
'I
--
DATA
GAUSSIAN FIT
,5
WIDTH;
a
I
T0
970eV
=
I
i
10
9,1cm
ro=-1.5cm
CENTRE;
CENTRAL TEMPERATURE;
=
0
-10
RADIUS (cm)
Fi g. (1)
Electron temperature profile with fit.
From
steady
state
portion of Hydrogen discharge:
60 kGauss, density = 2.5x1014 cm 3 line average.
Page 4b
Te
ELECTRON TEMPERATURE
KEV
vs RESISTIVE VOLTAGE
3.0
0
0
0
0
8
0
0
0
0
0
0
00
0
0
00
2.0
-0
0
00
7
00
=2
0
0
00
0~000
000
1.0
77
Vres
I
I
1.5
2.0
I
2,5
Fig.(2)
Resistive
voltage
incorporates
inductive
corrections.
These
data
from steady state
portions of sawtoothing hydrogen discharges in
80 kGauss magnetic field.
-1
VOLTS
1*
I
3.0
Page 46
RELATIVE ELECTRON TEMPERATURE
vs LINE AVERAGE DENSITY
Te
T*
0
00
E
T*
S0
-
2
380 V'q0
0
3.0
0
0
0
0
0
0
0
2.0
00
0
0
0
0
0
0
0
0
00
00
0P
00
0
1.0
-14
ne =10
A
1.0
f
2.0
cm
3.0
Fig.(3)
These data correspond to the same discharges as
th
previous figure.
The predicted temperature
Te supposes n/n*
= 1.
-3
Page 47
7*
(Te
,
PREDICTED RESISTIVITY
vs MEASURED RESISTIVITY
V)
/
7
//
Te
7*
7*
6
[T3
/
/
0
0
0
5
0
0
LEAST SQUARE FIT
SLOPE = .95
4
/
/
0
3
00
0/
00
2
0 00
0
/10
1
0
.914 Zeff
1,077 + Zeff
7
W
+ .58 Zeff
/
/
7*(Zef)
I
I
t
I
I
I
1
2
3
4
5
6
Fig.(4)
These data correspond to the previous two figures
where measured Zeff are available. The primary
contribution to Zeff comes from partially ionised
Molybdenum.
7
Page 48
a cm
TEMPERATURE PROFILE WIDTH vs LIMITER q
12
LEAST SQUARES FIT
a = 18.03 ql-,50
a 0 ==
q
8
=0qq = 1,85
6
2
q LIM
I
1
2
3
4
5
(5)
Data from discharges Fig
in Hydrogen and Deuterium;
magnetic fields ranging
from 50 to 100 kGauss.
The error bars represent a standard
deviation over
an averaged set
of data.
6
I
rEM R R
a CM
-ts
TEMPERATURE PROFILE WIDTH vs LIMITER q
12
0
0
I0-1
10
0
0
0
*0,
0
.
O
1.0
0
a.
0o=
0.8
8
6
* HIGHEST TEMPERATURES
4
o LOWEST
2
TEMPERATURES
q
I
i
I
1
2
3
I
4
5
Fig.(6)
These
data
represent
the
15%
most
extreme
temperatures
from the previous figure.
This
separation does not occur if data is selected
according to current, density, or working gas
LIM
Page 50
PROFILE CENTRE vs. FLUX CENTRE
CYCLOTRON ro
(cm)
000
1.5
0
0
UNITY SLOPE
0
0
0
0
1.0
0
0
0.5
0
0
0
0
0
0
0
0
0
00
0
0
010
0
000
-0.5
0
0
0
FLUX POSITION (cm)
I
-1.5
-1.0
I
-0.5
I
0.0
0.5
Fi g. (7)
The offset in position between the profile centre
and edge flux surface measurement most likely
results from errors in magnetic field calibration,
and should not be interpreted as resulting from a
Shafranov shift. For these data the same field
calibration was employed for discharges from a
single day at constant field, so only a constant
systematic error results.
1.0
Page 51
NON-GAUSSIAN PROFILES
SEC 3.0
During the steady state portion of plasma
from
different
very
Gaussian
are
discharges,
occasionally
radiation
portion -of
plasma
observed
from
profile
most extreme cases the profiles are hollow with
are
deviations
diffusion time of the ohmic flux.
finite
large
During the current rise
elements.
impurity
discharges,
the
to
attributed
by
These
observed.
deviations result from pathologic current distribution or
localised
profiles
a depressed
In the
central
While these unusual profiles may be transient, they must
temperature.
persist for the duration of the Michelson scan period in order
to
be
observed.
SEC 3.1
CURRENT RISE
The plasma current is
Because
the
hot
plasma
driven
by
an
induced
electric
field.
is a good electric conductor, magnetic flux
will penetrate inward over a diffusion time:
EQ (1)
T
=
2 a L2
=
.55 T3 /2 L2
(msec, KeV,cm)
where a is the plasma conductivity and
L is
During
time
steady
state
this
diffusion
the
is
diffusion
about
Te = 1.0 keV and L = 9 cm, while the ohmic flux changes
scale
of
400 msec.
be
nearly
constant
across
the
assumption of constant voltage made in
steady state.
45 msec,
over
for
a time
Because diffusion through the plasma is quicker
than the change in ohmic flux at the edge, the
will
length.
plasma
induced
cross
sec (1) is
loop
voltage
section.
justified
The
during
Page 52
During current rise (fall) the ohmic flux is being changed
times
comparable
to
the magnetic diffusion time, consequently flux,
and hence current, will be added
portion
outer
(removed)
preferentially
initial
ionisation
period
the
diffusion
steady
state.
Once ionisation
is
time rises as the temperature increases.
cannot penetrate the plasma at the rate of the initial
the
the
During
diffusion time is close to zero
because the plasma is a poor conductor.
the
from
The corresponding current distribution
of the plasma.
will be broader (narrower) than predicted for
the
over
complete
Because flux
current
rise,
additional current will flow along the outside of the plasma.
some instances a hollow current profile will be produced
In
by
a rapid
current
profiles
current rise.
The relationship between the temperature
expressed
in
eq (2.7)
implicitly
across the plasma cross section.
current
rise
This condition does not apply during
so a hollow current profile does not
a similar temperature profile.
with
the
indicates
density.
voltage
a comparable
current, since the increased conductivity at the
plasma edge resulting from the higher temperature
increased
necessarily
However, a hollow temperature
profile during current rise almost certainly
situation
a constant loop voltage
when the rapidly changing flux creates a higher voltage
at the plasma edge;
require
requires
and
can
only
result
in
an
Thus a measured hollow temperature
together
increased
profile
rise indicates the coexistence of a skin current.
with
any
edge current
during
current
Page 53
An example of this skin effect is shown in
evolution
rise.
of
the
is
not
continuous,
during
current
plasma breakdown.
feature
will
rise
peaked
but
disruptive rearrangement of the
shape
where
the
hollow profile can be followed through the current
The transition from hollow to
discharge
fig.(1),
is
during
such
accomplished
by
a sudden
distribution.
current
is
profile
determined
The
profile
during and shortly after
If the earliest measurable profile is hollow,
persist
during
current
a
rise.
this
If the first measured
profile is peaked, subsequent profiles will also be peaked.
The persistence of the
beyond
the
magnetic
hollow
current
(temperature)
profiles
diffusion time results from the localisation of
ohmic heating power to regions of highest current (temperature).
disruptive
conversion
from
ensues from the instability of
hollow
to
hollow
peaked
The
profiles undoubtedly
profiles
to
various
tearing
modesi.
SEC 3.2
Localised
IMPURITY RADIATION
energy
loss
by
impurity
significantly alter the temperature profile.
line
radiation
can
The energy loss rate per
free electron per impurity ion is represented by the
quantity
so
Lz
that the total radiated power is:
EQ (2)
Prad = ne nz Lz(T)
where ne,nz are the
electron
and
element
Lz
particular
impurity
impurity. ion
will
be
densities.
a sensitive
For
a
function of
electron temperature, so that different temperature regions will
have
Page 54
very
Unlike field diffusion the effect of impurity line radiation
density.
can
radiation losses, even if they have the same impurity
different
alter profile shapes during steady state.
Molybdenum is a common
impurity in Alcator and an effect upon the temperature profile can
attributed
to
this element.
be
A similar effect has been observed from
PLT resulting from tungsten radiation 2 .
Calculations 3 show that molybdenum radiated power is maximum for
coronal
temperatures
slightly
under 1 KeV and decreases sharply for
temperatures above or below 1 KeV.
when
Alcator
molybdenum
of
[ne < 1014 cm-31
densities are typically large, two distinct
profile shapes are observed;
value
At low densities
either a peaked
profile
with
central
several KeV, or a hollow profile with a central temperature
of 1 KeV or less.
Examples of such profiles are shown in fig.(2).
In
the case of the hollow profile, bolometrically measured radiated power
is greatly increased over the peaked case, and accounts for 70% of the
ohmic
input
power.
The
hollow
profile
discharge with the central temperature
central
soft
oscillations.
early
in
x-ray
oscillations.
is
reduced
throughout
exceeding
and
1 KeV.
shows
no
the
The
sawtooth
In the other case the central temperature exceeds 1 KeV
the
central soft
emission
never
endures
discharge,
x-ray
and rises to values of 2.5 - 3.0 KeV.
emission
is
large
and
shows
normal
The
sawtooth
For both cases the ohmic input power is about the same,
although the spatial distribution of the ohmic power may be different.
If this input power can be converted only to plasma energy or to
line radiation, for each case a balance of power requires:
EQ (3)
POH
+
Plasma + Radiation
=
neTe + nPT
T
+
Prad
Page t!
where the confinement time
line
radiation,
Furthermore,
and
is that expected for the condition of
T
is
assumed
to
be
if the impurity density
is
assumed
equal
for
both
constant
no
cases.
for
both
cases, the difference in measured plasma energy for the two situations
depends upon the difference in radiation rates ALz as follows:
{ATe + ATi) = nZT ALz
EQ (4)
Based
upon
the
calculations
for
molybdenum
and
the
measured
temperature differentials:
ATe = 2.2 KeV ; AT 1 = .7 KeV
+ ALz = 1.75 x 10-25 watts-cm 3
EQ (5)
assuming a confinement time of 10 msec the inferred
is
nz = 2.5 1011 cm- 3 with
density
and
resulting
measurements
of
a Zeff
Zeff
molybdenum
= 5.
impurity
density
These values for impurity
are
consistent
line
radiation
with
and
spectroscopic
spectroscopically
measured Zeff.
Because the radiated power increases with temperature for values
below
0.7 KeV
any
increase
increase in radiation loss.
radiation
term
in
If
determined
the
will be balanced by an
temperature
exceeds
I KeV
the
decreases with temperature so that the temperature is
free to rise to higher values.
is
temperature
quite
Which of the two profiles will develop
early in the discharge as no transition between
profile types has been observed.
In fig.(3). the
profile can be seen throughout the discharge.
hollow
temperature
Page 56
It is not clear that the hollow temperature profiles of
are
associated
sawteeth under
current
with
these
density
so
hollow
current
conditions
profiles.
only
requires
fig.(3)
The lack of visible
a reduced
central
that q0 < 0. That no disruptions occur suggests
that the profiles are not hollow, or if so, are somehow stable against
disruption.
Figs.(2)
and
(3) refer
to
a particular reproducible
phenomenon resulting from impurity radiation, and do not represent the
plasma
response
to
the
presence
of
impurities
in
Situations have been observed where a sudden influx of high
results
in
heating,
cooling,
non-reproducible events which may
disruption
occur
etc.
cases.
Z matter
These, however, are
randomly
condition.
SEC 3.3
all
REFERENCES & FIGURES
1.)
B. Carreras, H.R. Hicks, B.V. Waddell
Oak Ridge Report ORNL/TM-6570 (1979)
also:
Nuclear Fusion 19 1423 (1979)
2.)
Plasma Physics and Controlled Nuclear Fusion
I.A.E.A. Vol.III 10 (1978)
3.)
D.E. Post, R.V. Jensen
At. Data Nucl. Tables 20 597 (1977)
at
any
operating
Page 57
EXAMPLE OF SKIN EFFECT
DURING CURRENT RISE
Te (KeV)
1.5
1.0
0.5
I
10
i
i
0
i
-10
I
RADIUS (cm)
Fig.(1)
Temperature profiles measured during current rise
60 kGauss field;
400 kAmperes.
0 to
from
3
density = 1.6x1014 cm- line average.
Page 58
Te (KeV)
RADIATION DOMINATED TEMPERATURE PROFILES
3.0
2.0
1.0
10
0
-10
RADIUS (cm)
Fig. (2)
Profiles from steady state portion of
Hydrogen
discharges at 80 kGauss.
results from Molybdenum radiation.
consecutive
Hollowness
In both
the plasma
current = 350 kAmperes;
density = 8.5x1013 cm- 3 line average.
cases
Page 59
EVOLUTION OF RADIATION DOMINATED
TEMPERATURE PROFILE
Te (KeV)
1,5
1.0
10
0
RADIUS (cm)
-10
Discharge in Hydro en at 0 kGauss. Density never
exceeds 1.0x101 4 cm- 3 line average. No sawtooth
activity is observed for the entire discharge.
Hollowness results from Molybdenum radiation.
Page 60
THIRD HARMONIC EMISSION
SEC 4.0
Radiation at the third gyroharmonic rivals the
in
harmonic
intensity, and is the highest order of cyclotron emission detected
by
from thermal Alcator plasmas
the
frequencies
these
the
Michelson
is
plasma
not
emission
interferometer.
At
to
the
optically
extraordinary mode, but in a gray region where
the
second
T
thick
< 1. Because of this
will not be at the blackbody level and will depend upon
electron density as well as temperature.
This in contrast to the case
of the second harmonic where values of T > 1 allowed the assumption of
blackbody emission.
the
third
provide
analysis
While the treatment of sec (2) when
harmonic
information
of
the
can
not
regarding
provide
the
applied
to
temperature profiles, it does
plasma
density.
Conceptually,
second and third harmonic emissions together allows
for an absolute calibration of the diagnostic system, freeing it
from
dependence on other sources of temperature measurement.
SEC 4.1
REFLECTIONS & OPTICAL DEPTH
The optical depth for
the
extraordinary
polarisation
of
the
is
not
third harmonic isl:
EQ (1)
T3 = .236 neTe2
B
(1014 cm-3, KeV, Tesla)
The expected intensity from a thermal plasma is:
EQ (2)
13 = Ibb
{1
-
e-T3} + I e-T3
where Ibb is the black body level.
Because the optical depth
Page 61
large the I term must be included to account for additional radiation
reflected into the viewing angle from the vacuum chamber walls.
fraction
p of
each
ray
is
reflected
If
a
when encountering the wall,
summation over all reflections results in the following expression 2:
EQ (3)
13
=
Ibb
1
-e-
1
-
-T
pe
The effective reflection coefficient p must account for radiation loss
to the vacuum chamber, port extensions and windows.
with polarisation scrambling.
not
absorbed
by
the
Because the
plasma,
fig.(1)
blackbody
eq (1).
during
level
the
is
third
loss
to
plot
harmonic
or
intensity
relative
to
the
plotted versus the calculated optical depth from
run.
Any
systematic
errors
of
Inherent
errors
discharges
due to alignment or
The
is reasonable when the strong dependence of
and temperature is considered.
plasma
walls
(See sec (1.3))
calibration may be assumed constant for these data.
this
the
These data come from the steady state portion
a single
is
will reduce the effective
reflectivity by increasing the likelihood of
In
polarisation
ordinary
scrambling
rejection by the interferometer.
It must also deal
in
T
scatter
in
upon density
these
measured
parameters limit the accuracy of computed t to about 20%.
The
data can be described by eq (1) if an effective reflectivity of 0.5 is
chosen.
When
operating
with
various
toroidal fields or different
optical alignments, the value of p inferred from plots such as fig.(1)
changes.
Systematic
errors
in
calibration
making it a measure of more than reflectivity.
will
be absorbed by p
Page 62
In fig.(2) the measured electron temperature is
product
the
plotted
versus
the line average density and the ratio of second and
of
third harmonic intensities.
In this format the
potential
of the third harmonic as a diagnostic is illustrated.
usefulness
Provided that a
curve such as fig.(2) has been previously generated it can be used
infer
the
density from the measured temperature, or temperature from
the known density.
the
to
second
In the latter case a temperature
harmonic
can
be
achieved.
calibration
of
In practice this technique
provides a check of the calibration currently being used, but does not
free
the system from independent temperature measurements.
alignment requires a new curve like fig.(2).
be
realised
computed.
where
the
Every new
Ideally, a situation can
systematic error remains constant, or can be
A single curve such as fig.(2) would then be a useful
tool
for measuring temperature.
SEC 4.2
PROFILE SHAPE
Given that its intensity is proportional to the
the
third
optical
depth,
harmonic spatial profile becomes a function of the density
and temperature profiles:
13 (r) - ne(r) exp -3
EQ (4)
where the Gaussian temperature profile
example
third
harmonic
profile,
is
plotted
2
invoked.
case
an
The shape, as
in
of the second harmonic, is approximately Gaussian as shown.
The narrowness of the
dependence
is
in the fashion of sec (3)
where the frequency squared coefficient is removed.
the
Fig.(3)
profile
in
fig.(3)
upon the temperature profile.
results
from
the
cubic
Because of this the density
ragc u.)
profile contributes
profile.
In
nothing
to
the
shape
of
the
of
harmonic
fig.(4) the third harmonic width is plotted against the
corresponding second harmonic width for a number of
ratio
third
widths
is
nearly
/3 ,
discharges.
which reflects the fact that the
intensity is proportional to the cube of the temperature.
only
confirms
to empirically
between
This
plot
that the density profile is broad, and can not be used
describe
a parabolic
the
shape.
For
example,
the
difference
and square root parabolic density profile would
not be discernible in the third harmonic width.
SEC 4.3
The
REFERENCES & FIGURES
1.)
Engelmann & Curatolo
Any finite density effects will be incorporated into
the factor p so that only the nT2 dependence of the
optical depth is important.
2.)
D.A. Boyd
Int. J. Infrared & MM. Waves
1 45 (1980)
This reference contains a general description of the
process by which density profiles can be extracted
from the third harmonic emission.
Page 64
RELATIVE INTENSITY vs OPTICAL DEPTH
FOR THIRD HARMONIC
13
'bb
0?
07
o
0
07
0
13
- e
7P=O. 5
1-pe
'bb
0
0"
0
0 0
0
o
7
/
P0
000
0
0
0
O
0
od
0
0
/ 0
/
I
.10
0
0
/
0
0
0
0
0
0
I
.15
I
.20
Fig.(1)
from
Hydrogen
intensities
harmonic
Third
The optical depth is
discharges at 80 kGauss.
and
temperatures
measured
from
calculated
densities.
I
T
3
.25
Page 65
ELECTRON TEMPERATURE vs.
PRODUCT OF DENSITY & HARMONIC RATIO
0 0
00
Te(KeV)
00 0*
0
1.')
%0
00
0
0
0
00
00 00
1.0
000
0 00
00
0
0
0
.8
.6
.2
fe (fringes)
12
13
I
3
I
6
9
Fig.(2)
Data from Hydrogen discharges at 80 kGauss.
The
'fringe' is the ex erimental unit of density, 1
fringe = 5.6x10 13 cm- line average.
12
Page 66
GAUSSIAN THIRD HARMONIC PROFILE FIT
GAUSSIAN FIT
CONSTRUCTED DATA
WIDTH; a = 4.5 cm
CENTRE; ro=-1.5 cm
+A
10
0
-10
RADIUS (cm)
Fi g. (3)
Third harmonic profile from the steady state
at
80 kGauss.
discharge
of,
portion
Density = 2.6x,014 cm-3 line average.
The steep
increasing signal on the outer edge is due to
overlap of the second harmonic.
Page 67
a 2 CM
SECOND HARMONIC WIDTH
vs THIRD HARMONIC WIDTH
0
12
08
...
2 (r)
e (
/0
0
10
000
0
00
8
6
SLOPE =
1.67
I3 (r) e-
03
2
a3
I
I
I
1
2
3
I
I
I
I
5
6
7
Fig. (4)
Data from sawtoothing discharges at 80 kGauss.
Widths
are
determined
by Gaussian fits to
corresponding harmonic emissions.
CM
7
Page 68
TEMPERATURE FLUCTUATIONS
SEC 5.0
The Michelson interferometer requires an oscillation
generate
a single
interferogram.
The
resulting
period
spectrum
is not
measured at one time but is representative of this entire period.
is
to
Nor
the spectrum swept in time, with each frequency corresponding to a
different time during the sweep.
If the
emission
intensity
changes
during -the scan period, this change appears in the interferogram, and
consequently in the
spectrum
through
the
fourier
transform.
The
problem with temporal fluctuations is that there is no way the fourier
transform process can separate features in the interferogram resulting
from temporal variation from features which are produced by the action
of the interferometer.
will
For example, a single narrow spike of emission
produce a sinusoidal perturbation on the spectrum.
is sampled at
emission
irregular
intensity
time
will
not
intervals,
produce
periodic
periodic
Because data
fluctuations
in
fluctuations in the
interferogram, hence no sharp spectral features will result.
The study of emission
monochromator,
such
fluctuation
is
best
performed
with
a
as the Fabry-Perot interferometer which provides
adequate time resolution, while relying upon discharge reproducibility
to
measure
separate
frequencies
one
at
a time.
Two
common
fluctuations in cyclotron emission are due to the sawtooth instability
and to plasma motion.
SEC 5.1
FABRY-PEROT INTERFEROMETER
Page 69
The Fabry-Perot interferometer is
an
etalon
of
two
parallel
The separation between meshes is adjustable.
reflective metal meshes.
Radiation will be transmitted through the device if its wavelength
an
half
integer
times
the
mesh
separation.
Thus the device will
transmit a fundamental frequency and higher orders of this
So,
for
example,
when
fundamental frequency,
transmitted
using the
separate
viewing
radiation
in the second order.
Fabry-Perot.
polarising
grid
The
the
at
the
frequency.
gyroharmonic
fourth
harmonic
as
will
the
be
This effect must be considered when
device
must
second
is
be
is
non-polarising
so
that
a
used if only one polarisation is
desi red.
The resolution, or finesse, in the first order
of
transmission
is determined by the wavelength viewed and by the mesh spacingi:
F = w{2g ln(g/2wa)}- 2
EQ (1)
X2
where g is the mesh spacing (lines per inch)-and
While
a large
finesse
allows
high
amount of available incident signal.
more
desirable
so
that
a the
wire
width.
resolution it reflects a large
In practice a modest finesse
a reasonable
is
signal to noise ratio can be
maintained.
SEC 5.2
The
SAWTOOTH INSTABILITY
sawtooth
perturbation
instability
has
been
modelled 2
by
a growing
of the q = 1 surface resulting-in an internal disruption
which flattens the central temperature and densities to the values
the
q
= 1 surface.
at
The redistribution of energy extends beyond this
Page 70
surface, so that the density and temperature will increase
for
disruption
regions
just outside the q = 1 surface.
surface the sawteeth are 'normal', outside, 'inverted'.
surface
is
identified
with
the
change from normal to inverted.
point
the
after
Inside this
q =
The
1
of inversion where sawteeth
This is the model which will be
used
as a basis for comparison of measured quantities.
When observing sawteeth in
polarisation
of
the
the
optically
thick
extraordinary
second electron gyrofrequency, the amplitude of
the sawtooth is a direct measure of the temperature variation.
of
this
change
relative
to
the
A plot
time average local temperature is
plotted as a function of position in fig.(1).
Rather
than
absolute
magnitude a ratio of temperature change to average temperature is used
to characterise the sawtooth.
This eliminates the
need
for
precise
calibration and corrects for any shot to shot variation.
The model used to describe sawtooth temperature fluctuations
outlined
in
fig.(2).
Before
the disruption the profile is assumed
Gaussian, though not necessarily the same as the 'average' profile
sec (2).
and
For
narrower,
is
of
example, the pre-disruption profile may be more peaked
reflecting
a slightly
smaller
value
of
qo.
The
post-disruption profile is taken to be flat in the central region such
that q = 1 throughout this area.
Using
this
model,
the
predicted
fractional sawtooth amplitude as a function of position is:
EQ (2)
2
TI)= 1 - exp r a2
I
T(r)
where x is the 'inversion
point'
of
the
2
sawtooth.
This
position
depends upon -the temperature profile width a, and the q value on axis,
Page 71
qo.
The inversion point and
related,
but
not
equal.
q = 1 surface
pre-disruption
the
For
a peaked
current
are
profile
the
pre-disruption q = 1 radius will always occur slightly outside of
the
inversion
point,
according
to this model.
The radial extent of the
sawtooth perturbation
is
plasma
is approximated by rmax
density,
and
restricted
by
values of qo and a, as inferred from
requiring
measured
=
conservation
of
/2 x. For reasonable
temperature
profiles,
the inversion point occurs at radii varying from x = 2 cm to x = 5 cm.
The corresponding central sawtooth amplitude is expected to vary
from
7% to 20% according to the model.
A further
measuring
consideration
the
spatial
resolution
The Fabry-Perot is
used
the same optical path described in the appendix.
along the viewing axis is governed by the
being
of
the
instrument, which is not negligible compared to the size of
the feature being measured.
with
is
observed,
optics employed.
while
finesse
at
in
conjunction
The resolution
the
frequency
perpendicular resolution is determined by the
The measured fluctuation will be a weighted
spatial
average of the actual fluctuation amplitudes:
EQ (3)
AT(r) measured
T(r)
=
where g(r,r') is a normalised
J
g(r,r') 6T(r') actual dr'
T(r')
response
resolution limits of the apparatus.
function
g(r,r') =+
R[1 +.2F
T
the
The appropriate choice of g(r,r')
for the Fabry-Perot interferometer is:
EQ (4)
incorporating
2 2]1/2
sin2 R+r1
R+r
Page 72
where F is the finesse.
With
an
expected
finesse
of
18
for
the
Fabry-Perot at 420 GHz (B = 75 KG), the spot size (<2 cm diameter) and
diffraction width (.3 cm) are not limiting factors in the instrumental
resolution.
As shown in fig.(1) the measured central sawtooth amplitudes are
to
insensitive
changes
qlim,
in
variations in plasma current or
point
does
in
whether these changes result from
field.
magnetic
The
with decreasing qlim as expected, but the profile
change
width also changes, leaving the ratio x2 /a2 constant.
EQ (5)
EQ(5
inversion
AT2
x-
q=
qo
Since:
a.
T
(r=0)
=
1-exp2 2
a
the constancy of central sawtooth amplitude indicates an invariance of
q0 Within the context of the model of fig.(2) the central
amplitude
is
determined
relate a with qo and qlim.
a.
It
is
by
x and a, while the arguments of sec (2)
Finally, eq (5) identifies x with
the
uncertainties
of
in
it is possible to nearly satisfy them all.
in fig.(1) can be made consistent with the model if
value
qO
and
unlikely that a single value of qo will satisfy all these
relations, however, by invoking
quantities
sawtooth
qO = .80 is chosen.
a
since
between
sawteeth
the
observed
suggests
measured
The data shown
pre-disruption
The variation of qo between .8 and 1.0
during a sawtooth compares favourably with the 'average'
sec (2),
the
linear
a 3/2
temperature
power
current
qo = .85
change
density,
of
with time
or
qo,
rage I3
variation.
Fig.(3) shows the sawtooth amplitude for
otherwise
under
identical
as
conditions
for
those
helium
in
discharges
The
fig.(1).
inversion point is the same in the two figures, but the amplitude
in
helium is doubled what it is in hydrogen or deuterium. While doubling
the amplitude, the sawtooth frequency is halved, indicating
the
that
heating of the electrons occurs at approximately the same rate in both
The amplitude difference can not be related
cases.
to an
ion
mass
difference
since
no
dissimilarity was detected between hydrogen and
deuterium.
There
is
insufficient
regarding
data
to make
a similar claim
If the same sawtooth model is to apply to helium as
Zeff.
well as hydrogen it is necessary to require a smaller
qo (i.e.
higher current density) in helium.
SYMPATHETIC PLASMA OSCILLATION
SEC 5.3
Another source
motion
pre-disruption
produced
by
of emission
fluctuation
oscillations
of
results
from plasma
the controlling fields.
fields are powered by a solid state rectifier which imparts
ripple.
field.
When
polarisation
a 360 Hz
to noise pickup, the ripple produces a genuine
In addition
effect upon the
These
plasma
position,
presumably
through
the
vertical
viewing a portion of the second harmonic extraordinary
where
the
emission
profile
has
a
non-zero
spatial
derivative, plasma motion will produce the following fluctuation:
EQ (6)
-2 r Ar
iAT(r)
T(r)
where Ar is the amplitude
a2
of
plasma
motion.
Fig.(4
shows
these
Page 74
fluctuations
for
radial
positions
of
-10 cm
from two concecutive
discharges, and compares them to the loop voltage
two
signals
measurements 3 .
centroid
The
this
loop
proportional to the positive time derivative
ripple,
so
The
out of phase with respect to each other as expected
are
and both indicate a plasma motion of t3 mm,
x-ray
oscillations.
fig.(4)
indicates
field by about T/2 rads.
that
in
agreement
with
voltage
oscillation
of
vertical
the
is
field
plasma motion lags the vert.cal
This lag is
attributed
to
a 4 msec
flux
penetration time through the edge plasma 4 . No motion is observed when
viewing the plasma
Sudden
changes
in
centre
this
because
the
oscillation
profile
amplitude
alterations of the temperature gradient.
result
zero.
from
rapid
Fig.(5) shows this effect in
During the marfe the fluctuation level is 1% .
the
average
When the marfe
plasma
motion
ceases
temperature increases 20% and the fluctuation
level returns to 3% , as in the previous examples.
of
is
with a 'marfe'5 while viewing the -10 cm radial position.
conjunction
abruptly,
gradient
is
constant,
the
If
the
discontinuity
fluctuation amplitude represents a drastic change
in
amplitude
in
measured
profile
width;
from a = 15 cm to a = 10 cm.
These oscillations are an example where an apparent
fluctuation
temperature
in fact results from changes in position alone.
It would
be incorrect to assume that the fluctuations of fig.(4) resulted,
example,
from
fluctuation.
for
ohmic
heat.ing
variation
produced
by
the
for
voltage
Furthermore such fluctuations will complicate any search
small scale temperature profile variance, such as those suggested
to occur in conjunction with the onset of certain MHD activity6 .
Page /b
SEC 5.4
REFERENCES & FIGURES
1.)
K.F. Renk, L. Genzel
Applied Optics 1 643 (1962)
2.)
G.L. Jahns et.al.
Nuclear Fusion 18 609 (1978)
3.)
R.S. Granetz
Private Communication
4.)
P. Pribyl
Private Communication
5.)
The marfe is an episode of enhanced Ha emission
that often occurs during high density operation.
It is localised to the inside plasma edge, and is
accompanied by sudden increases of plasma density
at the inner edge.
The cause is unknown. The name
'marfe' results from the modesty of its two principle
investigators; E.S. Marmar & S.M. Wolfe.
6.)
R.S. Granetz
Submitted to Phys. Rev. Letters (1982)
Page 76
RELATIVE SAWTOOTH AMPLITUDE
vs RADIAL POSITION
0
---
ATe
%
-
ql~~m
-
3.6
.0im
Te
10
Q...-
lm
'
/0
/
I,
(I
/
/
5
'0
I
I
I
\
0
I
-5.0
I
I
I
I
I
+5.0
I
-
i
I
I~j
0
a CM
-*
00
07
0
-5
0
H-i
Fig.(1)
Data from discharges in Hydrogen at 80 kGauss.
Densities = 3.0-4.0x1014 cm- 3 line average.
No
difference in amplitude results from changes in
density, current, or magnetic field.
Page 77
SAWTOOTH MODEL
Te
T
NORMAL SAWTEETH
INVERTED SAWTEETH
--
.5
Fx x
PRE-DISRUPTION PROFILE-+
10
0
RADIUS (cm)
Fig.(2)
-10
A positive ATe/Te
iimpeies
a
reduction
in
temperature - normal sawteeth. Negative ATe/Te
implies an increase in temperature - inverted
sawteeth.
Page 78
RELATIVE SAWTOOTH AMPLITUDE
vs RADIAL POSITION IN HELIUM
10
15
0
0
Te
0
10
FOR HYDROGEN, DEUTERIUM
WITH SAME PLASMA PARAMETERS
/
/
/
/
-5
0
I
I
I
I
I
I
+5.0
e...%
-5.0
i
I
/
/
/
/
-~~0
0
5
Fig.(3)
Data from discharges at 80 kGauss. Change between
Hy.drogen and Helium was executed during single run
(day) with
immediate
doubling
of
sawtooth
amplitude in Helium observed.
CM
I
Page 79
CORRELATION OF CYCLOTRON EMISSION
WITH LOOP VOLTAGE OSCILLATIONS
I
I
2
e
1
6.0 7%
LOOP
VOLTAGE
INSIDE
OUTSIDE
r = -10 cm
r = +10 cm
24 e
61%
6.0
-F
LOOP
VOLTAGE
4.0
8,0
I
I
12.0
16.0
Fig.(4)
Data from consecutivedischarges in Hydrogen at
80 kGauss. Density = 3.3x1014 cm- 3 line average.
msec
Page 80
CORRELATION OF CYCLOTRON EMISSION
WITH LOOP VOLTAGE DURING 'MARFE'
I
I
LOOP
VOLTAGE
DURING 'MARFE'
AFTER 'MARFE'
6.0 %
2q e
2.0 %
I
10
I
20
I
30
Fig.(5)
Data from current rise portion
of
Hydrogen
discharge at 80 kGauss.
Density = 2.5x101 4 cm-3
line 'average; current = 250 kAmperes at time of
marfe.
I
40
msec
Page 81
SEC 6.0
DOMINANT PLASMA FREQUENCY EMISSION
'Dominant wpe emission' refers to a characteristic
radiation
near
the
plasma frequency.
electron
spontaneous
It is usually a few
times the thermal level in intensity and has spectral extent from
central plasma to the edge cyclotron frequency.
Dominant emission has
been observed from the Alcator Tokamak as well as other
is
most
noticeable
when wpe/wce
the electron distribution
nearly
devices,
highest
.6 and a non-thermal component of
present.
The
emission
intensity
Because the trend in Tokamak operation is to
is
density
possible
for a given magnetic field;
achieve
i.e., to
maximise a, the condition wpe/wce = .6 is not a usual situation.
this
and
constant in time and can easily be measured with the Michelson
interferometer.
the
is
the
reason
For
the appearance of dominant wpe emission is not expected,
except perhaps early in a discharge while the density is rising to its
steady state value.
The most successful explanation of this radiation
involves the Cerenkov
result
that
the
resonance
emission
with
intensity
runaway
electrons,
with
the
is proportional to the current
carried by these electrons.
SEC 6.1
SPECTRAL CHARACTER
Example spectra of dominant wpe emission are shown
In
in
fig.(1).
each case an intense feature at lower frequency rivals emission at
the second gyroharmonic
in
intensity.
Otherwise
the
spectra
are
completely normal, with second and third harmonic intensities equal to
the values expected from a usual discharge.
in
fact
occurs
That the emission feature
at the central plasma frequency is shown in fig.(2),
where the frequency of maximum
emitted
intensity
is
plotted
as
a
Page 82
function
of
line
averaged
fig.(1).
If a peak to averaged density ratio of 1.3 is
measured
frequencies
are
density
in
inferred
from
is
the
of
representative
measurements1
spectroscopic
assumed,
agreement with the expected
excellent
This ratio of 1.3
central plasma frequencies.
values
for many cases such as those of
and from fits to
multi-chord interferometric measurenents 2 of the plasma density.
Although apparently narrow in
gyroharmonic,
second
resolved by the
plasma
the
Michelson.
The
comparison
frequency
spectral
to
emission
the
is just barely
feature
width
at
is
equal
to
separation of the central plasma and edge cyclotron frequencies.
the
This
is shown in fig.(3) where the half-width of the feature is plotted
as
frequency
is
a
function
of
As
frequency.
the
edge
cyclotron
approached the feature width becomes progressively narrower.
same
time the emission amplitude progressively decreases.
the intensity relative to
frequency,
versus
and
the
thermal
this
plasma frequency exceeds the
blackbody
level
decrease can be seen.
edge
cyclotron
value
At
the
In fig.(4)
is
plotted
When the central
no
emission
is
observed.
Dominant emission usually occurs early in discharges which would
otherwise
be
considered normal with regard to macroscopic parameters
such as current,
dominant
cyclotron
emission
density,
are
radiation,
electrons, sec (1.4).
and
generally
indicating
However,
temperature.
preceded
the
by
presence
of
periods of broadband
of
relativistic
The emission intensity usually decays away over
a period of about 100 msec, but if the plasma density is
low,
episodes
sufficiently
so that wpe/wce is less than unity, the emission may persist for
the duration of the discharge with nearly constant intensity.
Page 83
When present, the emission is
For
higher
most
intense
for
Ope/wce = .6.
frequencies the intensity decreases, as shown in fig.(4),
while for lower frequencies the emission , if observed at all, is much
Fig.(5) is an example of dominant emission for the case
less intense.
exhibits
wpe/wce = .4. While the intensity is small, the feature now
a double
which
structure
humped
was
not apparent in the previous
examples.
The dominant emission feature occurs in both polarisations
no
difference
noticeable
extraordinary modes.
than
the
in
spectral
shape
between
with
ordinary and
The extraordinary mode emission is more
intense
ordinary, but the actual degree of polarisation is not well
measured.
REVIEW OF PREVIOUS EXPERIMENTS
SEC 6.2
Measurements on TFR 3 show
early in certain types of
of 100 msec.
that
the
dominant
emission
occurs
discharges, and decays with a time constant
These discharges are characterised by low densities
and
high temperatures with wpe/wce = .6. The emission occurs at about the
central plasma frequency and extends to the edge cyclotron
In
this
the results are identical to those presented in the
respect
The intensity of emission, while greater
previous section.
thermal
level,
is
not
than
the
as intense as the examples shown in fig.(1).
This situation also reflects the dominant emission seen
T-II Tokamak 4 .
frequency.
on
the
JIPP
Page 84
The Alcator A 5 results
emission
again
intensity
are
similar
being
less
to
than
those
from
double
humped
clearly
An
show
feature which is only just .noticeable in fig.(5).
This difference probably
reflects
between the two machines.
the
disparity
in
aspect
ratios
Alcator A has an aspect ratio R/rlim = 5.4,
while for Alcator C R/rlim = 3.9.
the
the
the Alcator C case.
important difference is that the Alcator A measurements
the
TFR,
For wpe/wce = .6 the separation
of
central plasma and edge cyclotron frequencies will be 20% greater
in Alcator A. It is reasonable to assume that
this
larger
spectral
region increases the likelihood of resolving any emission structure.
The Alcator A results also conclusively show the emission to
polarised,
with more radiation coming in the extraordinary mode.
be
The
degree of polarisation, about 3:2, is consistent with the polarisation
scrambling
model
used
to explain observed ordinary mode emission at
the second gyroharmonic.
SEC 6.3
CERENKOV EMISSION
Cerenkov emission occurs when the particle velocity
can
exceed
the local electromagnetic phase velocity, that is, when the refractive
index
is
situation
the
greater
can
vicinity
optically
thin
than
only
of
For
electron
plasma
relative
plasma
this
frequency.
The
plasma
is
this region, so the emission can be considered by
summing over all runaway electrons.
the
a magnetoactive
be realised by the extraordinary polarisation in
the
in
unity.
constancy
of
the
Because of
emissivity
velocities, under certain conditions the emission
proportional to the number of runaway electrons.
this
for
summation,
large
intensity
and
electron
will
be
Page 85
The plasma is described in the cold plasma approximation by
by
both
thermal
properties
of
This requires ignoring effects
dispersion relation.
Appleton-Hartree
and
electrons
relativistic
upon
the
dielectric
Effects caused by runaways can
the background plasma.
be ignored since the density of these electrons is assumed
than that of the thermal plasma.
the phase
velocity
of
much
In the frequency region
radiation
the
will
always
less
wpe< W <wce
exceed
thermal
the
electron velocity, so warm plasma effects may be ignored.
The emitted wave is required to be in phase
with
the
emitting
electron:
... =
EQ (1)
v cose
k
where v is the particle velocity, which
is
direction of the ambient magnetic field.
assumed
to
lie
with
the
electron velocity.
fig.(6).
The
is always less than unity for the ordinary mode.
condition can never be satisfied, so
expected.
angle
of
The.dispersion relation for a
particular choice of frequency is shown in
index
the
Combined with the dispersion
relation this coherence condition relates the frequency and
emission
in
no
emission
in
refractive
The coherence
this
mode
is
For the extraordinary mode the refractive index is greater
than unity everywhere, particularly for small angles with
respect
to
the magnetic field.
It has been
shown 6 ,7
for
reasonable
estimates
of
the
density and energy (nr/ne = 10-5, E = 200 keV) the emissivity
runaway
will be strongly peaked near
equal
that
to
or
the
plasma
frequency,
with
intensity
greater than the expected second harmonic emission from
Page 86
the 1 keV background plasma.
This
result
depends
upon
the
exact
used to describe the runaway population, but is shown to
distribution
be insensitive to the perpendicular component of the runaway velocity.
RAY TRAPPING
SEC 6.4
Although a sufficient emissivity exists, the radiation must exit
the
plasma in order to account for the observations.
While the phase
velocity of Cerenkov radiation will typically be within 15 degrees
the
of
magnetic field, the group velocity will be at 70 degrees or more,
that is, closer to the perpendicular direction.
This can be
seen
in
fig.(6), where the direction of the group velocity is indicated by the
normal to the dispersion
emitted
radiation
is
curve.
close
the
group
velocity
of
the
to perpendicular does not mean that the
radiation can be directly observed
refractive
That
from
the
outside.
Because
the
index for the extraordinary mode decreases moving from the
inside to the plasma edge, radiation may be reflected back toward
the
centre.
The plasma is homogenous in the direction of the magnetic field,
so
that
the refractive index component in this direction is constant
along a ray path.
In cylindrical coordinates (infinite aspect ratio):
n(rl) cosG(rl)
EQ (2)
n(r)
=
=
n(r2 ) cose(r 2 ) = constant
refractive index at r
At the point of emission this constant is determined by the
condition:
EQ (3)
{n coselemission point
I
[v-
)
coherence
Page 87
At the plasma edge the refractive
total
index
reflection . occurs.
internal
approaches
unity,
so
that
The situation is less severe in
toroidal coordinates, where the requirement for
radiation
to
escape
is 8 :
rlim
4 1 +
{n coseiemission point
EQ (4)
This condition is less restrictive on Alcator C than on Alcator A and
TFR,
which
may
in part explain the larger intensities observed from
Alcator C. For example, spectra similar to fig.(1) have been measured
DITE9 , which has R/rlim = 4.5.
from
If observed Cerenkov emission in
the ordinary polarisation results from multiple reflections within the
vacuum
chamber, the escape condition should be applied at the chamber
wall, if that is the source of 'emission.' Since the refractive
is
unity
index
at this point, from eq (4) it is seen that the radiation is
capable of escaping.
The most recent treatment 10 of Cerenkov
the
effect
of
multiple
reflections
emission over a 2w solid angle,
angles
of
emission.
This
that
work
by
by
is
limit
to
angles
over
all
successful
fig.(5).
The
in
spectral
of
emission.
Additionally
a
the considered runaway electron velocities is set by
the escape condition as determined
limit
included
a consequence of vacuum wall reflections, which necessitate
some type of average over all
lower
integrating
particularly
explaining the double humped spectra such as
shape
has
isotropising the reflected
is,
is
emission
from
eq (4).
The
chosen
lower
of .8 c restricts the velocity range of emitting electrons such
that only the total
number
of
electrons
in
this
region
has
any
Page 88
significant
effect
the
upon
intensity and the current
Using
emission intensity.
by
carried
these
Hence the average
are
runaways
related.
the result from ref(10) with the reflectivities of sec (1), the
runaway current is approximated by:
Ix
Irun(KAmp)
EQ
n11
n 1014 cm
-
4
m3-1/2T
Ix
-
Ibb
Te(KeV)
3
average intensity of extraordinary
mode Cerenkov emission averaged
over a small frequency interval.
This equation is meant to apply only to cases such
the
fig.(5)
where
emissivity is nearly constant over some small frequency interval,
and a good spectral fit can be obtained.
this
as
particular
Application
of
eq (5) to
example predicts a runaway current of 5 Kamps, which
amounts to 2% the total current.
This result is nearly
identical
to
As the plasma frequency approaches the cyclotron frequency,
two
the original treatment of Alcator A data.
competing
effects
occur.
The Cerenkov emissivity increases 1 l, which
enhances the emission intensity;
encroaches
upon
cyclotron
Thus the radiation resulting
wpe/uce = -4.
Fig.(5),
Under
this
interval over which the emissivity is
intensity
edge
from
for
example,
deals
a fixed
nearly
has meaning, and eq (5) applies.
15% the total.
with
the
condition there is a frequency
constant,
and
average
This is not the situation
in fig.(1) where eq (5) would predict a runaway current
or
frequency
current -will depend upon the particular values of the plasma
and cyclotron frequencies.
situation
the
the spectral region of emission, which decreases the
intensity by absorption.
runaway
and
of
50 Kamps,
Certainly it would be a mistake to apply eq (5) to
Page 89
fig.(4)
and
interpret
the
decreasing
emission
consequence of a diminishing runaway current.
SEC 6.5
REFERENCES & FIGURES
1.)
E. Marmar
Private Communication
2.)
S. Wolfe
Private Communication
3.)
A.E. Costley et.al.
Phys. Rev. Lett. 38
4.)
1477 (1977)
K. Sakai et.al.
Int. J. Infrared MM Waves
&1 77 (1980)
5.)
I.H. Hutchinson, D.S. Komm
Nuclear Fusion 17 1077 (1977)
6.)
H.P. Freund, L.C. Lee, C.S. Wu
Phys. Rev. Lett. 40 1563 (1978)
7.)
H.P. Freund et.al.
Phys. Fluids 21 1502 (1978)
8.)
M. Bornatici, F. Engelmann
Phys. Fluids 22 1409 (1979)
9.)
A.E. Costley
10.)
K. Swartz, I.H. Hutchinson, Kim Molvig
Phys. Fluids 24 1689 (1981)
11.)
A.G. Sitenko, A.A. Kolomenskii
Zh. Eksp. Teor. Fiz. 30 511 (1956)
Private Communication
intensity
as
a
Page 90
'DOMINANT
)pe
EMISSION'
EXAMPLE SPECTRA
ARB.
UNITS
80
KILOGAUSS
ARB
UNITS
60 KILOGAUSS
200
600
400
FREQUENCY
800
GHz
Fig.(1)
Extraordinary mode emission from times early in
discharges
when:
= I.ixi01 4 cm-3
(60 kGauss)
density
central
density = 1.5x1014 cm-3
(80 kGauss)
central
ragc .7
SPECTRAL FREQUENCY vs, LINE AVERAGE DENSITY
FOR 'DOMINANT Wpe EMISSION'
W/27r
P11-.
X
Gl
x
X X
X
XXXC
150 1
x
x
Xx
47r
X
n.
e2
X
x
x
-
100 I
x
XX
x
XX
50
1.0
2.0
3.0
Fig.(2)
Data from
ordinary
and
extraordinary
mode
emission;
magnetic fields from 60-100 kGauss.
The drawn curve assumes a peak to line average
density ratio of 1.3
101 cm
3
raye
t
SPECTRAL WIDTH vs. FREQUENCY
FOR 'DOMINANT cope EMISSION'
Co
I
I
8
Q =W
wce(r=a) - co
0
6
/e
p0
4F
/
/
40
/ 90
/ 0
/7W
0
21-
00
I
W/wce(r=O)
I
I
.5
.6
.7
Fi g. (3)
Spectral width is the FWHM of the plasma frequency
line.
Because of finite instrument resolution, Q
in excess of 8 is not achieved.
Page 93
INTENSITY RELATIVE TO THERMAL LEVEL
FOR 'DOMINANT
Ope
vs, FREQUENCY
EMISSION'
1bb
8
6
4F|
0
2 F
Wce (r=a)
0
%@
.60
0
Wce( r=O)
*.'D
I
f
I
,65
.70
,75
Fig. (4)
Data from repeatable discharges at 80 kGauss.
I
/Clce (r=0)
*
F
RESOLVED 'DOMINANT Wpe EMISSION'
pe
wce(r=alim)
pe
200
Wce
40
400
600
FREQUENCY
800
GHz
Fig. (5)
Extraordinary mode emission at 60 kGauss.
The
double humped feature of dominant emission is just
being resolved. In these cases the intensity is
reduced in comparison to previous examples.
~
-
sr
REFRACTIVE INDEX vs, ANGLE
FOR FREQUENCIES NEAR
Page 95
(0pe
nil
(= 1,0'4 (. pe
EXTRAORDINARY WAVE
ORDINARY WAVE
n_-
II
nil
EXTRAORDINARY WAVE
ORDINARY WAVE
= .60
= 1.02 wpe
-1
Nj
I
oce
=
.25
Fig.(6)
Only the extraordinary mode has phase velocity
greater than the speed of light, particularly for
direction.
field
the
to
near
propagation
Direction of group velocity is given by a normal
Page 96
SEC 7.0
FLUCTUATING PLASMA FREQUENCY EMISSION
Another
type
associated
with
emission.'
It
of
radiation
non-thermal
is
processes
characterised
feature,
near
the
electron
temporal
fluctuations.
The
observed
by
from
is
an
Alcator
termed
which
'fluctuating wpe
extremely
narrow
spectral
plasma frequency, which undergoes rapid
the
exceed
may
intensity
combined
While
radiation at the cyclotron harmonics by factors of ten or more.
sharing
similarities
theories,
the
with
results
from
various
experimental
Alcator
phenomenon
and
differ in some way from all of
In this sense fluctuating wpe emission is a novel
these.
is
result
not
previously documented.
The fluctuating emission is present to some degree in nearly one
half
of
all Alcator discharges.
deuterium,
and
100 KGauss.
the
helium
plasmas
It has been observed from hydrogen,
in
toroidal
fields
when
electrons.
conditions
However,
the
favour
the
production
fluctuation
may
reoccur
termination or may persist throughout the discharge.
not
related
with
of
to
suggests
runaway
during
plasma
The emission
is
fluctuations of other diagnostic signals,
such as current or density, however, occasional correlation
activity
40
The emission usually occurs during the early portions of
discharge
usually
from
with
MHD
that such activity may be involved in the emission
process.
The temporal behaviour
studied
with
a detector
submillimetre radiation.
of
the
operating
fluctuating
as
emission
a broadband
is
monitor
The large intensity of the feature makes
easily recognisable against the background level.
best
of
it
While the Michelson
of
interferometer as previously discussed is not suited for the study
rapidly
radiation,
varying
is
a
provide
does
amount of
limited
A scanning Fabry-Perot interferometer
spectral information.
is
used
to generate high-resolution spectra of the fluctuating feature.
TEMPORAL BEHAVIOUR
SEC 7.1
Examples of fluctuating wpe emission are shown in fig.(1), where
interferometer.
the
operating
is
detector
the
is
shortly
after
and
ionisation
persists
Comparison of these examples from consecutive
varying
levels
The
shows
discharges
that
of emission occur while macroscopic plasma parameters
As shown,
intensity of fluctuating emission can be associated with the flux
of hard x-rays.
in
bursting
about 50 msec.
for
such as temperature, density, and current remain unchanged.
the
on
evident
Superposed upon this background the fluctuating
emission appears as a series of high frequency bursts.
begins
Michelson
the
with
conjunction
The resulting interference structure
signal.
analog
in
the
presence
limiter.
emission
of
This suggests participation of non-thermal
process,
non-thermal
electrons
and
emission requires the
x-ray
their
collision
with
the
The absence of hard x-rays alone does not guarantee lack of
non-thermal electrons,
plasma
hard
since
electrons
discharge
however,
during
the
first
milliseconds
of
confinement of fast electrons will be poor, so that
the presence of these particles and the subsequent hard x-ray activity
can be related.
The early portion of a discharge is shown in fig.(2).
Fluctuating emission begins immediately after ionisation is
and
precedes the first intimations of x-ray flux by 1 msec.
voltage is a nearly constant 50 Volts and
relativistic electrons in .5 msec.
is
capable
of
completed
The loop
generating
Page 98
A notable feature of fluctuating wpe
with
periodicity
which
the
regular
the bursts of emission often occur.
Fig.(3)
show examples of such cases.
of
rates
5 - 20 kHz
emission
Typically these
of bursts.
narrow spikes
at
a total duration of several dozen cycles.
with
continue
with
a new
The fluctuations are best described as a series of
spaced
at
regular
The
oscillations.
sinusoidal
occur
fluctuations
The fluctuation may cease at this time, or may
series
is
time
intervals,
fluctuation
rather
spectrum
by
than
by a
produced
frequency analyserl will incorporate the non-sinusoidal nature of
fluctuations
by including overtones of the repetition rate.
Harmonic
structure on such a spectrum will reflect the structure of the
and
not
averaged
over
a small
during the time of
fluctuation
emission.
persists
bursts
necessarily the periodicity with which the bursts occur.
measure just this periodicity, the time
number
emission
frequency
is
of cycles.
shown
gradually
This property
can
interval
in
decreases
only
be
between
the
spikes
To
is
A plot of this frequency
fig.(4).
As
through
observed
seen,
the
the duration of
when
the
emission
for many oscillation periods and when the fluctuations are a
series of periodic spikes.
Often the spikes occur at
there
is
no
definite
irregular
fluctuation
frequency.-
explained by a superposition
of
differing
justification
await
frequencies.
discussion
frequencies.
of
The
spectral
intervals
several
in
Such
periodic
case
which
cases
may be
fluctuations
of
of this supposition must
measurements
of
the
radiation
Page 99
SEC 7.2
CORRELATED BEHAVIOUR
If the plasma density is sufficiently low so that wpe < wce, the
fluctuating
emission
in fact happens
at
may be observed throughout the discharge, if it
these
times.
When
this
occurs
the
temporal
behaviour changes dramatically, and a correlation between emission and
sawtooth disruptions becomes manifest.
This correlation is
fig.(5)
several
where
the
time
Whenever fluctuating
discharges
emission
is
observed
sawteeth is expanded.
during
the
spikes.
number
from
one discharge to another.
highest
fluctuation
frequencies
largest number of emission spikes.
fluctuation
one
frequency
sawtooth
interpreted
the
sawtooth.
observed
to
The fluctuation frequencies for
than
are
the
previous
associated
example.
with
the the
the
is plotted against the number of spikes during
period. for
as
is
This is seen in fig.(6) where
a number
of
discharges.
This
may
be
meaning the total time of emission is nearly constant
so that higher rates will produce more
rates
of
of spikes during a sawtooth cycle remains a constant over
sawtooth correlated emission are faster
The
of
However, they are fewer in number and are confined
many sawtooth periods during a single discharge, but
vary
portion
Like the previous examples the emission is a series
to a small time region surrounding the falling edge of
The
in
when sawteeth are present, it always appears in phase with
the sawteeth.
periodic
surrounding
shown
spikes.
However
the
faster
are also associated with higher plasma density, suggesting that
the frequency is somehow dependent on this or some
parameter.
associated
plasma-
Page 100
A closer examination of the sawtooth correlations
the
of
onset
emission
to
corresponds
that
reveals
of large m = 1 MHD
periods
oscillations which either precede or follow the sawtooth falling edge.
The emission is also seen to occur during periods of m = 1 activity in
This
the absence of sawteeth.
evident
that
the mode.
result
the
is
shown
in
fig.(7),
it
where
radiation occurs in phase with the oscillations of
These oscillations in the soft x-ray signals are thought to
rotation,
from
q = 1 flux surface.
either poloidal or toroidal, of the perturbed
The correlation of
fig.(7)
suggests
this
that
Fig.(8)
phenomenon is also related to the emission process.
shows
the time history of fluctuating emission from a single discharge,
displays
a large variety of correlated emission behaviour.
bits of emission are correlated with the small disruptions
with
the
is
and
The first
associated
After this the emission continues at a
current rise phase.
fairly steady rate until m = 1 oscillations become evident on the soft
x-ray
trace.
At this time the bursts of emission come in phase with
these oscillations.
Finally, once
sawteeth
is
largest.
Sawteeth and m = 1 activity
times
the q = 1 surface.
near
region
to
established
the
near the falling edge when m = 1 is
emission
confined
become
are
confined
to
the
plasma
Because of the correlation with this
activity the region of emission is
probably
also
localised
to
the
plasma centre.
SEC 7.3
COMPARISON WITH OTHER EXPERIMENTS
Fluctuating emission has
Although
behaviour
sawteeth.
also
been
observed
from
Alcator A.
less carefully studied, the emission shows the same temporal
as
the
Intense
Alcator C
bursts
of
results,
including
correlations
with
submillimetre emission have also been
vage iut
observed from DITE 2 and Microtor 3 , although it
of
consequence
interferograms
occur
useless,
fluctuating
briefly
from
measurements
with
fluctuating
signal
A
the
the
Since
signal.
fluctuations
during the early phase of discharges, they do
steady
Michelson
as
ionisation, and ignored.
of
since the Fourier transform technique cannot
not ruin the more important (from a confinement study point
interferograms
these
if
rendering
the
is
emission
fluctuating
a rapidly
deal with
usually
the
unknown
the same character as the Alcator results.
have
events
radiation
is
may
interferometers
a nuisance,
For
portion.
state
or
this reason,
this
regard
'noise' associated with
as
A submillimetre
view)
of
experiment
based
upon
a
heterodyne detector, or monochromator, concerned with radiation at the
second gyroharmonic will necessarily fail to detect
frequency
Furthermore, even if measured by a broadband detector, the
emission.
fluctuating signal may not be
display
plasma
and
storage.
For
noticed
example,
due
to
Alcator
the
format
of
data
data is digitised for
archival purposes at an effective rate of 3 kHz, making it useless for
the study of the much faster fluctuating emission.
SEC 7.4
REFERENCES & FIGURES
1.)
I.H. Hutchinson, S.E. Kissel
Phys. Fluids 23 1698 (1980)
2.)
A.E. Costley
3.)
R.J. Taylor
Private Communication
Private Communication
WITH CORRELATION TO SUBSEQUENT HARD X-RAYS
PLASMA
CURRENT
DENSITY
SUBMMI
.MICHELSON)
ARD X-RAY,
20
MSEC/DIV.
Fi g.(1)
Discharges in Hydrogen EaO kGauss from same run.
All pictures have the same scale.
Page 102
vage iu.
ONSET OF 'FLUCTUATING
ope
EMISSION'
LOOP VOLTAGE
50
VOLT MAX.
VISIBLE LIGHT
BROAD-BAND
SUBMILLIMETRE
.
HARD X-RAY
1
MSEC/DIV,
Fig. (2)
lonisation begins 1 msec from the left edge.
Visible light results from molecular and hydrogen
line emission. Decrease in visible light signal
indicates complete ionisation.
Page 104
PERIODIC BURSTS OF 'FLUCTUATING Wp, EMISSION'
.1 msec/div Ave. fluctuation rate = 40 kHz
central density = 3.0 1013 c1-3
.1 msec/div
Ave. fluctuation rate = 10 kHz
central density
=
1.9 1013 c1-3
Fig.(3)
average
higher
that
is
trend
general
A
density.
increased
with
occur
fluctuations
However, the rate tends to decrease in time for a
sequence of bursts. This is noticeable in the top
examole.
Page 105
PERIODICITY OF 'FLUCTUATING
RATE OF BURSTS
40
EMISSION'
ope
kHz
±
1.1
30
2
I
L
20
I
I
10
TIME FROM START OF EMISSION /ksec
I
I
200
I
I
400
I
I
600
Fig.(4)
Data. from two sequences of periodic bursting
emission from single discharge in Deuterium at
I
Page 106
EXAMPLES OF 'FLUCTUATING Ope EMISSION'
CORRELATION WITH SAWTEETH
PLASMA CURRENT
PLASMA DENSITY
SOFT X-RAY
CENTRAL CHORD
BROAD-BAND
SUBMILLIMETRE
50
MSEC/DIV.
2 MSEC/DIVI
50PySEC/DIV.
Fig. (5)
Successive expansions of two dischargj
Hydrogen at 60 kauss; 3 left ne = 7.3x10
right ne = 1.5x10 cm~
c n
cm,
Page 107
FREQUENCY OF BURSTS VS. NUMBER
DURING SAWTOOTH CORRELATED EMISSION
80
0
/
/
AVERAGE
FREQUENCY
/
kHz
/
601-
/
/
S
40
PREDICTION BASED UPON
CONSTANT EMISSIN TIME
t = 190 pksec
/
/
/
/
/
/
/
20
S
/
/
/
NUMBER OF BURSTS
I
I
5
10
Fig. (6)
Data from cases such as previous figure.
I
-I
15
20
Line
is
Page 108
CORRELATION OF 'FLUCTUATINGcope EMISSION'
WITH m=1 OSCILLATIONS
ASOFT
i'
X-RAY
CENTRAL CHORD
BROAD-BAND
SUBMILLIMETRE
.5MSEC/DIV.
Fig.(7)
The oscillations in the x-ray signal correspond to
rotation of the m=1 mode. Bursts of wpe emission
occur in phase with this rotation.
Page 109
EVOLUTION OF 'FLUCTUATING Wpe EMISSION'
CORRELATION WITH MHD ACTIVITY
UPPER TRACE:
BROAD-BAND SUBMILLIMETRE
LOWER TRACE:
SOFT X-RAY CENTRAL CHORD
Fig.(8)
Continuous picture of discharge
after ionisation.
1 MSEC/DIV.
starting
50 msec
Page 110
SEC 8.0
SPECTRAL MEASUREMENTS OF FLUCTUATING EMISSION
Spectral measurements of the fluctuating wpe emission show
radiation
occurs
in
an
that
extremely narrow bandwidth near the central
electron plasma frequency.
The spectral narrowness of the fluctuating
emission distinguishes it from the broader radiation feature discussed
in Sec.(6).
detailed
Because of the novelty
account
of
the
with
the
fluctuating
techniques
measurements will be presented in
comparison
of
this
and
emission
results
section
so
of
that
a
spectral
meaningful
other experimental results and discussion of various
theoretical interpretations can be made.
SEC 8.1
MICHELSON INTERFEROMETER
As mentioned in sec (5) the Michelson interferometer is not well
suited for the study of signals which vary during the 10-15 msec sweep
time of the vibrating mirror.
This
insensitive
fluctuations,
to
more
rapid
is
not
because
but
the
due
system
to
the
is
false
interpretation of these temporal fluctuations by the Fourier transform
routine
as
instrument.
resulting
In
the
from
case
the
of
changing
sawtooth
path
or
difference
loop
voltage
of
the
induced
oscillations, the amplitude of fluctuation is small enough so that the
effect on the generated spectra is very small.
fluctuating
thermal
wpe
emission
background
catastrophic.
Even
the
when
However,
because
the
is many times larger in intensity than the
effect
upon
the
resulting
spectra
is
the interferogram is smoothed prior to the
Fourier transform the spectrum is usually dominated
features resulting from the sudden bursts.
by
unreal
noise
Page 111
Fortunately
Michelson
it
is
not
interferometer
concerning
the
interferogram
necessary
signal
incident
occurring
in
to
order
radiation.
Fourier
transform
to
recover
Fig.(1)
is
the
information
an
example
during a period of fluctuating emission.
In
addition to the characteristic 10 KHz spiking there is a slower signal
modulation resulting from the action of the interferometer.
That this
effect is not the manifestation of a peculiar time history of emission
can
be
not
demonstrated
by
a single
example.
observations bear out the contention that this
indeed
caused
by
the
interferometer.
However repeated
modulation
effect
is
In this example, the minima
occurring at approximately 1.8, 3.5, 4.9, 6.3, 7.6 msec after the zero
path
difference
correspond
to situations where the path differences
are integral multiples of the
conditions,
Axi=NiL,
can
to
an
that
interval
motion is nearly sinusoidal
observed
case
the
minima
can
indicated
wavelength(s).
Since
these
be satisfied for only a single wavelength,
the recurring minima imply
restricted
emission
the
fluctuating
emission
about a single frequency.
with
known
amplitude
must
As the mirror
and
period,
be converted to a radiation frequency.
frequency
is
about
100 GHz,
be
the
In this
which
compares
favourably with the expected central plasma frequency.
The persistence of the modulations throughout the
is
a
reflection
of
the
spectral
narrowness
feature will no longer produce coherent
difference
feature:
is
equal
AX=c/Av.
interferogram
the
to
the
of the radiation.
interference
when
the
A
path
inverse of the frequency spread of the
Thus by measuring the 'coherence
spectral
interferogram
width
This technique was employed in the
length'
of
the
of the feature can be determined.
1 9th
century by Fizeaul and
others
Page 112
to
measure widths of optical lines resulting from atomic transitions.
A similar attempt to measure the coherence length of
emission
is
by
restricted
the
mechanical
fluctuating
the
limitations
instrument, however an upper limit of 5 GHz can
be
assigned
of
the
to
the
bandwidth of the emission feature.
SEC 8.2
In
FABRY-PEROT SYSTEM
order
fluctuating
wpe
spectra than
discussed
doublet
to
feature
the
in
investigate
any
spectral
structure
Michelson
allows.
As
an
example,
for
reasons
sec (9) the emission might be expected to occur as the
For
Alcator
wpi is approximately 1 to 2 Ghz, which is smaller than the
resolution capability of
measurements
system.
the
it is desirable to obtain higher resolution
wpe±wpi where wpi is the ion plasma frequency.
conditions
of
are
made
the
Michelson.
with
Because this device
High
resolution
spectral
a rapid scan Fabry-Perot interferometer
is
not
a standard
instrument
it
is
necessary to describe its operation before a presentation of resulting
measurements can be undertaken.
The experimental arrangement of the
system
is
shown
in fig.(2).
Fabry-Perot
Radiation from the Tokamak follows the
same optical path described in the appendix, and is
wire
mesh
This
polariser
polariser
is
which
oriented
to
couple
equal
upon
a
amounts
of
both
One of these detectors is a broadband
monitor having a nearly constant response from
coupled
incident
splits the signal between two detectors.
polarisations to each detector.
radiation
interferometer
50
to
200 GHz.
The
to the second detector must first pass through the
Fabry-Perot etalon which transmits
only
specific
frequencies.
The
Page 113
is
etalon
to that described in sec (5), and consists of two
similar
wire meshes of
loudspeaker 2 . When the loudspeaker is driven the
adjusting
by
the loudspeaker dome.
these
be
the position of the static mesh with respect to
three
generates
system
the
During operation
as a position monitor for the moving mesh.
from
of
from each of the two detectors and a third which serves
one
signals,
can
range
this
of
The location
for example from 54 to 68 GHz.
varied
amplitude
etalon can be swept over a small frequency range,
the
Thus
Typically the
oscillation
an
with
110 Hz
loudspeaker is driven at
between
separation
the two meshes varies with time in a sinusoidal manner.
.3mm.
is
a moveable stage, the other on the dome of a
on
mounted
statically
mesh
One
constant.
mesh
inch
per
lines
100
is
it
signals
position
frequency is related to the
to
necessary
In order to make any sense
know
monitor
how the transmitted
also
is
It
signal.
important to know the frequency resolution of the instrument.
This calibration is
with
made
an
Fig.(3) shows such a calibration.
source.
respect
the static mesh with
location
changing
to
the
83 GHz
klystron
radiation
By varying the location of
moving
mesh
and
noting
the
the 83 GHz signal with respect to the position
of
monitor signal, an accurate calibration of the position monitor signal
is
made,
frequency.
mesh
thus
allowing the conversion of this signal to transmitted
The position monitor signal is nearly proportional to
the
a small phase shift must be included.
The
separation,
though
frequency resolution is measured by noting the
line.
The
line
width
of
the
The
measured
of
the
83 GHz
source can be neglected so that the
measured width results entirely from
instrument.
width
finesse
the
finite
resolution
of
the
of 114 at 83 GHz is in agreement
Page 114
with calculated values 3 . When measurements are
other
made
at
frequencies
83 GHz it assumed that the finesse is proportional to the
than
square of the wavelength as dictated by theory and experiment.
The
relative
spectral
shape
of
the
emission
signals
obtained
feature
determined
by
detectors.
It is not necessary that the two detectors have
response,
the
only
frequencies.
of
the
from
since
fluctuation will be
results
both
lost
detectors
when
show
will
the
the
be
independent
occur
in
the
of
same
the
first
ratio
is
fluctuating
order
of
temporal
be effected equally and the
taken.
emission
Because
to
narrowband, radiation passing through the Fabry-Perot can
to
the two
that their responses be in the same proportion at all
The measured spectra will
fluctuation
Michelson
ratios
is
transmission
be
be
only.
the
quite
assumed
The signal
intensities are measured with respect to any existing baseline so that
non-fluctuating thermal components will be ignored.
SEC 8.3
HIGH-RESOLUTION SPECTRA
Examples of the three signals from the Fabry-Perot system during
the
occasion
of
fluctuating
emission
are
period occurs 10 msec from the initiation
During
of
shown in fig.(4).
the
plasma
discharge.
this interval the mesh motion corresponds to a frequency sweep
starting at 50.5 GHz and increasing to 60.5 GHz.
fluctuating
signal
remains
nearly
While the
sweeps through the emission feature.
the expansion of the region of emission
mentioned
that
a good
bit
of
broadband
constant, the Fabry-Perot signal
shows a maximum of transmission indicating that during this
etalon
This
chance
in
time
the
This is shown clearly in
fig.(4).
It
should
be
is involved in making these
Page 115
measurements.
Since the
preselected,
one
frequency
must
wait
region
for
to
be
scanned
must
be
the necessary conditions of plasma
density and presence of fluctuating emission in order to
measure
the
spectra.
The relative spectrum obtained from this example by dividing the
Fabry-Perot signal by the broadband signal is plotted as a function of
frequency,
determined
fig.(5).
The
spectrum
very narrow bandwidth.
primarily
from
from
the
calibrated
radiation
monitor,
in
shows an unambiguous transmission feature of
The
scatter
of
the
plotted
points
arises
uncertainties in reading the positions and amplitudes
of the various bursts of emission.
the
position
frequency
However, since it is assumed
that
of the emission remains constant during the
sweep time, the scatter might also result partially from small changes
in the frequency.
The residual level in the wings of the transmission
feature is due to leakage of
baseline
level
around
the
etalon,
so
the
should be regarded as slightly positive to compensate
for this problem.
Fabry-Perot
radiation
The resolution corresponding to
the
FWHM
of
the
is indicated on the spectrum and is such that the feature
is regarded as being unresolved.
The electron density at which the plasma
measured
frequency
of
54 GHz
frequency
is
not
too
far
from
the
approximately
typical
these
small
values
is
considered.
1.6
measured
particularly when the relative accuracy of the line
at
at
this
time.
the radiation is assumed to occur at the central plasma frequency,
a peak to averaged density ratio of
This
the
is 3.6 10-13 cm- 3 . The line averaged
density is about 2.8 10-13 cm- 3 and is rising rapidly
If
equals
is
required.
values
of 1.3,
averaged
density
Because accurate values of
Page 116
central
it
available,
emission
does
be
not
can
demonstrated
the central plasma
at
occur
However, no examples have been found
frequency.
frequency
the
that
conclusively
not
are
density
where
the
measured
The
of emission is not consistent with such an assumption.
measure
emission frequency conceptually provides a sensitive
of
the
plasma density since the instrumental resolution of 0.2 GHz translates
to density changes of typically 1%.
technique
The
accuracy
absolute
of
this
upon the association of the radiated frequency with
hinges
the central plasma frequency.
Occasionally the emission feature is observed
the
characteristic
situation where a non-fluctuating type of emission is
the
Fabry-Perot
system.
by the transmitted signal.
although
it
resolution
varieties
appears
at
are
Once
again
the
the
because
frequency.
measured,
encountered
by
The shape of the spectrum is directly shown
broader
higher
occur without
Fig.(6) illustrates the
behaviour.
fluctuating
to
the
Whenever
signal
fluctuating case, but they have the
of
feature
same
narrow
unresolved,
reduced instrumental
these
intensity
is
is
non-fluctuating
less
spectral
than
the
shape
as
their namesake.
SEC 8.4
SHIFTS IN EMISSION FREQUENCY
The assumption that the radiation frequency
constant
is
than
relatively
necessary because the Fabry-Perot requires a finite time
to sweep through a frequency interval.
more
remains
The
required
time
is
never
a few msec, which is small in comparison to the time scale
of density changes.
Because of this it is not likely that
variations
in plasma frequency caused by density changes will have an effect upon
Vage ii/
the measured spectra.
There is evidence of sudden change
successive
example.
bursts
of
fluctuating
in
radiation
emission.
frequency
of
Fig.(7) shows such an
The broadband signal shows a series of individual bursts
of
radiation, most of which belong to a sequence of evenly spaced spikes.
In addition there are a few spikes which occur outside this
The
Fabry-Perot
signal
shows
the
transmission
of
sequence.
the sequential
spikes, but also shows that the additional spikes are not transmitted.
The
situations
are
indicated where two successive bursts occur, but
only the one belonging to the evenly spaced sequence
The
frequency
content
of
these
sudden
shifts
transmitted.
bursts must differ by at least the
instrumental resolution, which is 0.3 GHz for this
demonstrating
is
in
radiated
example.
frequency
Besides
fig.(7)
suggests a connection between radiation frequency and the
also
periodicity
of temporal fluctuation.
It is difficult to make
where
the
bursts
are
good
regularly
spectral
measurements
of
spaced and few in number. This is
because the peak spectral frequency will most likely be swept
during
an
interval
are
irregularly
does not happen.
time
it will
spaced
In cases such as
fig.(4)
be
so
where
the
but much denser in time this problem
However, if the radiation frequency is
not
through
between bursts, and the few points obtained will
occur in the wings of the line.
bursts
cases
easily
noticed
in
changing
these cases.
in
If the
situation of irregularly spaced bursts is treated as the superposition
of
sets of periodic bursts, each with different periodicity, each set
may have assigned to it a different
spectrum
shown
in
fig.(5)
radiation
frequency.
Thus
the
may be some average of many frequencies.
Page 118
The problem of changing frequency makes it
that
every
in
case
the
However, no apparently broad
which
could
not
be
is
spectrum
spectral
difficult
to
be
certain
instantaneously very narrow.
features
have
been
observed
attributed to frequency changes during the scan
time.
Because
polarisation
the
two
equally,
from a polarisation
occurring
in
detectors
the
fluctuating emission
coupled
such
as
the
polarisations.
show
each
no
clear
always nearly equal in each mode.
preference
The
successive
two
of
mode,
absence
of
polarisation
of
the
nor
any
emitted
polarisation
at
assume
REFERENCES & FIGURES
H. Fizeau
Ann. Chim. Phys. 66
429 (1862)
2.)
D.S. Komm, R.A. Blanken, P. Brossier
Applied Optics 14 460 (1975)
3.)
Renk & Genzel
that
the detector implies the
similar situation at the region of emission within the plasma.
1.)
bursts
Because of the large effects of diffraction and
measured
SEC 8.5
of
The measured intensity is
reflection at these wavelengths it would be imprudent to
the
mode
Polarisation measurements of
evidence of sudden changes in polarisation.
radiation is unknown.
to
situation shown in fig.(7) can not result
effect;
different
are
Page 119
MODULATION OF 'FLUCTUATING wpe EMISSION'
BY MICHELSON INTERFEROMETER
2 MSEC/DIV.
Fig. (1)
The indicated point is the zero path difference of
the Michelson interferometer. Extremes of mirror
vibration occur 3 msec prior to and 12 msec after
this time.
Page 120
SCHEMATIC DIAGRAM OF FABRY-PEROT SYSTEM
BROADBAND
DETECTOR
DATA
* STORAGE
F DISPLAY
LOUDSPEAKER
DRIvE
RADIATION
FROM
TOKAMAK
DETECTOR
B
BEAM
SPLITTER
* DETECTOR:
*
MESH :
DRIVE :
*
FINESSE:
F XED
GI RID
MOV11
GRII
InSb crstal at 4.20 Kelvin
100 lines per inch electroform copper
110 Hz
amplitude
114
Fig. (2)
=
-
.3 mm
Page 121
CALIBRATION OF FABRY-PEROT SYSTEM
WITH 83.5 GHz KLYSTRON
1 msec/division
Fig.(3)
position
the Fabry-Perot
is
trace
The upper
transmitted
the lower trace is the
monitor,
The appropriate horizontal
klystron radiation.
scale is 1 GHz/division. This implies a FWHM'of
0.73 GHz, or a finesse of 114.
Page 122
'FLUCTUATING W0 pe EMISSION'
EXAMPLE SIGNALS FROM FABRY-PEROT SYSTEM
(b)
(a )
FREQUENCY
MONITOR
FABRY- PEROT
SIGNAL
UPW-WMWW- -
0.5 ms
W,
, WV
'W-=
EXPANSION
BROAD BAND
SIGNAL
0.1 ms
Fig.(4)
Emission from Hydrogen
plasma
in
60 kGauss
magnetic field.
Density = 2.8x1013 cm- 3 line average;
Current = 120 kAmperes at time of emission. a.)
The Fabry-Perot is sweeping from 50.5 to 60.5 GHz
left to right. b.)an expansion about the region
of maximum transmission.
Page 123
HIGH RESOLUTION SPECTRUM OF
'FLUCTUATING Cope EMISSION'
RESOLUT ION
0.
I-
0
C,,
UJ
F-
w
*
0
1-J
*0
w
0
4#
*
0
0e
.0
e0
55,0
54.0
FREQUENCY
(GHz)
Fig. (5)
Spectrum
obtained
from
previous
figure.
Instrumental resolution of .2 GHz is determined by
the meshes used in the construction of
the
Fabry-Perot interferometer.
Page 124
'FLUCTUATING
Wpe
EMISSION'
SPECTRUM OF NON-FLUCTUATING COMPONENT
BROADBAND SIGNAL
POSITION MONITOR
FABRY-PEROT SIGNAL
.25 msec/division
Fig.(6)
Horizontal scale corresponds to 1.3 GHz/division
with
the
feature occurring at 71 GHz.
The
instrumental resolution at this frequency
is
0.5 GHz, so that the feature is regarded as
unresolved.
Page 125
SUDDEN SHIFTS IN RADIATED FREQUENCY
OF 'FLUCTUATING W0 pe EMISSION'
POSITION MONITOR
BROADBAND SIGNAL
FABRY-PEROT SIGNAL
t
I
,imsec/division
Fig.(7)
Horizontal scale corresponds to sweep in frequency
from 55 to 52 GHz, left to right. At indicated
times only one of two consecutive bursts is
transmitted, indicating a frequency difference of
.3 GHz or more.
Page 126
SEC 9.0
FLUCTUATING EMISSION: PLASMA WAVES
The peculiar conduct of fluctuating wpe
numerous
explanations,
Tokamak environment.
untenable
when
particularly
when
emission
dealing
Nonetheless, explanations can
they
can
generate
with the complex
be
dismissed
predict behaviour contradicting the established
experimental results, or when they impose unrealistic restrictions
plasma
parameters.
as
to
A candidate process must require the presence of
runaway electrons, and must generate intense narrowband radiation near
the
plasma
frequency.
fluctuations
and
It
must
correlations
allow
for the variety of temporal
observed,
and
must
be
capable
of
operating under a continuous variety of plasma conditions.
SEC 9.1
STIMULATED EMISSION
An example of a process eliminated
spontaneous
from
consideration
Cerenkov resonance discussed in sec (6).
is
the
As demonstrated
by experiment and theory, the spectral width of this radiation is very
much
broader than measurements of fluctuating emission, sec (8).
frequency domain
dispersion
of
relation
Cerenkov
by
the
emission
is
determined
in
For
radiation
similar
discounted.
through
density and magnetic field;
which are constant in neither time or space
reasons spontaneous cyclotron
the
The
the
quantities
Tokamak
plasma.
by runaways is
This process, while capable of generating emission far in
excess of thermal levels, is fundamentally very broadband in nature.
These spontaneous emission processes produce broadband
because
the
dispersion relations governing electromagnetic waves are
-not particularly restrictive, allowing a single fast
resonant
emission
electron
with, and produce radiation at many frequencies.
to
be
Stimulated
Page i/
processes can produce narrow spectra by coupling the emission
back
with
the
emitting plasma, allowing the most unstable frequency
(the one with greatest growth rate) to dominate the spectra.
possibility
is
product
the
stimulated
emission
of
One such
extraordinary waves by
Cerenkov resonance with runaway electrons.
In analogy with an optical
laser,
medium
this
would
require
an
unstable
characterised by a
population inversion or negative temperature, a coupling process,
and
some effective cavity to provide the feedback between wave and medium.
Or, it is possible for the gain to be so large that in a single
pass
As
through the medium a dominant wave emerges, requiring no 'cavity'.
discussed in sec (6) trapping of extraordinary waves can
result
from
the total internal reflection due to the radial decrease in refractive
index.
However, this trapping appears to
be
weak
in
the
toroidal
case, by the very fact that the spontaneous emission is observed.
It seems unlikely
radiation
is
that
stimulated
responsible
for
the
observed
accompanying spontaneous emission is ever
fluctuating
emission.
of
emission
seen
extraordinary
feature,
in
since
conjunction
no
with
Furthermore, the Cerenkov resonance requires a
high electron energy, since the refractive index is
close
to
unity,
which contradicts the observation of fluctuating emission from plasmas
which would appear to have marginal runaway energies.
The considered explanations
involve
some
of
fluctuating
emission
all
intermediary 'mechanism, electrostatic plasma waves in
particular, which greatly restricts the range
eventually
wpe
radiated.
The
utility
their narrow frequency spectrum
marginally non-thermal electrons.
and
of
frequencies
to
be
of the plasma waves results from
their
possible
resonance
with
In addition, the trapping condition
Page 128
on electrostatic waves is much stronger, so that
and
electrostatic
plasma centre.
waves
a 'cavity'
exists,
can be described as modes confined near the
Given the existence of runaway driven plasma waves, it
must be shown how these waves give rise to electromagnetic radiation.
SEC 9.2
RELATED EXPERIMENTAL OBSERVATIONS
Related experimental observations of an extraterrestrial
concern
a
class
of
events
known
as
type
III
solar
characterised by radiation at the fundamental and second
the local plasma frequency.
origin
burstsi,
harmonic
of
Energetic electrons (100 keV) are ejected
from the sun during a solar flare and stream along field lines through
the
solar
waves.
would
wind
plasma,
where
they
drive linearly unstable plasma
Saturation of these waves by a quasi-linear plateau
limit
their
formation
propagation to only a few kilometers, conflicting
with the observation of waves in the vicinity of earth.
An
alternate
non-linear stabilisation mechanism has been proposed 2 which allows the
plasma waves to propagate as
This
mechanism
produces
localised
wave
packets,
fluctuations.
solitons.
a spread in phase velocity of plasma waves
through a parametric decay which generates an enhanced
acoustic
or
Radiation
at
the
level
of
ion
fundamental results from
scattering of electrostatic waves from the density fluctuations of the
ion
waves,
while
radiation
at
the
second
harmonic
occurs
by
coalescence of two plasma waves.
A somewhat
contradictory
model 3
of
solar
burst
radiation
involves the 'collapse' of the plasma wave packets on time scales much
shorter than
stabilisation
the
growth
of the mode.
time
of
the
beam
mode,
rather
than
a
The collapse of the wave packet is in the
Page 129
spatial sense, and is produced by a spreading
Radiation at wpe and
plasma waves.
2wpe
in
wavenumber
of
the
occurs by the same scattering
process described previously.
Regardless
of
what
model
is
used,
the
fact
remains
that
electromagnetic radiation associated with type III solar bursts occurs
at twice the
intensity
plasma
at
frequency
twice
the
as
well
as
the
fundamental.
plasma frequency is as or more intense than
that at the plasma frequency, which is not the case
where
results,
2 wpe
no radiation at
is seen.
for
the
relation
for
Alcator
Furthermore, the solar
wind plasma is not strongly magnetised, (wpe/wce = 100), so
dispersion
that
the
case
for
the
plasma waves is not strongly dependent upon
the wave vector direction with respect to the magnetic field.
not
The
wpe < wce,
when
the
appropriate
This is
branch of the
dispersion relation becomes 4 l
EQ (1)
2
2
h_
1
2
_ 4w2c2cos2e
"W
-
1
1/2
cos2e
This means that the envisioned decay to a low frequency wave can
occur
for
resultant
plasma waves propagating at small angles to the
field, reducing the problem to one dimension, whereas
models
are
two dimensional.
the
so that acoustic waves will be strongly damped.
into
ion
'collapse'
A final difference arises from the fact
that the Alcator plasma has comparable electron and ion
to proceed
only
quasi-modes,
and
Decay would then have
radiation
scattering from the thermal fluctuation level.
temperatures,
would
result
by
Page 130
Back
on
measurements
earth,
of
some
of
the
earliest
reports
type
of
spectrum
stellarator 5 ,6 .
A
microwave emission above the thermal level lies in
the vicinity of the electron plasma
noise
spectral
a fluctuating plasma frequency emission were made in
conjunction with turbulence studies on the Uragan
bursting
of
shows
while
frequency,
fluctuations
the
turbulent
at harmonics of the ion acoustic
frequency.
The radiation has an instantaneous spectral width
while
spectrum of fluctuations shows two peaks at twice and four
the
times the ion frequency.
and
The radiation can be described
by
of
10%
wpe+-wpi
wpe+-2wpi, and is explained by the nonlinear scattering of plasma
waves by density fluctuations at the ion acoustic frequency.
A fundamental
results
is
that
difference
the
Alcator
concerning
and
Landau
damping
of
ion
Alcator
Significantly, when auxiliary
Ti > Te, the
heating produces equivalent temperatures in Uragan,
fluctuating emission vanishes.
ion
Uragan
plasma is nearly isothermal, Te = Ti,
while for Uragan it is not, Te = 10 Ti.
ion
the
This can be explained
by
the
strong
waves which occurs when the ion thermal
velocity approaches the ion sound speed.
Also, the Alcator plasma
is
not in the same turbulent state, with the marked resistive anomaly, as
the Uragan plasma.
Finally, the spectral measurements
from
the
two
machines have a dissimilar character.
A more recent experiment
results
which
stellarator.
a uniform
W >
are
similar
on
a linear
to
those
reported
has
from
generated
the
Uragan
An electron beam is injected along the magnetic axis
plasma.
of
these
of
High frequency electrostatic waves with frequency
wpe are seen to grow and quickly saturate.
dispersion
device 7
Measurements
of
the
waves identifies them as electron plasma waves.
Page 131
Saturation of the plasma wave amplitudes corresponds to the onset
of
growth
low
density fluctuations, and the simultaneous
frequency
emission of electromagnetic radiation near the plasma frequency.
density
fluctuations
and
are
confined
The
to a relatively narrow frequency
band, and display the dispersion of ion acoustic waves propagating
the
direction
of
the
electron beam.
The frequency spectrum of the
electromagnetic radiation is a few percent wide, with centre
above
the
the direction of the incident beam (ordinary mode).
results
are
shown
to
agree
very
slightly
The radiation is polarised in
plasma frequency.
electron
in
well
with
The
experimental
a mechanism wherein
radiation results from scattering of intense plasma
waves
from
self
consistently produced ion acoustic waves.
In this and other linear experiments,
waves which are only moderately damped.
high
Experimentally,
beam
downstream tracing
convective
of
injection
wave
allows
acoustic
In addition, there is often a
at
radiation
even
and
turbulence,
of
degree
Te > Ti
2wpe 8
at one end of a linear device allows
growth,
so
that
distinctions
and absolute instabilities 9 can be made.
Tokamak, the 'beam' is injected continuously in
between
In the case of a
space,
so
that
the
distinction between a instability that travels spatially with the beam
and one that grows in time for all space can not be made.
Finally, some mention in passing should
class
be
made
concerning
a
of runaway instabilities noted in Tokamaks, which arise from an
effect1 0 .
anomalous Cerenkov resonance
produce
periodic
explain
the
simultaneous
bursts
radiation
of
that
fluctuations
plasma
is
While
waves
observed),
this
mechanism
does
(the mechanism does not
it
also
generates
in the loop voltage, temperature, current,
Page 132
hard x-rays etc.
The driving force of the
and
anisotropy,
temperature
the
periodic
sudden sudden attempt to equilibrate the
electron temperatures.
instab'ility
is
a severe
fluctuates result from a
parallel
and
perpendicular
This effect is not observed in Alcator, nor is
it expected, since the total energy
of
the
runaways
is
negligible
compared to the background plasma, which is well behaved.
SEC 9.3
An
RUNAWAY ELECTRONS
initial
experimental
circumstance
required
in
all
bursts
external
to
and
the
Stellarators
beam
plasma
plasma
with
In
the
case
of
studies, these electrons are created
which
they
interact,
whereas
in
and Tokamaks creation occurs within the very plasma.
these toroidal devices the source of
runaway
previous
results, including those from Alcator, is the generation
of a detached component of suprathermal electrons.
solar
the
phenomenon
electric field.
associated
with
suprathermal
the
electrons
current
driving
By equating the energy gain during free fall
is
In
the
induced
in
the
electric field with energy loss through coulomb collisions, a critical
electron runaway field is found'l:
47r e3 ne lnAc
Ec
-
EQ (2)
m vth
2
4 lnAc
ne
15
Te
{Volt cm~1 , 1014 cm- 3 , KeV}
The highest electric fields, .12 V/cm, are encountered
ionisation
and
shortly
thereafter.
during
These fields do not exceed the
critical runaway field (Ec = 2 V/cm) during this time, so
as
a whole
will
not
plasma
the
plasma
runaway, rather only electrons with motion in
excess of a critical velocity will:
Page 133
[L 1.2
Ec
.c
EQ (3)
vth
The rate of runaway production is
determined
by the
ratio
of
the
thermal electron velocity to this critical runaway velocity:
rvthl 3/4 exp .
EQ (4)
K ne v(vth) ---
-
c
2
/2 vc
2vth
vth
An estimate of the number of runaways can be made by
applying
[vC
to
period shortly after breakdown, while the voltage is high and
the
before hard x-ray activity indicates the loss
case
eq (4)
of
fig.(2),
of
runaways.
In the
sec (7)this approximation is between 106 cm- 3 and
107 cm-3.
The presence of runaway electrons alone does not a plasma
make.
The traditional linear formalism requires a positive slope to
the runaway distribution, as in the case of the well
tail'
problem.
This
type
critical runaway velocity
known
'bump on
of distribution can be expected when the
decreases
rapidly,
so as
to
detach an
isolated segment of velocity space from the bulk distribution.
situation can be realised during the early
Tokamak
current
rise
Such a
portion
of
discharges when the critical velocity changes from Uc = 16Vth
to Uc = 60Vth in a period of 3-5 msec.
the
wave
presence of neutral
ionisation
phase 12 , which
Bumps in velocity space due to
atoms
have
may
account
also been noticed during the
for
the
observation
fluctuating emission immediately following complete ionisation.
not unreasonable to suppose that a beam type
accompanies
electron
of
It is
distribution
fluctuating emission, but since no direct measurements of
the distribution have been made this can only be an assumption.
It is
Page 134
noteworthy
that
grossly
runaway discharges, produced by withholding
neutral gas injection, do not
suggests
the
necessity
of
generate
fluctuating
a subtle
perturbation
emission.
This
of the electron
distribution, rather than a wallop of runaway current.
SEC 9.4
PLASMA WAVE MODES
A beam of runaway electrons will drive linearly unstable
waves
which
satisfy
the
resonance
condition;
plasma
w - k vb = 0-
runaways are restricted to having only parallel velocity, the
use
If
of
the cold plasma wave dispersion, eq (1), yields:
EQ (5)
wp cose
=
Ik|
vb cosE
ki
;
vb-
This means that a particular beam velocity will be resonant with waves
having
the same magnitude wavenumber.
Since the duration of a single
burst of radiation is much greater than
time
of
the
driving
the
circumferential
transit
electrons, it is reasonable to assume that the
plasma waves can be represented by a toroidaily
uniform
waveform
or
mode.
Inspection of the dispersion relation, w = wb cosE shows that as
the
wave propagates radially outward into regions of reduced density,
cose must increase to
maintain
constant
frequency.
The
condition
cose = 1 limits the radial extent of the wave, so that peaked density
profiles will result in the trapping of plasma waves near
centre.
mode
plasma
If the torus is treated as a cylinder with periodic boundary
conditions over a length 2WR, undamped plasma waves can
by
the
numbers
be
described
representing the allowed perpendicular and parallel
Page ijb
components of the wave vector.
'i'(r,z,t)
=
Om,n(r) exp i{knz - wm,nt}
EQ (6)
kn = n/R
om,n(r) + 0 for r > 0
The parallel wavenumber must be an integer multiple of the
major
radius.
reciprocal
Because of the relative large size of the major radius
with respect to typical plasma wavelengths, the axial mode number will
be
quite
large, n > 800.
Changes
produce small changes in the wave
which
is
in
the
frequency,
axial mode number will
typically
Aw = 70 MHz,
smaller than the resolution of the measurement system.
The
perpendicular extent of the wave is determined by the density profile,
and
by
the
frequency of the wave with respect to the central plasma
frequency.
a
rr dr
k1
2
=
om, n~r
k (r)tm,n(r)
+
dr
=
0
k2k2 p(r)2
2
EQ (7)
1/
k 2(r)
=
where a square root
particular
k2{
[I
parabola
-
r2/rlim 2 ]1/ 2 _ w2
density
k, as- determined
by
the
2~
profile- is
number
radial
This
correspondence
frequency,
activity.
restriction 'offers
of
a
The
two
lowest
solutions are shown in fig.(1), where it is seen
that the radial extent of the plasma wave is confined
centre.
For
beam velocity, bounded radial
solutions will exist for discrete values of w only.
mode
assumed.
the
emitted
a possible
frequency
to
to
the
explanation
the
central
plasma
for the
plasma
and the correlation of emission with centrally located MHD
A shift in radial mode number, for example
m = 2, will
produce
a frequency
change
of
from
m = 1 to
430 GHz under typical
Page 136
This
conditions.
so
resolution,
is
difference
that
observed
the
frequency may be related to radial
in the emission
discontinuities
of
transitions
mode
instrumental
the
than
larger
the
plasma
wave.
OTHER WAVES
SEC 9.5
Fig.(2)
nearly
shows
parallel
the
Appleton-Hartree
propagation.
The
dispersion
waves
that
As
shown,
these
branches
branch
spontaneous
Cerenkov
are resonant
because
of
to
with
their
electrostatic waves.
electron
mode
velocities.
pass through the w = wpe region.
also
corresponds
ordinary
waves occur at or just slightly below the
plasma frequency, and can be resonant with many
Additional
for
have been discussed
belong to the electrostatic limit of the low frequency
branch.
relation
the
extraordinary
emission.
mode
involved
One such
with
the
It is seen that waves in this region
relativistic
electrons
large
velocity cannot be treated as purely
phase
Notice that
the
only,
1.0 >n> 1.1.
extraordinary
mode
and
resonance
with relativistic electrons extends nearly to the cyclotron frequency.
This
is
why
broadband.
the
spontaneous
Cerenkov
emission
is
fundamentally
Ignoring stinulated Cerenkov emission means that waves on
this branch are not considered.
Electrons resonant with the magnetised
resonate with
plasma
electron cyclotron waves near wce.
waves
made.
of
the
also
If the fluctuating
emission involves driven waves belonging to the wp branch,
consideration
can
then
some
higher frequency electrostatic waves should be
These waves have the following angular dependence:
Page 137
2h
EQ (8)
W2
=
w
2
1
2L1
1/2
W4 _ 4w2 W2 COS20
p
2
Wc2 w5j2 sin2e
ccoe
As the wave propagates outward to regions of decreasing magnetic field
and
density,
the
angular
term
increase to maintain constant
must
perpendicular
Eventually the wave travels
frequency.
to
the
field
it is absorbed at the upper-hybrid layer, which will always lie
where
between the
propagates
resonance
cyclotron
the
observer.
are
radiation
process
the
from
excluded
stimulated process is expected to occur with
were
to
frequency, it would not be observed from
since
the
wave
In contrast to the magnetised plasma waves, the
waves
cyclotron
non-thermal
If
the increase in field and density will decrease e
inward,
and reflect the wave.
electron
and
these
occur
the
Thus no
centre.
If
waves.
at
outside
the
in
some
cyclotron
any
case,
the fundamental is optically thick in the ordinary mode, and is
cutoff by the hybrid layer in the extraordinary mode.
SATURATION OF PLASMA WAVES
SEC 9.6
Given the instability of plasma waves to linear growth,
concerns the saturation mechanism of these waves.
question
mechanisms are beam trapping 13 , and
cold
small
the
beam,
initial
determined by trapping of beam
driven
wave,
while
decay
limitation
processes
Two known
instabilities1 4 .
decay
electrons
a fair
of
in
require
the
the
For
instability
potential
of
a
is
the
larger levels of beam
energies.
According to the trapping model, a single wave emerges from
initial
unstable
band
of wavenumbers.
the
If a beam has density nb and
velocity vb, this single wave will have the maximum growth rate:
Page 138
EQ (9)
Yx=
131/2 2-4/311
/3J
k
Wpne
k
P
p
vb
e
When the beam density is small, the wavenumber spread of
the
growin'g
waves will be negligible, and trapping of beam electrons will occur in
the potential wells of the
After
several
trapping
nearly
times,
sinusoidal
the
wave
wave
electric
energy
tends
field.
toward
a
meta-equilibrium:
2-4/3
nb m Vb
2
W =
EQ (10)
n
ne
In the most extreme case this energy is expected to be less than 1% of
the thermal electron energy, so in no way is this a strong beam plasma
case.
to
The meta-equilibrium will persist until other waves have
a similar
amplitude
as
the
original
unstable
quasi-linear description becomes appropriate.
broadening
This
mode,
will
grown
when
a
produce
a
of the wave spectrum and the formation of a plateau in the
beam distribution.
The decay
transfer
of
instability
energy
from
limits
growth
plasma
wave
decays
into
wavenumber, and an ion acoustic wave.
have
lower
of
the
non-linear
solar
another
bursts,
plasma
The product
where
wave
plasma
the
of larger
wave
will
phase velocity, so it will be non-resonant with the beam,
but may be damped
spectrum
to
beam resonant waves to non-resonant modes.
This is the process applied to the case of
original
due
high
by
the
ambient
frequency
plasma.
fluctuations,
stated which saturation process is occurring.
Without
it
measuring
the
cannot be definitely
However,
the
trapping
I
Page i39
model
only
requires
a coherent
linearly
operate for small values of beam energy.
an
upper
limit
wave, and will
unstable
Eq (10) may be
regarded
as
to the expected wave energy, since any decay process
will only reduce this value by transferring energy to other modes.
SEC 9.7
LINEAR RADIATION PROCESSES
A plasma wave by itself will not radiate.
test
electron
will
undergo
an
oscillation
Although
which
produce radiation, the particle under consideration
medium
of
changing
dielectric.
a thermal
in vacuum would
is
moving
in
Because of this, a dipole moment is
generated which cancels the effect of the electron motion, so that
radiation
results.
a
no
Alternately it can be said that since the density
and velocity perturbations of the plasma wave are out of phase, no net
oscillatory
current results.
If the test particle has a relativistic
mass, radiation can result from its motion through the plasma
However,
the
plasma
waves are being driven by non-thermal electrons
and have phase velocities close to the speed of light,
relativistic
particles
through the wave.
broadband
will
be
moving
in
spectrum,
so
that
most
phase with, rather than
Furthermore, this relativistic
radiation
wave1 5.
effect
produces
a
which is contrary to the experimental
measurement.
Another mechanism providing a direct conversion of plasma
waves
to electromagnetic radiation is linear mode conversion in the presence
of an inhomogeneity, such as the steady state
has
been
density
gradient.
It
shown by theory and experiment1 6 that such a process occurs
in an unmagnetised plasma, however, the treatment cannot
to the strongly magnetised case.
be
extended
The reason for this is the inability
Page 140
to match both frequency and wavenumber of the two modes,
by
the
linear
and the resonance condition:
2
k
W2
-
=
=
The conversion must occur
near
propagating
the
to
-
k
It is seen that for
to
C2
w = wpe
field,
into
that
waves
electromagnetic
is, into left circularly
W2
2
2
=
c
possible
2
The appropriate cold plasma dispersion is:
polarised waves.
EQ (12)
c2
-7 v
v
parallel
required
The plasma wave can be described by the cold
theory.
plasma dispersion, eq (1),
EQ (11)
as
1 I
a typical
<T2
W(W + WC) I
situation,
wce/wpe = 4, it
satisfy both conditions on the wavenumber.
is
not
The two mode
branches remain separate, and no linear conversion may occur.
It has been assumed that the density gradient is totally in
perpendicular
direction,
that
the
is , Vn - B = 0. This is expected to
hold in steady state since the rapid parallel diffusion will produce a
toroidally
symmetric
density
distribution.
The
treatment also assumes that the scale length of the
much
not
greater
true,
continuously
it
mode
inhomogeneity
than the wavelengths of the modes involved.
is
across
becomes appropriate.
no
longer
the
possible
to
treat
conversion
is
If this is
the wave
modes
inhomogeneity, and a scattering description
Page 141
REFERENCES & FIGURES
1.)
E.N. Kruchina, R.Z. Sagdeev, V.D. Shapiro
Pis'ma Zh. Eksp. Teor. Fiz. 32 443 (1980)
2.)
H.P. Freund, K. Papadopoulos
Phys. Fluids 23 732 (1980)
3.)
M.V. Goldman, G.F. Reiter, D.R. Nicholson
Phys. Fluids 23 388 (1980)
4.
This approximation to the actual cold plasma
dispersion is accurate to better than 5% for
all angles E under the Alcator plasma conditions.
It is better than 1% accurate for angle 0 < 300.
5.)
N.Perepelkin, V.Suprunenko, M.Vasiliev, A.Dikii
Paper 19 III Int. Sym. Toroidal Plasma Confinement
Garching, FDR (1973)
6.)
A.Longinov, N. Perepelkin, V.Suprunenko
Fiz. Plazmy 2 626 (1976)
7.)
D.A. Whelan, R.L. Stenzel
Phys. Rev. Lett. 47 95 (1981)
8.)
Y.Kalinin, D.Lin, V.Ryutov, V.Skoryupin
Zh. Eksp. Teor. Fiz. 55 115 (1968)
9.)
M. Seidl, W. Carr, D. Boyd, R. Jones
Phys. Fluids 19 78 (1976)
10.)
L. Muschietti, K. Appert, J. Vaclavik
Phys. Fluids 24 151 (1981)
11.)
H. Knoepfel, D.A. Spong
Nuclear Fusion 19 785 (1979)
Review paper on runaway electrons in Tokamaks
12.)
G.A.W.Lins, H.-J.Kunze
Phys. Fluids 24 839 (1981)
13.)
T.M. O'Neil, J.H. Winfrey, J.H. Malmberg
Phys. Fluids 14 1204 (1971)
14.)
A.L. Throop, R.R. Parker
Phys. Fluids 22 491 (1979)
15.)
A. Gailitis, V.N. Tsytovich
Zh. Eksp. Teor. Fiz. 46 1726 (1981)
16.)
R.W.Means, L.Muschietti, M.Q.Tran, J.Vaclavik
Phys. Fluids 24 2197 (1981)
Page 142
TWO LOWEST MODE RADIAL PLASMA WAVEFUNCTIONS
1.0
4(r)
a8
6
,4
.2
Radius (cm)
0.
:1.
2
3
4
-.2
-,4
-.6
Fig. (1)
Numerically evaluated solutions for radial langmuir
wavefunctions, based upon a square root parabola
steady state density profile.
Both solutions are
for the case kp =13cm- . The two modes are separated
by 430 MHz for a central plasma frequency of 60 GHz.
5
Page 143
DISPERSION RELATION FOR PARALLEL (0=10) PROPAGATION
kc
3
Electron Cyclotron Waves
WP coso
waves
2
Possible Cerenkov resonance o
1
0
-I
4
=3
p
I
I
I
1
2
3
Fig. (2)
Appleton-Hartree dispersion for cold plasma.
The two nearly electrostatic waves which may
be resonant with non-thermal electrons occur
P
Page 144
SEC 10.0
FLUCTUATING EMISSION: NON-LINEAR EFFECTS
The resonant three wave interaction is a non-linear coupling
which
modes
allows
oscillating waves.
emission
is
the
Pertinent
the
of
transfer
interaction
to
energy
a discussion
involving
The
former
wave
is
of
three
the
fluctuating
frequency
as
two
of
the
expected to exist by the linear
coupling to runaway electrons, and is described in terms of axial
radial
mode
numbers
per
the previous section.
wave is known to exist, since it is directly
having
exited the plasma.
feature.
of
and
by
the
and
The electromagnetic
measured,
albeit
after
The identity of the third wave is unknown,
however, its frequency and
conditions,
wp
magnetised plasma waves and
electromagnetic radiation near the plasma
constituents.
between
of
wavenumber
measured
are
spectral
restricted
width
by
resonance
of the radiation
Because of the numerous experimental and theoretical studies
three wave interactions involving acoustic fluctuations, this mode
is of particular interest as a possible third wave.
SEC 10.1
RESONANCE CONDITIONS
The process under consideration can be schematically represented
by the following expression:
[ (U, kl J+ [ ws, ks ]
EQ (1)
[ w1 , kl ]
where
subscripts
the
electromagnetic
--
i
l,t
(s,
ks
refer
wt,
t kt ]
] + E wt,
to
the
kt
J
plasma
(langmuir)
(transverse) waves, and s to the scattering wave.
restriction has yet been placed on the scattering wave, which in
can
and
No
fact
be a particular fourier component of a steady state perturbation;
Page 14!3
i.e ws = 0.
that
the
The resonance condition is. the fairly obvious requirement
sum
of
the
constituent
waves
and the product wave must
maintain a nearly constant phase in both time and space:
Wt
kl + ks 5 kt
W1 5 WS + Wt
ki 5 ks + kt
l + Ws
EQ (2)
Because the measured frequency of
equal
the
emitted
radiation
is
nearly
to that assumed for the plasma wave, the third wave must have a
relatively
example,
small
the
frequency
scattering
in
comparison.
process
involving
This
two
eliminates,
for
plasma waves.
The
resonance condition thus restricts the scattering wave:
EQ (3)
WtWl
While the close proximity of the transverse
frequencies
limits
the
is
that
the
allows
for
particular frequency.
depends
strongly
upon
the
allowed
wavenumbers.
within
a variety
the
plasma,
where
The
the
The refractive index near the plasma
the
strong
of transverse wavenumbers for any
frequency
wave polarisation, and upon the angle of
emission with respect to the magnetic field [see fig.(6)
As
wave
resonance condition applies to the waves at the
location of scattering, i.e.
dispersion
langmuir
magnitude of ws, this consideration alone is
far less restrictive when applied to
reason
and
of
sec(6)].
these quantities cannot be measured, the appropriate wavenumber of
the emitted transverse wave is uncertain.
density
is
not
known
well
enough
to
The
determine
transverse wave frequency lies above or below
the
central
if
plasma
the measured
associated
plasma
Page 146
frequency.
However, the separation of the lowest order langmuir mode
and the central plasma frequency is 430 MHz for a typical
Thus,
if
the
situation.
scattering wave has a frequency above this difference,
the condition wt > wp can be satisfied everywhere in the plasma.
a situation
ws = 1 GHz.
both
could
result
scattering
by acoustic waves where
The case wt/wp > 1 allows the transverse wave
polarisations
for
all emission angles.
alternate situation, wt <
centre.
from
This
can
Such
access
to
Since ws is small, the
p, can only be satisfied
near
the
plasma
be seen in fig.(1) of sec(9) where the inflection
point (change in sign of curvature) of the radial langmuir mode occurs
at
the
point
where
wl = wp.
Thus the condition wt < wp can only be
satisfied for a region within a few cm.
appropriate
to
characteristic
maintain
localise
dimension
less
emission
than
centre.
process
to
It
is
not
a volume
a few wavelengths
and
of
still
of
the wave as being everywhere a mode of a
continuously varying medium.
Even if the transverse wave frequency is
slightly
a description
the
of the
less than the central plasma frequency, in the plasma volume
outside of the region of emission the dielectric is still described by
the condition wt > wpThe inability to experimentally discern a preferred polarisation
for
the
transverse
the emitted wave.
effects
of
polarisation.
frequencies
wall
wave does not indicate a lack of polarisation of
As was the case for the
reflections
diffraction
This effect is expected to
when
the
wavelengths
dimensions of the flange and
consideration
and
of
the
are
be
of
gyroharmonic,
will
more
comparable
bellow
internal
polarisation
second
the
scramble
severe
to
the
lower
the physical
assemblies.
non-linear
at
the
Although
oscillating
Page 147
current
favours
justification
radiation
into
the
ordinary
mode,
there
is
no
for disallowing the extraordinary mode, particularly if
the waves are emitted with small angle to the magnetic field.
SEC 10.2
DISPERSION RELATION
Fig.(1) shows the wavenumbers as a function of angle for the two
polarisations
above
the
of
transverse
plasma
dispersion
waves
frequency.
relation
is
the
emitted
The
at a frequency slightly
Appleton-Hartree
cold
source of these curves.
In addition, a
region of k-space is indicated where the langmuir waves are
as
plasma
expected,
determined by their required resonance with non-thermal electrons.
These electrostatic modes are also solutions to
the
equation
refractive index.
for
the
special
case
of
a large
Appleton-Hartree
satisfy the resonance condition, the scattering wave must connect
wavevectors
of
the
langmuir
and
transverse
possible scattering waves are indicated.
fairly
restricted
in
direction,
so
waves.
Fig.(1)
that
shows
To
the
Several such
that
ks
is
the scattering wave must
propagate mostly along the magnetic field, and in a direction opposite
to
the
plasma
waves.
This
means that a purely perpendicular wave
cannot satisfy the resonance condition.
unrestricted.
For
values
The magnitude of ks is
of ks ranging from near 0 to greater than
2.5 wp/c a possible
scattering
solution
allowed
wavenumber
apply
scattering
quite
exists.
only
to
The
emission
extraordinary mode, ks must satisfy .68 wp/c<ks<2.2 wp/c in
produce radiation in the ordinary mode.
extrema
into
order
of
the
to
Page 148
The escape condition
can
emission
Transverse
be
also
discussed
applied
emitted
waves
to
close
in
conjunction
to
field
the
Cerenkov
transverse waves.
possible
the
with
direction
in
the
extraordinary polarisation will be internally reflected and trapped in
the plasma centre.
occurs
angles
for
shown
For the
conditions
less
than .14 rad.
propagates along the magnetic field,
in
trapping
fig.(1),
Thus, if the scattering wave
it
is
possible
for
both
langmuir and transverse waves to be trapped in the same volume.
the
W.hile
this trapping may provide for strong interaction between the waves, it
forbids
of emitted transverse waves unless some disturbing
detection
allows
perturbation (MHD activity?)
course,
if
the
escape
of
radiation is emitted in the ordinary polarisation no
the
such trapping may occur, regardless of the emission angle.
the
Of
radiation.
Because of
small perpendicular scale of the langmuir modes, it can be argued
that coupling is more efficient for transverse waves
nearly
which
propagate
parallel to the magnetic field direction, this because in such
a case the interaction volume of the waves is maximised.
Fluctuations which
electrostatic
frequencies
ion
near
considerably
the
by
frequencies
and
plasma
and
and
include
waves,
drift
with
waves
at
Measurements of low frequency density
perpendicular
-
fields
magnetosonic
frequency,
scattering
w = 300
magnetic
400 kHz.
indicate
a broad
spectrum
wavenumbers;
These
of
typically
fluctuations
are
during most discharges, showing no correlation with plasma
frequency emission.
plasma
ion
across
waves
frequency.
laser
ki = 0 - 25 cm-1,
continuous
cyclotron
lower
fluctuationsl
propagate
While the fluctuation level
is
highest
at
the
edge, a tentatively identifies secondary maximum occurs at the
I
Page 149
It
centre, with typical magnitude of n/n = 3% .
is
that
suspected
these fluctuations result from drift waves, though this identification
is very uncertain 2 . Regardless of their identity
can
these
fluctuations
only satisfy the resonance condition if the parallel component of
the wavevector is sufficiently large.
been
placed
on
this
parallel
An upper bound
component,
that
6 cm-1
of
is,
the
wavevectors are preferentially perpendicular, and cannot
resonance
condition.
typical
satisfy
the
However, the range of measured values of k1 and
kp allow for possible
mentioned
has
solutions.
These
possible
drift
waves
are
since they constitute the only measured fluctuations in the
Alcator Tokamak.
It seems that they are unlikely candidates
for
the
non-linear emission process.
NON-LINEAR CURRENT
SEC 10.3
When all three waves are plasmons
the
resonance
condition
is
equivalent to a statement of conservation of energy and momentum 3 . In
the case considered here, the necessity
follows
directly
source term
distribution,
three
wave
oscillating
for
from
the
of
the
resonance
condition
mechanics of the radiation process.
electromagnetic
waves
is
an
oscillating
current
which cannot be produced by the plasma wave alone.
interaction
allows
the
generation
of
a
The
The
non-linear
current, which results from the mixing of the density and
velocity perturbations from the coupled waves:
EQ (4)~~
Enl(r,t)
=
where only the electron
-e nl(r,t) vs(r,t)
fluctuations
need
e vl(r,t) ns(r,t)
-
be
considered.
If
the
scattering wave has a small phase velocity in comparison to the plasma
Page 150
wave (such as an acoustic wave), or if
perturbation,
the
it
is a
steady
state
density
first term in eq (4) is negligible compared to the
second, so that the current density becomes:
-e w
n1 ns
i0w1 + ws] t}
exp {i[k1 + ks]r
EQ (
Jnl(r,t)
=
JO exp i[kj r - wj t)
The radiation field resulting from the non-linear current distribution
is found by solving for the vector potential:
EQ (6) A(r,t)
=
iC-r - w
c
-exp
cr
t]
exp i[k
ik
-
c
n
r' d dr'
The oscillating coefficient determines the frequency and wavenumber of
the
radiated
wave,
i.e.;
wt = wj = wl + ws and kt = wj/c, and the
integral insures that kt = kj = kl + ks
These
.
are
precisely
the
resonance conditions of eq (2).
The degree to which the resonance condition must be satisfied is
determined
by
the
extent
of
the integral eq (6), which in turn is
controlled by the degree of coherence of
waves.
wave
For
packet
Akj = 2w/L,
The
measured
a&t/wt < 1%,
example,
of
plasma
extent
L,
a
spread
scattering
in
wavenumber,
thus radiated frequency, Aot = wt* Akj/kt, results.
upper
limit
of
the
radiated
frequency
bandwidth,
indicates a minimum wave packet extent, L = 50 cm, which
is comparable to the toroidal circumference, so
representation
and
if the current density is treated as a spatial
parallel
and
the
becomes unacceptable.
mode description of
sec (9) is
that
a wave
packet
For this reason the plasma wave
favoured.
The
narrowness
of
the
bandwidth of radiated waves also restricts the frequency extent of the
Page 151
scattering wave by a similar argument.
wave
is
known
to
have
However, since the
scattering
a low frequency anyway, applying a 400 MHz
bandwidth for example, does not require an especially coherent wave.
SEC 10.4
ABSORPTION
When the size of the interaction region becomes large enough
that
the
radiated wave amplitude is comparable to that of either the
plasma or scattering wave, it is necessary
processes,
so
whereby
energy
is
returned to the plasma modes.
removed
This
to
consider
the
inverse
from the transverse wave and
situation
is
analogous
to
the
treatment of emission and absorption of cyclotron radiation, in that a
transport equation results.
When wave-particle interactions can be ignored,
the
transverse
mode can be described by4:
d
d
+ Vt (group) L
EQ (7)
=
Nt
Atls
where Na is the number density of mode
energy
density
by
Wa = waNa.
It
NlNs - NtNl - NtNs
a, related
is
assumed
to
in
the
spectral
eq (7) that the
resonance condition is satisfied by the three waves, so that frequency
and
wavenumber
indices
are ignored.
the evolution of the other waves.
upon
the
identity
of
the
Two similar equations describe
The interaction kernel Atls depends
scattering
wave,
relationship between the non-linear current and the
and
of the plasma and scattering waves.
expresses
energy
the
densities
Page 152
By eq (7) it is evident that a quasi-equilibrium
solution
can
If the density of scattering waves is small, so that N1 >> Ns,
exist.
the last term in eq (7) is ignorable, so that for steady state:
EQ (8)
Nt = Ns
+
Wt =
This has been considered in the
Ws
context
of
explaining
dominant
wp
emission by the scattering of plasma waves from a thermal level of ion
acoustic fluctuations 5, the intensity
being
determined
by
the
of
equipartition
the
acoustic
The result is an
theorem.
intensity of emitted radiation which is independent of
plasma
wave
fluctuations
the
level
of
energy, and is described by the effective temperature of
the acoustic fluctuations:
EQ (9)
it
=
Ibb(Te)
; Te
w
Ti] 1 /2
ET
For the specific case of ion acoustic waves, an intensity
than
40
times
the
thermal
level
results,
which
representative of typical fluctuating wp intensities.
the
damping
(second)
of
greater
is
fairly
Provided
that
term in eq (7) is large enough, the results of
eq (9) will be true regardless of the type of
scattering
wave
used.
Accordingly, other branches of the ion spectrum will produce a similar
result.
also
If the damping term is small, then
presumably
will
be
equal
to
the
or
effective
greater
temperature, and the frequency of the scattering
the
emission
term
will
be small, so that a negligible amount of radiation results.
radiated intensity will be determined by
which
the
scattering
than
wave.
The
temperature,
the
plasma
From
eq (3)
wave is known to have a low frequency compared to wp,
Page 153
so the predicted emission intensity will
thermal
Drift
level.
be
much
than
the
waves, because of their lower frequency, will
The
generate an even greater effective temperature.
waves
greater
case
of
drift
in fact cannot be treated by eq (8) because the drift waves are
saturated modes.
The saturation mechanism, whatever it may
be,
must
be included in eq (7),- so that the process will no longer involve only
three waves.
RELATION TO EXPERIMENT
SEC 10.5
A
consequence
intensity.
Quite
of
eq (9) is
often
the
a constant
observed
described as having a nearly constant
expected
fluctuating
level.
This
emission
emission can be
is
particularly
true with regard to radiation which accompanies MHD activity, or which
occurs in
examples
periodic
of
this.
bursts;
figs. (3),
(The discerning
(5),
(7) of
experimentalist
sec (7) are
refrains from
citing those cases where constant intensity results from saturation of
detector
and/or
electronics.) According to the scattering model, the
emission fluctuations result from like oscillations in the
current.
This
requires
either
thermal
level,
hard
x-ray
an
then only fluctuations of the plasma waves
can produce the observed emission
correlated
the density of plasma or scattering
If the scattering waves are populated to
waves to change rapidly.
effective
non-linear
emission,
behaviour.
the
Because
oscillation
in
there
plasma
is
no
wave
amplitude probably does not result from the wholesale loss of resonant
runaway particles, but rather some subtle alteration of their velocity
distribution which destroys the positive growth of the wave.
Page 154
The correlated behaviour
activity
m = 1
with
emission
fluctuating
result either from a direct involvement of the MHD mode
can
in the radiation process, i.e.
indirectly,
of
for
example
by
the m = 1 is the scattering
wave,
or
trapping of the transverse and langmuir
waves in the local density maxima.
If the m = 1imode is regarded as
perturbation
of
a few
percent,
a nearly
with
stationary
density
infinite parallel extent and
perpendicular scale length of order .5 cm, reference to fig.(1)
that
a ksJ. = 3 cmnC
shows
could satisfy the wave matching condition only if
a similar or larger ksl were present.
However, an
extent implies a 0 parallel wavevector component.
infinite
parallel
The MHD mode cannot
satisfy the resonance condition because it is a purely
perpendicular
perturbation.
If the scattering wave is independent
radiation
can
correlation
untrapping
be
with
of
imagined
state
m = 1 activity,
as occurring continuously.
m = 1 rotation
may
result
from
The observed
trapping
density
perturbation.
could
and
If
the
density profile is peaked enough on axis, the radiation
may escape only when a large rotating mode flattens the profile.
then
the
the emitted transverse wave in the region of increased
refractive index associated with the
steady
of
explain
the
This
result that during steady state emission is
only observed in phase with large m = 1 oscillations which precede
follow sawtooth disruptions.
or
Unfortunately, the absolute phase of the
rotating mode relative to the emission is unknown.
vage it0
SEC 10.6
1.)
REFERENCES & FIGURES
R.L. Watterson et.al.
Bull. Am. Phys. Soc. 26
885 (1981)
2.)
C.M. Surko, R.E. Slusher
Phys. Fluids 23 2425 (1980)
3.)
R.C. Davidson
Methods in Non-Linear Plasma Theory
Academic Press, New York (1972) p.117
4.)
V.N. Tsytovich
Nonlinear Effects in Plasmas
Plenum, New York (1970) p.9 0
5.)
I.H. Hutchinson, Kim Molvig, S.Y. Yuen
Phys. Rev. Lett. 40 1091 (1978)
Page 156
SCATTERING GEOMETRY
OF WAVEVECTORS
LANGMUIR WAVES
Kj
POSSIBLE
SCATTERING
WAVES
KL[WP}
&-
U
_____________________
1.0
TRANSVERSE WAVES
EXTRAORDINARY
ORDINARY
-1,0
_
Wc
t
3CO
Fig.(1)
Because of only approximate knowledge of the
plasma frequency (central density) the choice of
Wt/wp
is somewhat
arbitrary.
Dashed
line
indicates
trapped transverse waves.
Langmuir
waves are assumed to be resonant with runaway
electrons; ve Z c, v, >> v,
- 1.02
Fage i:>/
CONCLUSION
SEC 11
The
diagnostic
utility
of
of
submillimetre
plasma
emission
measurements
as
a
condition should be manifest from the previous
sections.
As a technique for determination of
evolution
of
temporal
and
spatial
electron temperature the measurement of emission at the
optically thick- second gyroharmonic is demonstrated to be particularly
well
suited.
The ability to describe the temperature profile by the
simple Gaussian expression should assist those persons
modelling
transport
properties
in
interested
in
Tokamak plasmas, or to anyone in
need of any analytic expression for the spatial temperature variation.
That the temperature and profile width are consistent with a classical
resistivity model indicates a relative small
and possible neo-classical effects.
influence
by
anomalous
In addition, the repeatability of
the scaling of profile width with limiter q allows the
prediction
of
results in new parameter ranges, such as high current and field.
Radiation at the third gyroharmonic has been shown to be
as
a method
for
self
consistent calibration of the entire system.
This is not a trivial matter, as calibration
remains
difficult
for
problem.
As
a possible
suggested that a meticulous attempt to
undertaken.
useful
topic
calibrate
a serious
future
the
and
work, it is
diagnostic
be
This would probably require the development of an intense
broadband reference source, and possibly the resolution
of
the
wall
reflection, polarisation scrambling problem.
Sawtooth measurements have been made which do
ambiguities
emission.
inherent
not
involve
the
in chord-average techniques, such as soft x-ray
These results together with profile scaling suggest a range
Page 158
of
central current density such that qO fluctuates between .8 and 1.0
during the
peculiar
sawtooth,
with
time
average
value
qo = .85.
of
The
behaviour of sawteeth in helium plasma should be of interest
to persons attempting an explanation of the sawtooth phenomena.
The observation of non-thermal
used
as
an
indicator
of
the
emission
of
the
from
equilibrium
extremely
Furthermore, the
emission
events.
others
is
be
uncertain
a subtle
In fact it would appear that only
electron
distribution
fluctuation emission occurs in such a narrow
for
can
process, at least in the case of fluctuation wpe
radiation, is uncertain.
perturbation
emission
a non-equilibrium electron distribution.
However, the degree of the deviation
since
submillimetre
accurate
interesting
density
is required.
bandwidth,
measurement
correlation
behaviour
Since the
a technique
suggests
of
itself.
fluctuating
with m = 1 modes may provide new information regarding these
It is hoped that in
will
be
stimulated
reporting
to
this
duplicate
novel
radiation
event
the measurements on other
devices, and perhaps to offer explanations regarding the origin of the
process.
Page ib9
OPTICAL PATH
APX 1
construction
Optical access to the Alcator plasma is limited by
to
six
diagnostic
Each port offers vertical and horizontal
ports.
views of the plasma through narrow gaps in the toroidal
called keyways.
field
The horizontal view is preferable for the measurement
of cyclotron radiation since it allows sampling of the full
generated
spectrum
magnet
by
the magnetic field variation.
frequency
Because of the
narrowness of the keyway, only radiation emitted
within
the
When viewed from the
perpendicular
can
be
observed
directly.
outside this keyway is the limiting constriction that
observable
plasma
area.
This
.10 rads
determines
of
front surface mirrors as shown in fig.(2).
mirror images the detector
keyway.
the
situation is illustrated in fig.(1).
Having passed from the torus, radiation is brought to the detector
a series
of
aperture
onto
the
inside
by
A spherical
edge
of
the
Since the wavelengths involved are not ignorable compared to
the physical dimension of the optics employed, every
mirror, window,
and aperture is a source of diffraction.
The effective aperture diameter of the Michelson
is
5 cm.
Plane radiation incident on this area will be focussed into
the detector.
be
ignored,
In the very short wavelength limit when diffraction can
this 5 cm diameter would approximately equal the area of
the plasma viewed, since
between
the
plasma
interferometer.
second
and
the
the
spherical
entrance
For a-wavelength
gyrofrequency
of
mirror
is
aperture
.67 cm,
located
of
the
corresponding
halfway
Michelson
to
the
at 80 KGauss, diffraction effects will increase
this spot size slightly.
incident
interferometer
Tracing backwards from
the
detector,
rays
upon the spherical mirror will be nearly plane waves, so the
Page 160
resulting
diffraction
will
be
the
usual
circular
aperture
Airy
pattern.
dP .
a = 1/2 ka sine
2 Jj(a)
dn
a
k = 95 cm-1
EQ (1)
a = mirror diameter-
15 cm
Jj= first order Bessel function
The image upon the 10 cm vacuum window
determined
by
will
have
a maximum
extent
the imaged aperture size plus the diffraction from the
spherical mirror.
At 1.5 m from the mirror the first zero of the Airy
pattern will be determined by:
Ar= Lemin = .4 cm
L = 1.5 m
EQ (2)
emin = 2.7 x 10-3 rad (first zero of Bessel function)
The spot size is 7.5 cm, so no diffraction effect from the
expected.
The
window
is
keyway is 4 cm wide by 30 cm tall, so it will produce
the characteristic diffraction pattern of a vertical slit:
dP
sina
d12
a
2
a = 2 ka sine
EQ (3)
k = 95 cm-1
a = slit width = 4 cm
This will effect only the horizontal direction.
a horizontal
diffraction
By analogy to
eq (2)
size is found at the plasma centre a 20 cm
distance from the 'slit' exit.
Ar = Lemin = .16 cm
EQ (4)
emin = sin- (7r/ka) = 8 x 10-3
L = 20 cm
Page ibI
This distance is added in the
dimensional
spot
size
separate horizontal and
at
horizontal
the
direction
only.
The
two
plasma centre is found by adding the
vertical
diffraction
effects
the
to
5 cm
aperture image:
Ar = Aemin(mirror)x3m = .81 cm
Vertical:
EQ (5)
Horizontal:
Ar e Aemin(mirror)x3m + Aemin(slit)x2Qcm =.97cm
The spot size is an oval with first minimum defined by 6.6 cm vertical
and 6.9 cm horizontal diameters.
To determine the temperature profile resolution that
size
mandates
it
is
toroidally symmetric.
using
the
assumed
that
determined
magnetic
This
ripple).
is
unmeasured and assumed small).
effect
by
the
is
will
straight
doppler
ignorable
volumes.
spot
any
(or
by
resolution
The effect upon
the measured central temperature can be ignored since the
correct
(and
When viewing a
spatial
produce a 5% overestimate of the profile width.
will
width
The axial (in-out) extent is fixed
20 GHz Michelson frequency resolution, &r = 3 cm.
process
forward;
In the horizontal
totally
temperature profile of 20 cm diameter, the finite
spot
is vertically and
criterion, Ar = 3.3 cm.
direction the resolution is
the
profile
The vertical resolution
Rayleigh-Jeans
field
the
this
calibration
error resulting from averages over finite
In the case of the Fabry-Perot, sec (5), both
the
aperture
size and the frequency resolution were much better, resulting in
a more accurate measurement.
Page 162
REFLECTIONS & POLARISATION
APX 2
None
of
optics
the
employed
can
from
result
may
will
polarisations
mirror
remain
distinct
reflections,
in
this
but
orthogonal
process.
Reflections
within the keyway will retain their polarisation
identity
extraordinary (ordinary) mode polarisation vector.
mixing
mentioned
effect
the
The
polarisation
in sec (1) must occur elsewhere if it is to account
In an effort to demonstrate
for the observed ordinary mode radiation.
this
since
surfaces are planes parallel (perpendicular) to the
metal
reflective
the
Rotation of the electric
polarisation state of the emitted radiation.
vector
alter
significantly
a reflective mirror was affixed to the back wall of the
vacuum chamber [see fig.(1)].
This mirror was designed to
vacuum
so that no randomly reflected radiation
window
onto
could be observed.
by
this
state.
itself,
Reflections within the keyway are
addition,
but
The effective
improvement
in
of
presumably
this
was
reflective
had
no
prohibited
will not effect the polarisation
dump
was
only
a modest
A further
refinement
involved
of a glass liner on the inside of the port extension.
intended
to
absorb
polarisation
effect
the
10%
the degree of polarisation,.with a significant amount
of ordinary mode still present.
placement
beam
not
image
upon
the
any
incident
radiation,
thus
mixing in the port extension.
measured
degree
of
the
This
preventing
This absorber
polarisation.
In
retrospect, neither experiment provided a severe enough test to either
confirm or disprove the polarisation scrambling model.
The beam
dump
was not large enough, and unfortunately could not be aligned properly,
while the glass covered only a small
surface.
portion
of
the
exposed
metal
I
Page 163
THE MICHELSON INTERFEROMETER
APX 3
The plasma emits a broad spectrum
the
intensity
varying
in
time
of
during
radiation,
submillimetre
a tokamak
discharge.
function of the Michelson interferometer system is to measure as
of
spectrum
this
possible.
include
the
For
the
as
Alcator
first
three
Tokamak
beyond this since intense radiation
The
encountered.
the
cyclotron
mechanics
of
measured
harmonics,
higher
at
much
the best temporal resolution
with
necessary,
The
spectrum
should
but need not extend
harmonics
is
rarely
the interferometer inversely relate
time and frequency resolution, so that improvement of one comes at the
expense of the other.
important.
Unless
fluctuation
times,
Of the two, the spectral resolution is the more
beyond
improvement
such
as
sawteeth,
reason for the pursuit of higher temporal
emission
characteristic
can be achieved, there is no
resolution.
However,
the
spectral resolution should allow determination of temperature profiles
with reasonable spatial resolution since these profiles are the raison
d'etre of the interferometer.
In
simplest
form
the
Michelson
splitter with two reflective beam paths.
interferometerl
is
Radiation from the two paths
recombines constructively or destructively depending upon
difference
of
the
two
interferogram is generated.
recover
the
radiation
The
interferogram
spectrum.
In
the
can
case
be
transmits
or
reflects
of
couple
equal
intensities
a wire
according to polarisation.
into
to
the polarising
wire mesh selects one polarisation and the beam splitter
to
length
analysed
Michelson interferometer, Fig.(3), the beam splitter is
which
the
varying this path difference an
By
paths.
a beam
the two beam paths.
mesh
An initial
is
arranged
If the path
Page 164
difference
is
polarisation,
mesh.
zero,
the
beams
recombine
to
form
the
and are rejected from the interferometer by the initial
If the path difference is one half wavelength,
beam
is rotated
reflected
by
conventional
originiA
the
recombined
w/2 with respect to the original polarisation and is
the
initial
device,
mesh
into
the
detector.
Unlike
the
the polarising interferometer detects a minimum
at the zero path difference.
The length
vibrator
of
attached
one
to
path
is
a reflecting
varied
by
mirror.
motor is driven at 35 Hz to produce a swing of
path
length
can
Two
electromechanical
Typically the vibratir
+-5
mm.
The
second
be varied manually and is adjusted so that the zero
path difference state occurs near the extreme
excursion.
an
complete
interferograms
of
the
result
moving
mirror
from one vibration
period.
The position of the moving mirror is measured
HeNe
laser,
so
by
auxiliary
that the path difference is known at all times.
data is sampled at equal intervals of path difference,
of
an
which
The
because
the sinusoidal mirror drive does not correspond to equal intervals
of time.
The sampling interval is variable, but is
usually
126.5 um
Data sampling is controlled by a Z-80 micro-processor.
Input to
of path difference
the
Z-80
includes
the
analog interferometer signal, a mirror drive
reference signal, and the output of the auxiliary HeNe interferometer.
The number of complete vibration periods to be sampled is preselected;
typically about 12, which corresponds to
plasma
discharge.
The
entire
sampling
the
full
process
duration
of
the
is initiated by a
Page 165
trigger from the
sequencer.
Alcator
After
sampling
the
data
is
transferred to a PDP 11-55 computer for treatment.
APX 4
FOURIER TRANSFORM INTERFEROMETRY
The output of the
upon
the
idealised
Michelson
interferometer
depends
intensity I(w) and the path difference of the two
spectral
arms x:
J(x)
EQ (1)
I(w) cos(wx/c) dw
1
= J-
0
J
is the average level of the interferogram J(x) and is equal to one
half
the
total
intensity of all frequencies.
Experimentally U
the non-fluctuating baseline of the interferogram.
is
I(w) is related to
0(x) by the inverse transformation 2:
I(W)
EQ (2)
f
4
=
cos(wx/c){J(x)-J} dx
0
Provided that J(x) is measured for a semi-infinite
entire
spectrum
can
be
recovered.
measured for a finite range of path
In
range
practice
difference,
of
x, the
J(x) can only be
hence
limiting
the
accuracy of the deduced spectra.
A standard
technique
for
treating
extent is the process of apodisation.
interferograms
of
According to this procedure the
inverse transformation is:
L
EQ (3)
fA(xL) cos(wx/c){J(x)-J} dx
1(w) =
car
0
finite
Page 166
the true spectrum with an instrumental line shape which
finite
correspond
to
different
function
is
fairly
functions
apodisation
A
shapes.
line
standard,
the
reflects
Different apodisation
capabilities of the system.
resolution
the
called
is
A(x,L)
Apodisation may be considered a convolution of
function.
apodisation
and
difference
path
where L is the maximum
and
triangular
been
has
employed
throughout this thesis:
L
EQ (4)
where L may
be
chosen
equal
According
difference.
to
or
to
less
eq (3),
the
the
than
spectrum
maximum
produced
triangularly apodised interferogram resulting from a single
path
by
a
frequency
source, IO(wO) is:
I(w0 ) = JO L sinc 2 [(W-W 0 )L/c]
2wc
EQ (5)
By way of comparison, the unapodised spectrum is a sinc
sinc 2
function.
condition that
Applying
two
the
frequencies
Rayleigh
of
criterion
Aw
separation
to
be
rather
than
eq (5), the
resolved
is
expressed by:
Aw =
EQ (6)
Thus the resolution is inversely
difference,
displacement.
difference)
which
in
turn
2wc
L
proportional
to
the
maximum
path
is the amplitude of the vibrating mirror
For example, a 1 cm
total
mirror
travel
provides a conceptual 15 GHz resolution.
(2 cm
path
Since data from
Page i)/
the entire mirror oscillation may not be
usually
be
less.
For
used,
the
will
resolution
data presented in this thesis, the frequency
resolution is typically 20 GHz.
The interferogram is actually a series of sampled points, rather
a
than
continuous
curve.
Hence the integral over x in eq (3) is in
reality a summation:
N-1
EQ (7)
1(w) = d
cos(wjAx/c) A(jAx,L) [J(jAx)-T]
j=0
where N is the total number of interferogram points
can
sampled.
Eq (7)
be used to generate spectral intensities I(w) for all frequencies
w, so it must be borne in mind that the resolution is still determined
by
eq (6).
As
the
frequency in eq (7) is increased, at some point
I(w) = I(w-2wc/Ax), and the spectrum will repeats itself.
frequency
that
may
be
recovered
is
determined
by
The maximum
the
sampling
interval:
=c
Wmax =....
EQ (8)
Ax
Thus the standard 126.5 um separation
implies
a maximum
measurable
frequency of 1186 GHz.
Although eq (7) as it stands is a perfectly reasonable means
of
generating
spectra,
derivative
expression
employed.
Called the fast fourier transform, this routine introduces
it
is' rarely
based
upon
used
the
in
the
favoured
process
in
form.
Cooley-Tukey
a significant savings in computation time, and
become
this
for
Instead a
algorithm
this
reason
fourier transform spectroscopy.
is
has
A
Page 168
description of the Cooley-Tukey algorithm can be found elsewbere 3 , and
for
the
sake
A requirement of the
of brevity will not be included.
fast fourier transform routine is that the number of sampled points to
be transformed be an integer power of 2, i.e.:
N =
2 n.
To satisfy this requirement a zero-filling technique is used
generate
additional
interferogram points.
seen
The additional points all
generated
have zero value, and do not effect the
to
This
spectra.
is
in eq (7) where the summation can be extended to larger integers
without effect provided that the
cbrresponding
J(x)-~J
'are zero.
This is precisely what zero-filling accomplishes.
A complication
interferogram
arising
from
the
nature
of
the
is a phase shift that results from not sampling a point
precisely at the zero path difference.
sampled
discrete
For example,
if
the
nearest
point is measured at some displacement 6 from the actual zero
point, eq (7) should be recast as:
N-i
I(W)
=
A(j,L) cos[(jAx-6)w/c] [J(j)-~T]
-4
j=0
N-I
EQ (9)
.Q-4 cos(w6/c)
cT
A(j,L) cos(wjAx/c)
2
{J(j)-~J)
j=0
N-i
+ 4 sin(w6/c)
A(j,L) sin(wjAx/c) [J(j)-~]
j=0
Thus if 6 is known, the spectra can be
recovered
provided
that
the
I
Page 169
and
cosin
transforms
sin
are
Fortunately, this is
both computed.
manifest in the fast fourier transform.
The raw digitised data from a typical
discharge
is
shown
in
fig.(4), where the entire analysis process is schematically followed.
individual
The data is segmented into blocks corresponding to
(1)
oscillation
each
of
direction
backwards
Each block contains two interferograms, one for
periods.
with
mirror
respect
motion.
second
The
interferogram
to the first, and must be reversed.
is
Now all
interferograms may be treated equally.
(2)
The I
level
is
level is subtracted
found
interferogram.
if
the
by
fitting
a
from
each
to
line
interferogram.
the
flat
part
This
slope
The line may have a slight positive or negative
emission
intensity is increasing or decreasing.
the
of
The data is
scaled accordingly, under the assumption that all frequency components
change in the same way.
Subtraction of the base level is necessary to
avoid generation of large zero frequency spectral components when
fourier
transform
is
performed.
the
Because of the inverting effect of
the polarising meshes, the interferogram is next reflected
about
the
zero
path
new zero level.
(3)
Because the sampling of
data
occurs
before
the
difference is encountered, and since it is unlikely that a -sample will
be taken exactly at this time, the
interferogram
must
be
three largest
data
points,
location
approximated.
and
the
center
The interferogram is next apodised.
function is used.
this
point
in
the
A parabolic fit is made to the
identified with the zero path difference.
(4)
of
of
this
parabola
is
6 is determined.
A triangular
apodisation
Because the employed fast fourier transform routine
Page 170
requires 2n data points, the zero filling technique is used, typically
to
create
29
points.
Zero
filling
interferogram to larger values
of
path
involves the extension of the
difference
by
adding
data
points of zero magnitude.
(5)
Since the fourier transform
starting
from
the
zero
requires
to
the
actual
spaced
zero
path
data
point.
point
The
and
small
the
which
is
the
of
other
allowed because of the periodicity of interferograms.
Because the first data point is not the actual
interferogram
point
section
interferogram occurring before this point is reattached at
end,
points
path difference point, the interferogram is
shifted by the spacing between the first
closest
equally
is
zero
not symmetric about this point.
path
point
the
Both cosin and sin
components should result from the fourier transform.
(6)
The fast fourier transform is now computed, and
cosine
components
are
the
combined as indicated in eq (9).
sin
and
Corrections
due to calibration of the detector response, or absorption in
windows
or other effects may be included.
The spectra are now available for further analysis.
or
third
harmonic
may
be
extracted
components may be investigated.
and
studied,
The
second
or non-thermal
For purposes of data storage, the raw
interferometer signal is used, thus allowing for retrospective changes
in calibration etc.
The time required to reprocess the data is
compared to the benefits gained in this caution.
APX 5
DETECTOR
small
Page 171
Regardless of which, if any, interferometer system was used, the
This consisted of an InSb crystal mounted
same detector was employed.
The crystal was
in a liquid helium cooled cryostat.
QMC Ltd 4 . of the United Kingdom.
manufactured
When biased with 100 vamps current,
the detector has a sensitivity of about 1 Volt per mWatt
radiation.
by
of
incident
For typical emission power levels of a few vWatts incident
level
on the detector, a 1000 gain amplifier produced a signal
of
a
Calibration with a modulated klystron source show that the
few volts.
temporal frequency response of the detector is adequate
to
at
least
100 kHz, so that in general, finite resolution limitations will be set
by the data gathering system rather than by the detector itself.
APX 6
CALIBRATION
The problem of calibration is two-fold.
spectral
features
such
as
cyclotron
When
harmonics,
measuring
broad
which may vary in
frequency due to changing magnetic field, the relative sensitivity
of
the detection system to the complete range of radiation frequencies is
required.
is
to
be
Furthermore, if the emission at some
equated
with
a physical
particular
parameter,
such as the second
harmonic intensity with electron temperature, an absolute
the
system
sensitivity is needed.
the easier, since
it
requires
relative spectral shape.
frequency
measure
of
Of these two, the first is by far
only
a radiation
source
of
known
An absolute calibration requires a source of
known intensity, as well as
a
precise
determination
of
the
angles subtended by the source and by the plasma being measured.
solid
Page 172
Relative calibration is performed using a mercury arc lamp which
a
approximates
blackbody
of
thousand
a few
degrees temperature.
Additional sources such as liquid nitrogen cold loads have
used
with similar result.
also
The. problem with this technique is that no
available broadband radiation source begins to approach the
of
plasma.
the
To
overcome
this
intensity
inherent source weakness during
calibration, many interferograms are averaged by the Z-80.
of
resulting spectrum is shown in fig.(5).
the
An example
Bearing in mind that
it
this spectrum represents the emission from a blackbody source,
400 GHz.
is
the system response begins to decrease substantially above
that
seen
been
This rolloff results
mainly
from
the
detector. response,
at high frequencies the interferometer wire meshes will begin
however
to fail as the radiation wavenumber approaches the mesh constant.
problems
at
low frequency evident in fig.(5) result from the Fourier
transform process.
due
to
The noticeable features near 560 and 760
absorption
by
molecular
water
vapour
present
interferometer chamber, a topic of later discussion.
field,
and
the
are
in
the
magnetic
emission
scaling
square of the frequency, and by the increase of temperature
with magnetic field [sec
emission
As the
GHz
consequent cyclotron frequencies, is raised, the decrease
in the detector response is offset by the blackbody
with
The
(1)].
For
example,
the
second
harmonic
will increase as w7 /2 , so that the detected signal intensity
will not degrade until about 700 GHz,
corresponding
to
a field
of
120 KGauss is reached.
Also shown in fig.(5) is a calibration spectrum
QMC
detector
before
its modification.
made
with
the
The spectral fluctuations at
400 GHz result from channelling in a 1 mm thick quartz disc
on
which
Page iij
the InSb crystal was mounted.
Modification involved wedging this disc
so the faces were no longer parallel, thus preventing channelling.
can
be
For the
this simple expedient eliminated the problem.
seen,
As
same reason the window on the detector cryostat is also wedged.
By dividing the spectrum of fig.(5)
then. used by the
spectra
from
computer
cyclotron
the
Not
plasma.
the
assumed
to
routines
correct
this
by
considered
blackbody
This curve is
curve is obtained.
sensitivity
a relative
spectrum,
by
measured
technique
is
absorption by the .25" fused quartz vacuum window, an important factor
at
higher
frequencies.
This
effect
need
dealing with a small frequency range, such
not
as
be considered when
the
second
or
third
harmonic alone, because the attenuation coefficient of fused quartz is
a weak function of frequency.
of
sec
However it has been included in fig.(1)
(4) where emission at the two harmonics is compared, the net
effect being a 15% increase in the relative
harmonic
window
emission.
will
become
Above
800 GHz
particularly
intensity
absorption
severe,
and
of
the
third
by
the fused quartz
a
crystal
quartz
replacement window should be used.
It has been found that the system response is different for
two
polarisations.
This is due in part to the differing condition of
the meshes used to select the incident polarisation.
fig.(2)
of
the
Consequently
in
sec (1) a separate calibration was used for each mode to
account for this difference.
Absolute calibration is performed by scaling the optically thick
second
harmonic
extraordinary
mode
measure of the electron temperature.
emission
with
an
The temperature as
independent
measured
by
Page 174
soft
x-ray pulse height analysis is the usual basis for this scaling,
although the results of sec (4) have been used succesfully in a pinch.
The
calibration
factor
arrived
at in this fashion remains constant
with plasma parameters such as density and current, so
that
once
agreement
same
electron
the
temperature.
the
two
techniques
always
generate
the
in
The calibration factor does change when modifications of
optical system due to deliberate or accidental realignment occur.
However, if the system
is
left
undisturbed,
the
calibration
will
remain in effect indefinitely.
APX 7
ATMOSPHERIC WATER VAPOUR ABSORPTION
The absorption lines seen in
from
rotational
transitions
transitions at 557 and 752 GHz
intense
lines
at
380
the
of
are
calibration
molecular
spectrum
water
particularly
to
be
ignored.
cyclotron harmonics
problem
exists.
evident,
but
and 450 GHz can also be detected.
less
Five other
sufficiently
As long as the magnetic field is such that the
avoid
At
The
vapour.
lines occur in the interval from 100 to 900 GHz, but are
weak
result
these
certain
absorption
fields,
lines,
no
significant
particularly above 90 KGauss,
atmospheric water vapour alters
the
dramatically.
to alleviate this problem, the entire
optical
system
plexiglass
ruining
attempt
including
and
the
this
Fig.(6)
temperature
is
spectra
interferometer
volume
was
A noticeable' improvement
remained.
the
an
chamber,
nitrogen gas.
still
In
measured
an
profile,
example
the
was
and
profiles
enclosed
in
a
continuously purged with
resulted,
but
distortions
of such an effect.
absorption
feature
While
does
demonstrate the spatial resolution of the system, since the line width
of the feature is due entirely to the finite instrumental
resolution.
Page 175
It is useless to hope for a dry day, or to move the detector closer to
the vacuum chamber to avoid this problem.
absorbent,
The 557 GHz is so
that under typical conditions a 1 cm air path will produce
nearly a 5% reduction in signal.
APX 8
strongly
REFERENCES & FIGURES
1.)
Principles of Optics
M. Born, E. Wolf
Pergamon Press New York
(1964)
2.)
Interferometry
W. Steel
Cambridge University Press, Cambridge (1967)
Contains pertinent discussions of transformations
apodisation, phase shifts, zero-filling .........
3.)
R. Bell
Introductory Fourier Transform Spectroscopy
Academic Press
(1972)
Contains a graphic demonstration of the Cooley-Tukey
algorithm for a case of N=8.
4.)
QMC Instruments Ltd.
229 Mile End Road
London El 4AA
Trivia question: What does QMC stand for?
11
Page 176
OPTICAL PATH TOKAMAK INTERIOR
Quartz Vacuum Windnw
Glass Absorb er
Port Extension
Scale 10 cm
Keyway
Bellows
Plasma Chamber
Concave Mirror
Vertical
through
view
the
Fig. (1)
of torus section slit
mid-plane.
Internal
structure is entirely stainless steel
except for absorber and window.
Page 177
OPTICAL PATH
/
2
PLEXIGLASS
ENCLOSURE
TOKAMAK
VACUUM
WINDOW
3 M RADIUS
SPHERICAL MIRROR
FLAT MIRRORS
F-
SCALE IM
-H
Fig.(2)
All mirrors are front surface aluminium coated
reflectors. The enclosure is continuously purged
with nitrogen gas, when necessary, to minimise
radiation absorption by water vapour.
INTER :EROMETER
ENTRANCE
APERTURE
Pave 178
THE MICHELSON INTERFEROMETER
SCALE 10cm
POLARISER
STATIONARY MIRROR
X1
CIDENT RADIATION
X2-
FOCUSSING LENS
BEAM SPLITTER
OSCILLATING MIRROR
Path Difference of Two
Interferometer legs = 2(X1- X2 )
DETECTOR
Fig. (3)
Both beam splitter and polariser are wire meshes.
Mirrors are flat front-surface aluminised.
The
initial polariser is selected according to the mode
of radiation (Ordinary, Extraordinary) to be viewed.
The beam splitter couples equal amounts of either
polarisation into the two legs.
If the path
difference is zero, a minimum signal is detected.
By varying the path difference via the oscillating
mirror, an interferogram is generated.
Page 179
DATA PROCESSING
.J I
& Invert
& temn Phs Sif
ion
I
.Zero-Fill To Here
Spectrum
Fast Fourier Transform
Fig. (4)
Reorder Data From Sampled
Point Nearest Zero Phase
Page 180
CALIBRATION SPECTRA
BLACK BODY (1)
BLACK BODY (2)
SYSTEM RESPONSE
FREQUENCY
500
1000 GHz
Fig. (5)
First two spectra are produced by black body (a w2 )
mercury lamp.
Periodic structure in first spectra
results from channelling in detector, which has been
corrected; second spectra. Relative system respon e
is determined by dividing black body spectrum by w .
Vertical scales are arbitrary linear units. Absolute
calibration results from scaling 2wc emission to an
independent temperature measurement.
Page 181
MOLECULAR WATER VAPOUR ABSORPTION
T(arb units)
DIFFERENCE BETWEEN MEASURED
AND ASSUMED GAUSSIAN PROFILE
RESOLUTION
500
550
20 GHz
600
GHz
RADIUS (cm)
10
0
-10
Fig. (6)
Example of 557 GHz absorption feature due to molecular
water vapour in the air path between Tokamak and detectoe.
The line width is determined by the instrumental resolution, and
is consistent with the 20 Ghz resolution determined by the
apodisation technique employed in the fourier transform process.
BIOGRAPHY
B Y
WALLACE STEVENS
One must have a mind of winter
To regard the frost and the boughs
Of the pine-trees crusted with snow;
And have been cold a long time
To behold the junipers shagged with ice,
The spruces rough in the distant glitte-
Of the January sun; and not to think
Of any misery in the sound of the wind,
In the sound of a few leaves,
Which is the sound of the land
Full of the same wind
That is blowing in the same bare place
For the listener, who listens in the snow,
And, nothing himself, beholds
Nothing that is not there and the nothing that is.
THE SNOW
Fig. (0)
MAN
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