FAST COMPRESSIONAL ALFVEN
WAVES IN A TOKAMAK
Hironori Takahashi
PFC/JA-80-7
March 1980
Fast Compressional Alfven Waves in a Tokamak*
Hironori Takahashi
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, Mass. 02139
U.S.A.
Enhancement of cyclotron
damping
is
theoretically
evaluated
based upon the wave E-field polarization which is modified by the
two-ion
hybrid
resonance.
The
degree
of
enhancement and its
asymmetry about the plasma center is consistent with experimental
observations. The enhancement factor is very large at low
plasma
densities at which most of the past damping strength measurements
have been reported, but approaches unity at high densities.
implies
the
predominance
of
pure
This
cyclotron damping in future
reactor-like deuterium
plasmas
minority
simple antenna and a pair of limiters are
protons.
A
in
sufficient to couple rf power with an
efficiency
of
about
75 %.
spite
of
overall
the
presence
power
of
deposition
There is no evidence that parasitic
loading accounts for a significant amount of the power.
*This work was supported
by the United
Energy through Contract EY-76-C-023073.
1
States
Department
of
1. IntroductiQn
One promising method
tokamak
plasma
of
additional
heating
for
the
near
cyclotron
cyclotron frequency generates in the
ion
plasma fast compressional Alfven waves whose energy
the
by
(and
ions
field-particle
range
frequency
The impressed radio
.of frequencies ("ICRF" heating)*.
field
a
thermonuclear temperatures is irradiation of
to
the 'plasma by electromagnetic fields in the ion
(rf)
bringing
possibly
interactions.
by
absorbed
various
through
electrons)
One
is
possible
mechanism
for
dissipating wave energy is ion cyclotron damping.
In
one-ion
B-field
of
species
a
plasmas
immersed
fundamental
tokamak,
in
and
the
inhomogeneous
cyclotron
harmonic
given
resonances occur at different major radius locations for a
wave
frequency.
Damping
of
small left-handed component (E+)
resonance
layer.
the
of
fast wave is governed by the
the
wave
E-field
in
In the fundamental resonance layer, where the
wave frequency equals the fundamental cyclotron frequency of
constituent
ions,
the
the
ions
experience acceleration proportional to
the magnitude of the E+.
However, their coherent motion tends to
shield the E+-field nearly completely from the
resonance
layer.
The fundamental cyclotron damping is consequently weak in one-ion
species
plasmas.
In
the
second harmonic resonance layer
acceleration is proportional to the ion Larmor radius
(
e*
)
ion
and
to
the
perpendicular
gradient
of
the E+. The gradient can be
approximated by kjE+Iin one-ion species plasmas,
and
the
wave
damping strength is proportional to 1/2e(kJE)Z.
In' recent tokamak experiments using deuterium
.gas,
small
concentrations
impurities.
Such
situation:
the
a
of
mixture
proton
protons
were
represents
fundamental
as
the
always
a
more
cyclotron
found
furthermore,
a
new
resonance
between
as
complicated
and the deuteron
second harmonic cyclotron resonances occur at the same
and,
working
location,
these ion species
("two-ion hybrid resonance") comes into existence. Because of the
proximity of the cyclotron and
influence
of
these
hybrid
resonances
resonance
plays
an
layers
important
mutual
role
in
determining the dominant damping processes and
division
of
absorbed
In
mixture
power
among
different
species.
shielding of the E+ from the proton fundamental
is
incomplete
and
spiraling motion.
proportional
stronger
than
concentration
to
the
the
is
resonance
layer
protons keep accelerating in a familiar
This so-called
the
this
the
minority
species
damping
is
magnitude of the E+.-field, and can be much
second
small
(a
harmonic
damping
when
the
proton
few percent). However, the relative
damping strengths depend strongly on the parallel phase speed.
The two-ion hybrid resonance surface lies
cyclotron
surface
for
low
proton
principal effects on the waves:
polarization
in
the
ion
very
concentrations
alteration of the
cyclotron
3
close
to
the
and has two
wave
resonance
E-field
layer
and
introduction of mode conversion near the hybrid resonance
Mode
conversion
is
important whenever cyclotron damping at the
hybrid surface is weak. This situation
proton
concentration
is
are
nearly
arises
either
when
resonance
layer,
or
when
when
significant
E+-field
the
perpendicularly incident on the hybrid layer.
Alteration of the polarization caused by the hybrid resonance
important
the
large and the hybrid resonance surface
lies well outside the cyclotron
waves
layer.
the
proton
increase
in
concentration
the
magnitude
is
and
is
a few percent. A
gradient
of
the
on the side. of the ion cyclotron resonance layer closer
to the hybrid
surface
results
in
strong
enhancement
of
the
deuteron second harmonic damping. The minority species damping is
also
significantly
greater
than
in
the absence of the hybrid
resonance.
In their pioneering work
second
harmonic
Adam
cyclotron
and
damping
Samain 2)
in
two-ion
species
plasmas.
temporal evolution of the ion velocity
wave
the
strength in one-ion species
plasmas and the fundamental cyclotron
species
calculated
damping
by
Stix 3),
distribution
the
minority
who
analyzed
under
fast
heating, obtained similar results for the damping strength.
The roles
played
resonance
were
Perkins 5).
by
mode
analyzed
conversion
at
the
independently
by
two-ion
hybrid
Swanson 4)
Enhancement of the cyclotron damping caused
by
and
the
two-ion hybrid resonance through modification of the wave E-field
polarization was first recognized by Takahashi 1,6).
The present
paper extends these earlier results to quantitative evaluation of
4
the enhancemen.t of the cyclotron damping.
A series of ICRF heating experiments was conducted in
tokamak 7),
and
the
was the first under the conditions in which the
ion heating could be critically evaluated. The principal
.of
these
paper
generation,
those
results
cyclotron
contrast
to
and
are
letter 8).
In
the
absorption
are
reported.
These
interpreted in terms of the theory of
damping.
other
short
aspects of the experiments concerning wave
propagation
experimental
enhanced
results
experiments, showing substantial heating of the plasma
core, have already been reported in a
present
ATC
These
experiments
interpretations
explained
in
terms
are
of
in
mode
conversion damping at the two-ion hybrid resonance.
In this paper the phrase, "proton
abbreviated
as
PFC
as
fundamental
"two-ion
is
in PFC resonance. The phrase, "deuteron
second harmonic", is shortened as DSH as
phrase,
cyclotron",
hybrid",
in
DSH
heating.
The
is represented by IHB as in IHB mode
conversion.
2.1 Wave Coupling System and RF Instrumentation
A schematic of the wave coupling system used in the ATC tokamak
is shown in Fig. 1. The overall electrical system and
structure
used in
internal
exterior
the
to
earlier
structure
the
tokamak vessel are similar to those
experiments
of
the
mechanical
ATC
5
in
the
coupler
ST
is
tokamak 9).
different
The
in two
important respects: the absence of an electrostatic shield;
the
presence
of
limiters
to
protect
and,
the antenna from direct
contact with the dense plasma. The absence of a shield makes
the
ATC structure considerably simpler to fabricate and install.
The central element in the coupling
located
on
the
outer
side
structure
(larger
plane
about
the
plasma
an
of
135
center,
degrees
which is
located at a major radius of 0.9 m, and has an inside
0.19 m.
The
antenna
is
in
poloidal
plane and
are
planes
located
in
the
nominally
radius
of
made of a copper strip, 1 in. wide and
1/4 in. thick, and is encased in a ceramic sheath.
lie
antenna
major radius side) of the
plasma torus. It subtends an arc-angle
poloidal
is
separated
The
limiters
by 0.2 m along the meridian
symmetrically
on
either
side
of
the
antenna. The sides of the limiters facing the antenna are covered
by
ceramic
plates.
The inside radius of each limiter is 0.17 m
and the ceramic plate is recessed by
electrically
grounded
to
the
1/8 in.
The
limiters
vessel and have slits cut in the
radial direction to reduce the rf return current near the
edge
that
would
tend
to
are
cancel
the
plasma
rf field created by the
antenna current.
In
order
to
achieve
efficient
coupling
in
the
face
of
relatively small radiation resistance it is important to obtain a
high
antenna
current.
This is accomplished through a resonant
impedance matching network which is adjusted to bring
maximum
(voltage
minimum)
of
a
current
the standing wave pattern to the
6
antenna center. and is tuned for a resonance.
this
becomes
As
plasma
detuned
and
reactive
the
effects
antenna
the
matching
antenna
remains
current
200-300 amp.
The
good.
reaches
10 % of
the
network
experiment
conditions
The measured root-mean-square
but
is
typically
incident and reflected powers are measured by
the
incident
The reflected power is
power.
coupling efficiency (net coupled
ICRF
vary,
450 amperes
two 50 db directional couplers.
than
of
current diminishes. However,
these effects are small under the heating
(RMS)
impedance
tuned circuit is matched to the line impedance by a pair of
stub tuners.
.and
The
This
no
more
means that the power
power/incident
power)
of
the
antenna in ATC is better than 90 %. A net rf power of up to
200 kw is coupled into ATC by this system without encountering
a
limit set by electrical breakdown.
The rf current in
magnetic
loop
the
which
antenna
is
half
is
an
monitored
inch
3/4 in. behind the antenna slightly below
(see Fig. 1).
by
a
one-turn
in diameter and placed
the
antenna
midpoint
This current transducer has a sensitivity of about
0.1 volt/ampere
measured across a 50 ohm termination.
null probe is also located 1/2 in. behind
the
A voltage
antenna
slightly
above the antenna midpoint. The probe, similar in construction to
a
spherical
Langmuir
primarily to locate
estimate
the
probe
the
plasma
interpretation of the ion
leads
to
an
(1/16 in.
voltage
density
minimum
behind
saturation
in
diameter),
point,
the
current
but
antenna.
from
is used
also
to
A
crude
this
probe
estimate of plasma densities of less than 1017 /m 3
7
behind the antenna.
There are arrays of probes that measure the toroidal
of
the wave B-field.
The relative phase of two probe signals is
determined either by observing them directly on
an
oscilloscope
of an analog phase comparator constructed for this
means
by
.or
component
purpose. The phase measurements yield information on the poloidal
mode number and toroidal wavelength of the fast wave.
One
important
equivalent
measure
of
the
antenna
performance
rf
current
(RMS),
square
in
this
paper.
rapid variation of the
eigenmodes.
The
plasma heating
absence
of
The
resistance
caused
by
toroidal
resistance is typically 1-2 ohms under
conditions
plasma.
(RV-)
computer is useful in following
loading
loading
the
of
and is determined by means of an analog
device which will be referred to as the loading resistance
computer
its
series loading resistance. It is defined as the ratio
of the net rf power (RMS) fed into the antenna to the
its
is
compared
to
about
0.1 ohm
in
the
Thus, only about 5-10 % of the coupled
power is dissipated in the circuit.
2.2 Discharge Characteristics
ATC is a tokamak with toroidal compression capability 7).
ICRF
heating
is,
however,
applied
plasmas with a major radius between
generator
frequency
to
0.84 m
All
uncompressed deuterium
and
0.9 m.
The
rf
is fixed at 25 MHz which corresponds to the
8
DSH cyclotron frequency at 1.64 T.
plasma
are
performed
All
at
attempts
the
plasma
center.
In
this
paper
both
the
lasts
placed
refer
For wave
to these
measurement
field and density are varied over much wider
ranges: n,,
0 =0.5-4.2x10 %/m3
typically
we
the
densities
layer
resonance
conditions as "plasma heating conditions".
purposes
heat
moderate-to-high
(ne,>2x10 19 /m3 ) and with the cyclotron
near
to
40 ms
and
with
a
Bo=1.2-1.9 T.
10 ms
The
discharge
rf pulse usually applied
20-25 ms aftef discharge initiation.
The most important impurity
for
the
considerations
of
wave
absorption is hydrogen. An accurate measurement of its density is
difficult,
but
the proton-to-deuteron ratio is estimated 10) to
be at most 5 %.
This upper bound is obtained from the fact
no
recognizable
at
a
location
peak exists in the wing of the DI spectral line
where
proton-to-deuteron
the
ratios
Ho
enhanced
the
observed
cyclotron
Throughout
damping
this
line
is
and are
wave
damping
rather
paper
among
the
assumed
to
be
3 %,
the
strength
than
(proton-to-electron ratio) is taken to be
arbitrarily
expected.
Small
found in ATC are in contrast to those
(~20 %) in TFR experiments 11),
interpreting
that
mode
proton
small
reasons
for
in terms of
conversion.
concentration
(<5 %)
and
is
whenever a numerical value is
needed. More detailed descriptions of the experimental facilities
and environment can be found in a laboratory report 12).
9
.3.1 Fast Wave Propagation in ATC
In ATC plasma the wave damping length is large compared to
(minor)
radial
propagation
extent
and
of
damping
the
plasma.
amenable
This
to
makes
separate
the
the
wave
analyses:
propagation characteristics and field structure of undamped waves
are
first obtained and the damping strengths are then calculated
based upon the known field structure. We first consider the
fast
waves in a straight cylindrical waveguide partially filled with a
cold
uniform
plasma
magnetic field.
and
Eigenmode
immersed
solutions
in
a
for
uniform longitudinal
such
a
system
were
obtained by Bernstein and Trehan 13) for general non-axisymmetric
modes.
A
theory
of
excitation
of
azimuthally
symmetric
eigenmodes was developed by Stix 14). The theory 15) used here is
an extention of the latter to more general non-axisymmetric cases
using the eigen-solutions of the former.
Poynting
energy
density,
flux, group velocity, and radiation resistance are also
calculated.
antenna
Wave
(The term, "radiation resistance", is used
loading
is
computed
as
averaged
the
ratio
over
the
waveguide
of the time-averaged
Poynting flux integrated over the waveguide cross-section to
wave
the
resistance in the absence of toroidal effects.)
In particular, the group velocity
cross-section
for
energy per unit length of the waveguide.
the
Both the Poynting
flux and wave energy consist of contributions from the plasma and
vacuum regions within
the
waveguide,
and
the
computed
velocity depends upon the size of the vacuum region.
10
group
The ATC vacuum vessel is oblong in cross-section (see
encloses
Fig. 1),
a large vacuum space on the small major radius side and
is difficult to model accurately in theoretical calculations.
On
one hand, theoretically calculated group velocity tends to be too
smal'l because of the failure to
space.
Calculations
of
account
the
loading
hand, depend critically on the
antenna and vacuum vessel.
torus
of
circular
for
vessel,
relative
vacuum
sizes
of
the
plasma,
The vacuum vessel is represented by a
cross-section
its
large
resistance, on the other
with a minor radius of 0.21m.
Were the theoretical vessel to enclose the
actual
the
equivalent
radius
same
volume
as
would be 0.27m.
the
Eighty
percent of the actual vessel walls are made of bellows structure,
but the theoretical model assumes smooth
The
stainless-steel
walls.
inhomogeneous plasma is represented in theory by a plasma of
constant density equal
density.
Ion
to
the
volume
average
of
the
actual
and electron temperature variations with time are
taken from a typical discharge at each density level.
A discrete fast wave eigenmode * in a plasma
can
be
numbers.
identified
by
filled
a set of radial(l) and azimuthal(m) mode
It is convenient to speak in terms of the perpendicular
(NLkVA/a)
indices 3),
and
where
parallel
kL,
k11,
(N1 =k11VA/w)
VA
and
(4
Alfven
frequency.
refractive
are, respectively, the
perpendicular and parallel wave numbers, Alfven speed,
angular
waveguide
and
wave
At sufficiently low densities the fast waves
*See, e.g., Fig. 2 of ref. 3 or Fig. 6 of ref. 12.
11
in a completely filled waveguide suffer a cut-off (N,<0) and
unable
to
propagate along the waveguide.
waveguide the cut-off exists
azimuthal
modes
(for
for
the
modes
propagate
In a partially filled
m=O
field
exp l(k 1 z+mO -wt)). However, the
.positive-m
the
and
all
variation
lowest
radial
negative-m
of
the
modes
very low densities *.
at
are
form
of
all
(They can
presumably
propagate
increases,
a cut-off mode may become propagating, at which point
at
N=1 and N =O. We term
propagating mode.
wavelength
all
densities.)
condition
this
the
As
the
"onset"
density
of
a
new
As the density increases further, the parallel
becomes
progressively
shorter.
Because
of
the
different dependence of the PFC, DSH and IHB damping strengths on
the parallel wavelength, the mode goes through
in
each
of
wave damping.
of
a
mode
which
different
a different mechanism is responsible for the
The exact plasma conditions under which the
takes
place
density
profile 17)
Because
of
the
phases
and
onset
is sensitively dependent on the radial
the
presence
of
the
vacuum
space.
approximate nature of the representation of the
actual ATC vacuum vessel and the uniform plasma model used in the
present theory, the model's prediction of the onset conditions is
not expected to be accurate.
Toroidal effects on
superposing
the
wave
propagation
are
accounted
for
by
fields of the waves circumnavigating around the
*This was demonstrated by numerical examples by Paoloni 16)
for
m=O and +1 modes. Similar calculations indicate that the same is
true for all positive-m modes.
12
torus.
The resultant fields depend upon both the phase shift and
damping that the waves suffer as they go around the torus.
such
conditions
processes
of wave generation, propagation, and
damping are closely intertwined
each
Under
and
isolated
process is difficult in practice.
investigation
of
Nevertheless, studies of
the loading resistance and wave amplitude under different
plasma
conditions provide evidence for wave generation and clues for the
physical mechanisms of wave damping.
Suppose that the antenna carrying unit rf current
wave
field
effect.
of
In a
experience
a
amplitude,
toroidal
complex
the torus, where kU
s,
in
geometry
the
with
generates
a
absence of the toroidal
dissipation,
the
waves
phase shift ik11 L each time they go around
is the complex parallel wave number and L the
toroidal circumference.
The total field for an rf current
of
I
for the waves propagating around the torus many times is obtained
by summing an infinite series,
S = sI(1
+ exp ik 1L + exp 2ikL +...)
= sI/(1 - exp ik1,L)
When the plasma conditions are
interference
appropriate
(1)
for
a
constructive
of the wave fields, the cavity becomes resonant. As
the conditions (principally the density) vary with time, a series
of such toroidal cavity resonances
resonance
and
at
an
should
be
observed.
At
a
anti-resonance (n is a half-integer) S is
pure real and is given respectively by,
13
SA
=
( )
(2)
+ O( )
(3)
-
SR= sIR/(1
SIA/(1
where c( =exp(-ktL) and kL is the imaginary part of
wave
number.
respectively.
experimentally measureable.
can
be
a
can
cause
consequences 1).
measuring
the
energy
more
a
Its
two
quantities
the
(as
waves
resonance.
opposed
travelling
This
can
to
be
has
poloidal
a more usual
in
only
some
determined
one
practical
simply
by
phases of the wave field at two or more
torus.
The
travelling
waves
also
deposit
uniformly around the torus than the standing waves.
If the so-called mode splitting, caused by unequal
the
are
It should be noted that the toroidal
wavelength
relative
points around the
These
travelling
standing) wave resonance:
direction
parallel
IR and IA are the rf currents at the resonance and
anti-resonance,
resonance
the
B-field
indeed present, all
on
influence
of
the left- and right-running waves, is
experimentally
observed
resonances,
other
than the m=O modes with no mode splitting, must be the travelling
wave resonances.
(No experimental demonstration of this fact has
been reported in the literature.)
The oscillogram in Fig. 2 shows
amplitude
(rf
envelope)
measured
time
by
degrees around the torus away from the
above(T)
and below(B) the plasma axis.
14
variation
of
the
wave
the probes located at 90
antenna
nearly
directly
The different amplitudes
NOMMONNOWNIM"
of the two signals are caused by a slightly off-centered
The
plasma.
peaks and troughs of the wave amplitude evidently correspond
to the occurence of toroidal resonances and anti-resonances.
The
theoretical calculations of the wave amplitudes corresponding
the ' experimentally
.presence
of
the
correspondence
observed
peaks
of
and
these
density
variations
troughs.
confirm
However,
to
the
one-to-one
peaks cannot be established in the ATC
experiments, becuase of the approximate nature of the theoretical
model.
It was shown in the
experiments 9)
earlier
in
the
ST
tokamak, however, that the density and B-field dependences of the
loading
resistance
dispersion
peaks
relation.
waveguide
For
closely
these
followed
the
experiments
fast
the
wave
circular
model was a better representation of the actual vacuum
vessel and the theoretical calculations accounted for the density
inhomogeneity.
3.2 Wave Damping Measurements
A set of Re-computer signals is shown in Fig. 3.
B-field
is
different
conditions are
density
also
increases
The
for each of the oscillograms. (The plasma
slightly
different.)
In
each
the density reaches a maximum during the pulse.
DSH
(deuteron
case
the
monotonically by a factor of two or less over
the pulse period, except in the left-most oscillogram
the
toroidal
for
which
The locations of
second harmonic) cyclotron resonance surface
are indicated in the figure.
The
data
levels sufficiently low (neo=7.5-9. 6 x10
15
are
/m3
taken
at
density
at the mid-point of
the
pulse)
that
the
m=O and all negative-m modes are cut-off.
The prominent peaks in these oscillograms are
the
m=+1
mode
determined
to
be
by phase measurements of magnetic probe signals.
(However, small-amplitude, higher azimuthal modes (m>+1) may also
be present simultaneously in the signals.)
measurements
of
selected
numbers of 2-4.
the
plasma,
Toroidal
resonant peaks indicate toroidal mode
When the cyclotron resonance surface
the
wavelength
toroidal
is
within
resonance is broader, indicating the
presence of damping processes on or near the resonance surface.
Quantitative evaluation of the damping
from
the
magnetic
strength
can
be
made
probe signals such as those shown in Fig. 2.
The quantity, s, in eqs. (2) and (3), which
is
proportional
to
the radiation resistance, is in general, not directly measurable.
However,
this
can be eliminated by taking the ratio of the wave
amplitude at a resonance to that at an
if
the
adjacent
anti-resonance,
variation of the radiation resistance for a small change
in plasma parameters is neglected.
The ratio is related
to
the
wave damping length(LD) through *,
LD/L
[ln(T
+1)/( T -1)]
(4)
where T =IASR/IRSA. The normalized damping length (LO/L)
number
of
times
amplitudes e-fold.
the
waves
go
around
The wave damping length
is
the torus before their
is
related
to
*Typographical errors in eq. 9 of ref. 1 are corrected here.
16
the
the
resonant cavity quality factor (Q) through,
LD
=
1/k; =2Qu1/C4
(5)
where u3 is the parallel group velocity.
It should be noted that
the group velocity varies widely depending upon the mode and that
a large Q mode does not necessarily imply a large
Lt
mode.
In
experiments either the damping length or the Q-factor is directly
determined
and
the
other
is
derived
ambiguities would be introduced through
through
this
eq. (5).
conversion
Some
unless
the group velocity is accurately known.
The damping length determined using
Fig. 4(a)
for
the
shown in Fig. 3.
the
DSH
(or
in
The abscissa is the major
is
the
range
of
include
2-4.
in
radius
location
of
different
When
toroidal
mode
the resonant surface is
side
(R(S2.=2)=70 cm),
the
weak and the waves go around the torus some 13 times
before their amplitudes e-fold.
the
shown 18)
PFC) resonance surface. The observed modes are the
outside the plasma on the high field
damping
is
conditions corresponding to the oscillograms
m=+1 azimuthal modes, but may
numbers
eq. (4)
This damping is
damping caused by the wall resistance.
length is one-twentieth the theoretical
interpreted
as
The measured damping
wall
resistive
damping
length computed for these modes in circular cylindrical waveguide
made
of
damping
increased
perfectly
smooth
stainless-steel walls.
length
appears
reasonable,
path
lengths
of
the
17
wall
however,
currents
The observed
because
due
of
the
to (1)the
elongated vacuum vessel cross-section
poloidal
circumference),
(2)the
(a
1.6-fold
partial
increase
bellows structure (an
average of five-fold increase in toroidal circumference),
presence
of
ports
and
other
(3)the
obstructions to the wall current
paths and (4)the surface roughness (0. 8 -13.0pm) * comparable
the
skin
depth
damping strength.
(8.5}.m),
in
all
of
to
which tend to increase wave
In the following calculations the wall damping
length is taken to be one-twentieth the
cylindrical surface.
When
the
value
resonant
for
layer
the
is
perfect
within
the
plasma, the normalized damping length is reduced to about 2.
The
ratio,
13/2=6.5,
of
the
damping
lengths with and without the
resonance is a measure of the effectiveness of energy
by
the
fast
wave: about 90%
(6.5/(1+6.5)=0.87) of the power is
deposited within the plasma and about 10% in
observations
deposition
the
walls.
These
lead to an optimistic view for fast wave heating of
tokamak pasmas.
When the resonant layer is outside the plasma on the low
side
(R(.a =2)=110 cm),
the damping is stronger than on the high
field side indicating the presence
mechanisms.
The
understood. There
charge
exchange
nature
are,
(CX)
non-linear processes
surface
field
of
some
additional
damping
of these damping mechanisms is not well
however,
analyzer
taking
indications
signals
place
on
observed
that
or
on
the
suggest possible
near
the
cyclotron
in the tenuous plasma outside the nominal plasma radius.
*The range of roughness of
stainless-steel sheets.
commercially
18
available
cold
rolled
In Fig. 5 oscillograms of the charge
and
R.7-computer
showing
the
signals
relative
are
exchange
shown
locations
neutral
together
of
the
analyzer
with a schematic
plasma,
antenna
and
cyclotron resoanace layer. Very large-amplitude, sharp spikes are
observed
on
the
high-energy channels of the CX analyzer nearly
exactly coincident with the appearance of
peak
in
the
loading
resistance.
a
of
the
observed phenomenum is taking
the
resonance
the
time
constant
analyzer, indicating that the energetic ions
are contained only for a very short time.
When
resonance
The CX signals rise and fall
very rapidly with a time scale comparable to
(300.sec)
toroidal
layer
place
is
This suggests that the
in
the
plasma
periphery.
deep inside the main body of the
plasma, or, outside the plasma on the high field
side,
no
such
spikes are observed.
One possible explanation for the absence of
spikes
conditions can be either that the energetic
under
these
ions are not contained long enough to reach the
they
are
created
created upon charge
penetrate
through
right
in
exchange
the
front
of
dense
analyzer without being reionized.
wave damping strengths when the
sides
of
body
resonance
plasma
ions
to
can
not
reach the
is
on
the
opposite
the plasma suggests that the observed phenomenum takes
of the spikes with the occurence
the
energetic
However, the difference of the
place mostly on the low field side.
which
unless
of it, or that the neutrals
the
main
analyzer
of
Furthermore, the coincidence
a
toroidal
resonance,
at
wave electric field becomes large, is indicative of a
threshold nature of the phenomenum.
19
A notable fe.ature of the damping curve in Fig. 4(a) is that the
region of the maximum damping is not symmetric
the
axis
plasma
major radius side 19).
expeiiments 20).
in
shown
Another
The
Fig. 4(a).
also
abscissa
pulse.
The net difference between discharges
due
loss
is
plotted.
and
with
of
to
without
A broad peak is located on the
larger major radius side of the plasma axis where the
the strong damping region is also located.
the
is
radiation integrated over the period of rf
and
exchange
pulse
TFR
that
to
identical
charge
heating
to
the output voltage of a bolometer
is
ordinate
is
facing the plasma and is proportional to the power
the
in
observed
manifestation of the asymmetry 21)
The
Fig. 4(b).
was
shift
This
respect
and is shifted toward the larger
RO=87 cm
at
with
center
of
Under these conditions
rf power is evidently more efficiently absorbed by the ions,
some of which become lost through charge exchange and intercepted
by the bolometer.
Another set of output signals * from the R..-computer
in Fig.
vicinity
6. The resonant surface (R(.C.=2)=88 cm)
is
shown
is located
in the
of the plasma center (Ro=87-89 cm) for all cases except
The electron density levels are
the lowest density case (No. 1).
The
different for each case and are shown in the figure.
density
case
(No. 1)
is
taken
under
conditions
identical to those of the center oscillogram of Fig. 3.
resonance peaks of the loading
resistance
become
lowest
nominally
Toroidal
progressively
*The incorrect label of the ordinate of Fig. 4 and numbering
oscillograms in Fig. 5 of ref. 1 are corrected here.
20
of
broader at higher densities and are difficult to recognize at the
highest
density
level.
The latter is a typical condition under
which most heating experiments are undertaken.
isolated
resonant
experimental
peaks
in
determination
these
of
the
high
In the absence of
density
cases
the
damping strength based upon
*eq. (4) is not possible.
3.3 Cyclotron Damping
We first compare the experimentally observed time variations of
the loading resistance and wave damping strength with predictions
based upon the theory of cyclotron damping. One low density case,
for
which
strength
direct
are
made,
experimental
and
one
measurements
high
of
the
damping
density case, for which the
measurements are not available, are examined in some detail.
electron Landau damping and transit time magnetic pumping
are
also
weak
needed
under
in
the
ATC
eqs. (1)-(3)
conditions.
The
are
from
results 3) given in terms of the quality
cavity.
(TTMP)
included in the calculations, but their effects on the
fast waves are
strengths
The
They
derived
factor
of
a
damping
Stix's
resonant
are for the minority (Q1 ), second harmonic (Q
and Landau/TTMP (Q
),
) damping given respectively by,
22 (a
n.
+ 9 1nHu1)2]
21
(6)
2
II
(7')
(±D2[1rrs
e )2}/2
2pD
(k
(8)
2
ge
/2u2
W 12Bu1
exp-
e
Here,
u
e
are respectively the plasma minor radius and the
a and r
n
axis.
densities.
the deuteron and proton
respectively,
u1j, vH and v. are, respectively,
proton,
speed,
phase
are,
nH
and
deuteron
and
plasma
the
and
layer
distance between the cyclotron resonance
parallel
wave
the
(3e is the
thermal speed.
ratio of the electron thermal pressure to the magnetic pressure.
In deriving these results Stix accounted for the
the
at
E+-field
the
evaluated
across
the
surface due to the presence of
cyclotron
resonant minority protons.
the E+ and its gradient
However,
thin
for
These
layer.
resonance
species
one-ion
enhancement of the cyclotron damping
nor
plasmas,
the
reasonable
were
mode
but
neither
conversion,
caused by the presence of the two-ion hybrid resonance, was
included.
were
were
at the resonance surface and were taken to be constant
approximations
both
of
reduction
the
Further approximations introduced in
assumptions
that
resonant surface and that N
the
=1.
Stix's
analysis
E+ was also constant along the
The latter would result
in
an
over-estimate of the second harmonic damping strength by a factor
up to four for those modes that are not near their onset.
22
It is often stated that minority damping is much stronger
second
harmonic damping when the minority concentration is a few
percent. However, the
eq. (6)
strong
should be noted.
wavelength
dominant
damping
kg
and
small
harmonic damping.
Q
of
in
rising
plasma
density,
mechanism rapidly changes from the second
harmonic to the minority damping.
large
dependence
As the wavelength of a new propagating
mode becomes progressively shorter with
the
than
Higher
radial
modes
with
a
k1l are more strongly damped by the second
Near the onset,
both
damping
strengths
are
proportional
to
sufficiently
high
temperature,
which may have interesting consequences because the
the ion temperature. Away from the onset and at
temperatures,
QE
becomes
independent
second harmonic damping becomes dominant for all
high
temperature reactor-like plasmas.
wavelengths
of
in
Note the symmetry of the
second harmonic damping with respect to the relative locations of
the resonant
layer
contrast
the
to
and
plasma
axis
experimental
(eq.
(7)).
observations
This
discussed
is
in
earlier
(Fig. 4(a)).
Theoretical loading resistance variation for
case
is
shown
in
Fig. 7(a)
oscillogram at the center
of
and
should
Fig. 3.
the
low
density
be compared with the
Toroidal
resonances
are
designated by sets of radial(l), poloidal(m) and toroidal(n) mode
numbers.
The
radiation resistance is first computed based upon
the theoretical model
outlined
effect
through eq. (1).
is
intrduced
in
parallel wave number needed in this
23
Sec. 3.1
and
the
toroidal
The imaginary part of the
equation
is
obtained
from
Q-factors give.n in eqs. (6)-(8) by converting them to the damping
length
through
eq. (5).
By
using
effects on the loading resistance
different
of
each
mode
density, the
going
phases of its "life span" can be simulated.
only' two poloidal modes (m=+1
.power.
time-varying
The
loading
and
resistances
+2)
that
carry
through
There are
appreciable
of the individual modes up to
m=+3 as well as the total resistance are shown in Fig. 7(a).
The
damping lengths predicted from theory are too large, so that
the
theoretical
toroidal resonant peaks which can reach 6.3 ohms are
much larger than the largest
Note
the
the
ambiguities
damping
Q-factors
small group
offsetting
the
by
velocity
effect
values
of
0.5 ohms.
introduced by converting the Q-factors to
length:
predicted
experimental
discrepancies
the
used
the
theory
in
the
caused
by
too
large
are partly offset by a too
conversion.
Without
this
difference between experiment and theory
would be bigger.
The theoretical values of the minority(I), second harmonic(II),
electron Landau/TTMP(E), wall resistive(W), and total(T)
lengths
damping
are tabulated in Table I for each toroidal mode shown in
Fig. 7(a).
The damping lengths
comparable
for the PFC and DSH damping for these toroidal modes,
predicted
by
this
and are much greater than the wall resistive damping
these
predictions
were
true,
then
theory
length.
the
If
most of the power would be
dissipated in the walls and there would be no significant
in
are
change
measured damping length as the resonant layer moved into
the plasma. The previous section shows that this is not the case:
the measured damping length is smaller than the tabulated
by
factors
between
18
entries are made for the
table
because
and
120,
values
depending upon the modes. No
electron
Landau/TTMP
damping
it is completely negligible (L./L>10
in
the
) for these
cases.
Fig. 8(a) shows the computed loading resistance
variation
for
the high density case which should be compared with the top right
oscillogram
(No. 5) of Fig. 6.
There are five propagating modes
that carry appreciable power (the
through
m=+2
and
the
first
radial
modes
of
second radial mode of m=+1). The loading
resistance of each mode and the sum over all modes are
the
figure.
(The
radial modes.)
shown
that
The theoretical radiation resistance of a mode is
the
long
wavelength
n=1
toroidal
resonance immediately following the onset.
the
by
mode comes into
mode
of
m=+1
onset at nearly identical times (at about 23 ms).
n=1 toroidal modes of these poloidal modes create a high peak
the
loading
resistance.
resistance
are
considered
to
model.
than
not
be
Such
observed
due
to
abrupt
in
the
the
The first radial mode
of m=-1 poloidal mode and the second radial
have
in
m=+1 curve is the sum of its first and second
in general highest at its onset, and is further enhanced
fact
m=-1
changes
mode
The
of
in the loading
experiments,
and
are
the deficiency of the uniform plasma
The other resonant peaks in Fig. 8(a) are more
prominent
the experimentally observed ones, indicating actual damping
strengths stronger than the theoretical predictions. However, the
differences are not as pronounced as
25
in
the
low
density
case
(Fig. 7(a)).
Theoretical values of
electron
the
Landau/TTMP(E),
minority(I),
second
harmonic(II),
wall resistive(W), and total(T) damping
strengths are tabulated in Table II for each toroidal mode
shown
in Fig. 8(a).
The DSH damping is the strongest mechanism for all
low
number modes (n .< 3).
toroidal
The m=-1 poloidal modes and
the 1=2 radial modes appear only at high densities and
more
strongly
are
much
absorbed by the DSH damping than the PFC damping.
These modes would deposit most
of
their
energy
directly
into
deuterons and make fast wave heating less susceptible to "thermal
runaway"
of
the
minority
species 22).
Unlike the low density
case, the theoretically predicted total damping is now
by
dominated
power dissipation in the plasma rather than in the walls. The
theoretical power deposition efficiency ranges from 74 % to
for
these
modes.
The
97 %
efficiency is generally better for those
modes that carry large shares of the
total
power.
Although
no
direct measurements are available, comparisons of the theoretical
loading
curve
with
the
experimental
computed damping strengths are
actual
still
one
an
indicate
underestimate
that
the
of
the
values. The actual efficiency is therefore expected to be
greater than these
completely
figures.
negligible
Electron
(L0 /L>10
Landau/TTMP
damping
is
) except where entries are made
in the table.
The theory in this section is based upon a uniform plasma model
with the cyclotron and Landau/TTMP damping strengths obtained
26
by
Stix.
The
mode
uniform plasma model is inaccurate in predicting the
onset,
and
one-to-one
correspondence
of
the
toroidal
resonant peaks cannot be obtained. This discussion has shown that
the
predicted
damping
lengths
at
low
(neo=0.5-1.0x10 9/m 3 ),
where the experimental
individual
be
modes
can
densities
measurements
for
made, are too large by a large factor
(18-120). The dependence of the damping length
on
the
resonant
layer location also fails to predict the observed asymmetry about
the
plasma
modes
center.
cannot
overlapping
be
Although the damping strength of individual
measured
modes
are
at
higher
densities,
where
many
present, the predicted time variation of
the loading resistance indicates that the discrepancies are
much
smaller.
3.4 Enhanced Cyclotron Damping
We next consider whether better agreement can be found
theory
and
caused by
included.
experiments
the
presence
For
proton-deuteron
distance (,v 1 cm)
small
hybrid
between
if enhancement of the cyclotron damping
of
the
proton
two-ion
hybrid
concentrations
resonance
away from the
surface
cyclotron
lies
resonance
(Cm~ 3 %)
only
resonance
a
is
the
small
surface.
*
The cyclotron and hybrid resonance layers overlap with each other
and
their mutual influence must be taken into account.
The fast
*The geometry of the two-ion hybrid resonance surface, cyclotron
resonance
surface
and fast wave cut-off surface can be found,
e.g., in Fig. 6 of ref. 1 and in ref. 5.
27
wave
has
only
component
a
in
sinall
one-ion
left-hand
species
polarized
plasmas.
electric
However,
field
the
hybrid
resonance (an electrostatic resonance) tends to produce a
linear
E-field
polarization
magnitude
of
(i.e.,
E+-field.
.electromagnetic
The
resonance)
E+=E..)
and
proton
cyclotron
tends
(The
theories
by
Adam
reduce
resonance
surfaces
both
enhance the
resonance
(an
the
magnitude
of
and Samain 2) and by Stix 3)
include only the latter effect.) Within the
two
to
to make the polarization both
circular and right-handed and thus to
E+-field.
thus
region
between
the
the magnitude and gradient of the
E+-field are greater than in the absence of the hybrid resonance.
The magnetic field inhomogeneity across the plasma
cross-section
must
treatment
be
included
into
the
analysis,
cylindrical geometry becomes difficult.
and
a
However, because of
in
the
relatively thin and straight geometry of the cyclotron and hybrid
resonance
layers
situation may be
in
the
important
amenable
to
an
central plasma region, the
approximate
treatment
in
a
simpler slab geometry.
Boundary value problems in slab geometry with an
B-field
and a non-uniform warm plasma were solved numerically by
Takahashi 1) to obtain the E-field
based
inhomogeneous
upon
a
second-order
structure.
differential
The
theory
equation (eq.
was
(23) of
ref. 1) that treated the PFC damping in a self-consistent manner,
but the DSH damping
strictly
valid
was
not
included.
Thus,
the
theory
only when the latter had small effects.
of the proximity of the cyclotron
28
resonance
layer,
the
was
Because
hybrid
resonance
is heavily damped: the real and imaginary parts of the
perpendicular
properties
wave
were
number
computed
for
the
local
comparable and the real part of the coefficient
equation
of the second order derivative term in the differential
did 'not
plasma
vanish
at the hybrid layer.
.resonances was obtained earlier 1).
A criterion for such weak
The resonance is
weak
when
the following inequality is satisfied:
2
1.082
H
Here, CO=nD/ne,
are,
ao=.:
e/(o ,
Da
2
19)
eL D aPDJ
and Wg ,
D,
cJ,
and
c
respectively, the deuteron cyclotron and plasma frequencies
and the speed of light in vacuum.
In the present
paper
we
are
cases
are
also concerned with such weak resonance cases.
In
Fig. 9
reproduced
the
from
E-field
structures
for
specific
ref. 1. The x-coordinate is in the direction of
inhomogeneity with its origin at the slab center. The abscissa of
the figure is R,-x, where R,=0.9 m is the
scale
length
of
the
B-field inhomogeneity ("major radius") at the center. The tokamak
axis
is
to the right of the figure. The slab half-width ("minor
radius") is, a=0.17 m, and neo=2.5x10
/m
variations of the E.-field across the slab
29
.
The top figures are
for
three
different
values
of
magnetic
field:
one
for which the PFC resonance is
located at the center and two others for which the
displaced
by
0.06m
on
either
side
figures are variations of
the
three
ten-to-one
cases.
between the
(Note
two
cyclotron
the
groups
(ICR)
of
of
E+-field
resonance
is
the center. The lower
for
the
corresponding
difference in the ordinate
figures.)
Locations
of
the
ion
and two-ion hybrid (HBR) resonance surfaces are
indicated in these figures.
Effects of the weak hybrid resonance appear as small
the
E.-curves (The E
of
the
evidently
as
there.
resonance
Et
the
However,
is
influence
of
the
hybrid
significant. Compare the CH=O and 3 %
cases shown in the lower central figure. The large difference
a
a
fact that the field and density gradients are in
the same direction
on
on
is little affected) in Fig. 9. The effects
are stronger on the low field side of the axis,
result
bumps
consequence
of
the
fact
strongly altered by the IHB
that
and
the
PFC
field
is
polarization is
resonances.
Particularly
noteworthy here is the appearance of a very steep gradient of the
E+
over
a
narrow
region
resonances. In the case
gradient
in
this
between
shown
region
in
the
the
cyclotron
lower
is
then
greater
figure
the
hybrid
resonance.
expected that the DSH damping should be enhanced by
two orders of magnitude.
be
hybrid
is enhanced by one order of magnitude
over the value obtained in the absence of the
It
left
and
than
without
Since the magnitude of the E+ can
also
the hybrid resonance, the PFC damping
could also be enhanced. Furthermore, the dependence
30
of
both
of
these
damping
asymmetric
strengths
on
the
resonance
layer
respect
to
the
plasma
center.
presented
in
with
quantitative
analysis
is
the
location is
A
more
remainder of this
section.
Ions that move across the cyclotron resonance layer
kick
in
their
perpendicular
energies
due
to
acceleration.
The power absorbed by the ions can
by
statistical
taking
gains.
a
average
of
the
be
receive
a
cyclotron
calculated
these incremental energy
For a slab of unit height and depth the power absorbed by
the protons and deuterons is given respectively by 6),
2
II
pD
C
1)
4
(x)12
J
2
Xx
+
(10)
I
C
Here,
x. and R. are the x-coordinate and the
the
cyclotron
resonance
surface.
"major
radius"
Nand
P
of
are,
respectively, the proton plasma frequency and the deuteron Larmor
radius.
The integrals, J,
J,
and JI
,
are given by,
rdC.I E(.)12exp - 2
dH
ir
31
H(1
1/2
0(12)
(
D
71/2 fo
II
3
where
-
(u11/vM)(x
=
(x)
xC)/RC
(14)
E+(x)/E+(x)
(15)
0(x) = a[dE+(x)/dx]/E+(xc)
Dimensionless quantities,
analysis
this
In
ions.
E+-field
and
to
perpendicular
refers
to
or
proton(H)
either
is the most probable speed of the M-species
v.
and
deuteron(D)
eq. (14)
in
M,
subscript,
are the
normalized by the E+at the resonance surface. The
gradient
its
E(x) and *(x),
(16)
number,
wave
the
axis
(y
k ,
the x-coordinate and- B-field) is assumed to be
zero.
If the E+ is evaluated at the
constant
across
the
integrates to unity.
to
correspond
"enhancement
E+
the
J,=1
2A
=1,
resonance
Thus,
and
layer,
gradient
and
surface
then
8(x)=1
the approximation used by
the
Stix
assumed
and
J3
would
actual value of JI represents the
factor" of the PFC power absorption.
Similarly,
if
is approximated by kiIE+j and evaluated at the
resonance surface, JI
NJ
resonance
equals to (kia)
.
Stix
further
assumed
or equivalently, k =k~l where kA is the Alfven wave number
( W/VA)-
JI/(kAa)
is therefore the enhancement factor for the
32
DSH power absorption.
specific
cases
are
The results of the calculations
for
some
presented in the remainder of this section.
For these calculations the integrals in eqs. (12)
and
(13)
are
evaluated numerically using the field variation obtained from the
solutions
of
the boundary value problem discussed in an earlier
are:
cases
.paragraph. The parameters common for all
a=0.17 m,
R0 =0.9 m, CH=3 %, and the on-axis ion tempeature, T,. =200 eV.
Several computed quantities for four different locations of the
cyclotron surface are presented in Table III.
respectively,
the
power
absorbed
by
PL
the
and
Pi
are,
PFC and DSH damping
within a slab of unit height and depth.
These values are for the
waves travelling in one direction only,
and
such
are
normalized
in
a way that the Poynting flux is 1 MW per unit height of the
slab.
Power absorption by both mechanisms
is
strong
when
the
resonance surface is located in the low field side of the center,
which
is
consistent
damping (Fig. 4(a)).
DSH damping.
resonance
with
observed
asymmetry in the wave
The asymmetry is particularly pronounced for
If the deuterons absorb much more
surface
under
power
when
the
is in the low field side, power loss from the
plasma through deuteron
greater
the
these
charge
exchange
conditions.
This
should
also
expectation
be
is
much
not
inconsistent with the observed asymmetry of the bolometer signals
discussed earlier (Fig. 4(b)).
refractive
index
correspond
to
considerably
(n,,)
The eigenvalue for
the
parallel
is about 30 for all cases and thus would
the
toroidal
greater
than
mode
the
33
number
toroidal
of
15.
This
is
mode numbers shown in
Figs. 6
and
wavelength
7.
(The
slab
considerably
shorter
with the same peak density.
k
is taken to be zero.)
is
about
one
geometry
than
leads
to
the
parallel
the cylindrical geometry
This is partly due to the fact
In these computed cases the PFC damping
order of magnitude stronger than the DSH damping,
and thus the assumption underlying the theory is valid.
resonance
surface
enhancement
is
located
at
the
plasma
from
When the
center,
the
factors are respectively 1.8 and 5.2 for the PFC and
DSH damping, but if the resonance surface is
away
that
the
plasma
center
on
the
located
low
field
0.04 m
at
side,
the
shown
for
enhancement factors are respectively 2.8 and 13 *.
In Fig. 8(b) variation of the loading resistance is
the
same case as (a),
but with the PFC and DSH damping strengths
multiplied respectively by 1.8 and
factors
should
be
different
for
given here are only approximations.
the
modest
enhancement
5.2.
Since
the
enhanceme-nt
different modes, the results
However, they indicate
that
factors expected from the theory of the
two-ion hybrid resonance are sufficient to suppress the prominent
toroidal resonance peaks
loading
curve
reasonably
and
to
make
the
appearance
of
the
similar to experimental observations.
The abrupt rise of the loading resistance at the onset of the two
*When the resonance surface is further away from the center and
comes
too close to the fast wave cut-off surface, these
enhancement factors become very large, but the normalizing factor
(the E+ at the resonance surface) tends to zero. The enhancement
factors lose their significance in these cases, since Stix's
results were obtained based upon the plasma properties of the
central core.
34
new modes and .subsequent gradual
fall-off
rise
resistance
seen
in
Fig. 6)
of
the
(instead
are
of
the
gradual
principal
discrepancies that are still evident.
Each mode with a given set of radial and poloidal mode
.has
a
given
radial
variation
of
the E+-field, and hence the
radial power deposition pattern. The lowest radial
m=+1
poloidal
periphery.
mode
This is
low-density
has
the
tokamaks.
the
most
numbers
pattern
peaked
prominent
mode
near
mode
in
of
the
the plasma
small-sized,
Higher radial modes of any poloidal mode
have in general much more
favorable
power
deposition
pattern.
These modes become propagating in larger, denser tokamak plasmas.
The
total
radial
power deposition pattern is obtained from the
sum over all modes weighted
in
accordance
with
their
resistances such as those shown in Figs. 6 and 7.
the
loading
the
Enhancement of
resistance by toroidal eigenmode resonances must be
taken into consideration in calculations
since
loading
loading
of
the
weighted
sum,
resistance of a lightly damped mode (such as
the m=+1) is enhanced strongly by the resonance.
In Table IV the results
densities
are
summarized.
of
computations
At
high
for
densities
four
different
the enhancement
factors are close to unity. This shows that the damping strengths
at high densities predicted by the
those
of
Stix's
present
theory
tend
toward
theory for "pure" PFC and DSH damping. At high
densities (and also at high ion temperatures)
cyclotron
damping
at the location of the IHB resonance is strong enough to suppress
35
it
unless
(i.e.,
it
high
implication
far
proton
for
spite of the
protons
is
away
concentration).
This
has
a
presence
deuterium
of
plasmas,
a
the
few
percent
dominant
.essentially pure cyclotron damping. DSH damping,
energy
directly
in
deuterons,
should
damping under reactor-like conditions.
Poynting
damping.
The
power
dissipated
tabulated
which
is
deposits
a
given
amount
of
by
strong
for
the
this mechanism reaches
density.
The
enhancement
are 15 and 590 for the PFC and DSH damping for this case.
However the latter is likely to be an
significant
E-field
For
mechanism
dominate over PFC
This tendency is particularly
nearly 15 % at the lowest
factos
also
impurity
flux, greater power is dissipated at lower densities by
both mechanisms.
DSH
significant
future ICRF heating of reactor-like plasmas: in
inevitable
in
from the cyclotron resonance layer
effect
of
calculations
the
and
DSH
the
overestimate
because
the
damping is not included in the
validity
of
the
theory
is
questionable.
In Fig. 7(b) the time variation
enhanced
damping
is
shown.
of
theoretical
loading
The enhancement factor for the PFC
damping is taken to be 15 from Table III, but the factor for
DSH
dampng
is
arbitrarily
indicated in the table.
individual
peaks
is
appearance and the level
closer
taken
Although
again
of
to the experimental
with
to be 100 rather than 590 as
one-to-one
impossible
the
loading
results (Fig.
experimentally observed damping strengths
36
the
to
correspondence
of
obtain, the general
resistance
3).
are
are
much
This shows that the
not
incompatible
with enhanced.cyclotron damping.
The lack of mode structure in the loading curve was interpreted
by
some
researchers
as
an
unrelated
to
fast wave generation. Such "parasitic"
depo'sition
indication
of
anomalous
.loading was thought to be especially severe for antennas
electrostatic
shields.
However,
experiments is different.
the
only
mode
that
interpretation
power
without
of the present
At low plasra densities m=+1
mode
is
can carry significant rf power, whereas at
higher densities more propagating poloidal modes become available
to carry away rf power.
become
propagating,
Higher
are
radial
especially
modes,
case
also rises as seen in Fig. 6.
(No. 5)
overlapping
of
the
apparent
strongly
lack
damped
of
of
resistance.
0.8-2 ohms is in good
radiation
structure
modes.
The
observed
agreement
with
resistance.
It
is
that
most
of
the
is
When
loading
the
recalled
loading
that
caused
resistance
most
by
damping is
approach
theoretically
heating experiments are conducted at these high
indicates
also
In the highest density
sufficiently strong, the loading resistance should
radiation
may
efficient in this respect.
Thus, as the density level increases, the level
resistance
which
the
of
predicted
ATC plasma
densities.
This
the rf power is indeed radiated as the
fast wave, and that it is not necessary in the ATC experiments to
consider anomalous phenomena to account for the bulk of the power
coupled into the tokamak vessel.
In
the
preceeding
paragraphs
37
interpretations
of
the
experimental
.results
are
cyclotron damping due to
resonance.
presented
the
in
presence
terms
of
of the enhanced
the
two-ion
A few words of explanation are in order at this point
regarding their
relation
to
interpretations
based
conversion damping at the two-ion hybrid resonance.
.analysis
hybrid
is
upon
mode
The previous
based upon a simple model for the plasma dielectric
tensor that results in a second order differential
equation.
In
the limit of the cold plasma approximation the coefficient of the
highest
order
derivative in the equation vanishes at the hybrid
resonance,
and higher order terms must be
retained
the
propagation
of
wave
in
the
vicinity
analysis in the previous section deals with
under
which
would
than
arise
temperature
the resonance.
physical
the
weak
The
conditions
the
second
order
term
remains
more
the higher order terms. Such physical situations
when
is
that
the
high,
proton
or
concentration
the
parallel
is
small,
wavelength
Equation (9) expresses a quantitative criterion for the
of
describe
thermal effects, primarily PFC and DSH damping, are
strong enough such
important
to
hybrid
category under most
resonance.
plasma
is
ion
short.
presence
The ATC plasmas belong to this
heating
conditions,
and
the
wave
damping data are interpreted in this light * (However, modes near
their
onset with a long parallel wavelength may not satisfy this
criterion and the mode conversion may be important for them.)
*There are theoretical treatments 4,5,23) based
upon
mode
conversion. However, none of them solved boundary value problems
to obtan the E-field structure in a bounded plasma, and so were
only applicable when the wave absorption was so strong that no
waves reflected from the boundaries need be considered. The ATC
experimental conditions clearly do not correspond to such cases.
38
I.
4. Conclusions
The damping length (rather than the
fast
cavity-Q
factor)
of
the
wave propagation in ATC tokamak is measured directly at low
plasma densities. The wave damping strength
.measurement
is
found
deduced
from
these
to be much greater than that predicted by
the "pure" PFC and DSH damping theory. Asymmetric location of the
region of the strongest damping about the plasma center
in
contrast
to
the
predominance
reactor-like
also
the theoretical predictions. The discrepancy in
the damping strength is much smaller at higher
implies
is
deuterium
of
pure
plasmas
in
densities.
This
cyclotron
damping in dense
spite
the
of
inevitable
presence of a few percent impurity protons.
These
discrepancies
attributed
to
interpretations of
conversion
at
the
between
theory
the
two-ion
other
experiments
hybrid
and
hybrid
resonance.
based
resonance,
experiments
the
upon
are
Unlike
linear
mode
current results are
explained in terms of enhancement of the cyclotron damping:
hybrid
resonance
alters the polarization of the wave E-field in
such a way as to enhance
enhancement
These
is
both
asymmetric
explanations
concentration
the
are
the
with
PFC
and
respect
applicable
only
DSH
damping.
The
to the plasma center.
when
the
proton
is small and the hybrid resonance layer lies close
to the cyclotron resonance layer.
Satisfactory performance can be obtained
39
from
a
simple
wave
coupling
strticture without an electrostatic shield commonly used
in the past ICRF heating experiments.
antenna
represents
a
step
The
forward
success
towards
of
a
the
more
(presumably all-metal type) structure which could be
ATC
rugged
used
in
a
thermonuclear environment.
The major part of the power coupled into the tokamak vessel
radiated
as fast waves and there is no evidence that significant
power is lost through so-called "parasitic loading". This is
to
is
not
say that no rf power is channeled into plasma phenomena other
than the fast
effects,
wave
generation.
If
there
are
any
parasitic
they are not of major consequence in dissipating the rf
power.
A coupling efficiency (net
excess
90 %
of
is
coupled
routinely
power/incident
attained.
resistance is 1-2 ohms under typical plasma
The
power)
total
heating
in
loading
conditions,
and the wave generating efficiency (wave power/net coupled power)
of
90-95 % is
achieved.
deposited in plasma/wave
efficiency
of
The power deposition efficiency (power
power)
is
about
90 %.
The
overall
delivering the power to the plasma is the product
of all these efficiencies: approximately 75 % of
the
output power is actually deposited in the plasma.
Only that part
of
the
power
which
is
deposited
contributes significantly to plasma
efficiency 8)"
is
lower
in
generator
the central plasma core
heating,
and
"the
heating
than "the power deposition efficiency"
quoted here.
40
5. Acknowledgement
This work was done
research
staff
at
while
the
author
of
the
the Princeton Plasma Physics Laboratory.
The
author would like to acknowledge many
-Drs.
F.W.
Perkins
Drs. H. Hsuan,
providing
him
information.
and
T.H.
S. Suckewer,
T.
Stix.
was
helpful
a
member
discussions
with
He would also like to thank
Nagashima
and
F.J.
Paoloni
for
bolometric, spectroscopic, charge exchange and rf
The author is indebted
to
Messrs. A.J.
Sivo
and
R.S. Christie for construction of the rf equipment and to the ATC
operating crew for the machine operation.
41
References
J.
H.,
(1)Takahashi,
Physique,
de
Suppl. au n12,
Coll. C6,
Tome 38(1977)171
.(2)Adam,
J.
and Samain,
Nucl.
T.H.,
D.G.,
(4)Swanson,
(6)Takahashi, H.,
Phys.
Rev.
Lett.
36(1976)316
Fusion 17(1977)1197
Bull. Am. Phys. Soc., 22(1977)1165
Furth,
Ellis, R.A.Jr., Eubank, H.,
(7)Bol, K.,
E.L.,
Assoc.
Fusion 15(1975)737
(5)Perkins, F.W., Nucl.
R.A.,
EUR-CEA-FC-579(1971),
Rep.
Fontenay-aux-Roses
Euratom-CEA,
(3)Stix,
A.,
Johnson,
L.C.,
Jacobsen,
and Tolnas,
Stodiek, W.
E.,
Mazzucato,
H.P.,
Phys. Rev. Lett. 29(1972)1495
(8)Takahashi, H., Daughney, C.C., Ellis, R.A.Jr., Goldston, R.J.,
Hsuan, H.,
S.,
Nagashima, T., Paoloni, F.J.,
Sivo, A.J. and Suckewer,
Phys. Rev. Lett. 39(1977)31
(9)Adam, J.,
Hooke,
W.,
Chance, M.,
Hosea,
J.,
Eubank,
F.,
Jobes,
Sperling, J. and Takahashi, H.,
Getty,
H.,
Perkins, F.,
in Proc.
42
W.,
of
the
Hinnov,
E.,
Sinclair, R.,
5th
Conf.
on
Plasma
Physics and Cont. Nuclear Fusion Res., Tokyo, 1974, (Int.
Atomic Energy Agency, Vienna), 1975, Vol.II, p.65
(10)Suckewer, S., Princeton Plasma Phys. Lab., private comm.
(11)Jacquinot, J.,
Lett.
McVey, B.D.
and
Scharer,
Phys.
J.E.,
Rev.
39(1977)88
(12)Takahashi, H.,
Rep. PPPL-1545(1979), Princeton Plasma Physics
Lab.
(13)Bernstein, I.B. and Trehan, S.K.,
(14)Stix, T.H., The Theory
of
Plasma
Nucl. Fusion 1(1960)3
McGraw-Hill,
Waves,
New
York, 1962, Chapt. 5
(15)Takahashi, H., Bull. Am. Phys. Soc. 20(1975)1241
(16)Paoloni, F.J., Phys. Fluid 18(1975)640
(17)Colestock, P., Princeton Plasma Physics Lab., private comm.
(18)Takahashi, H.,
R.J.,
Hsuan,
H.,
Daughney,
Nagashima,
C.C.,
Ellis,
R.A.Jr.,
T., Paoloni, F.J.,
Goldston,
Sivo, A.J. and
Suckewer, S., Bull. Am. Phys. Soc. 21(1976)1157
(19)Greenough, N.,
Paoloni, F.J. and
43
Takahashi,
H.,
Bull.
Am.
Phys. Soc. 21(1976)1157
(20)TFR Group, 3rd Int.
Meet.
on
Theo.
and
Exp.
Aspects
of
Heating Toroidal Plasmas, Grenoble, 1976
(21)Studier, P.,
Hsuan, H.,
Smith, R.R. and Takahashi, H.,
Bull.
Am. Phys. Soc. 21(1976)1157
(22)Adam, J.,
Italy,
1972,
Symp. on Plasma
Heating
and
Injection,
Varenna,
C.N.R., Lab. di Fisica del Plasma e di Elettronica
Quantistica, Milano, Italy
(23)Jacquinot, J.,
Paper D4-1, 3rd Topical
Heating, Pasadena, Calif., 1978
44
Conf.
on
RF
Plasma
Figure Captions
Fig.
1
Schematic of the wave coupling structure (shown in part).
Fig. 2
Time variation of the wave amplitude (rf
envelope)
observed
by
two magnetic probes located 90 degrees around the torus away from
the
antenna
nearly
directly
and below(B)
above(T)
the plasma
axis.
Fig. 3
Oscillograms of the loading
toroidal
magnetic
field
resistance
computer
outputs.
The
is different for each oscillogram. The
locations of the DSH or PFC resonance surface are indicated.
Fig. 4
(a)Variation
toroidal
of
the
wave
damping
normalized
layer.
The
region
location
of
of
the
(b)Variation
of
bolometer
through charge exchange and
with
and
signal
which
radiation.
without
the
center
registers
The
extent of the plasma region is indicated.
45
at
or
PFC
The outer
87 cm.
power loss
difference
rf pulse is plotted.
positioned at 107 cm.
DSH
the minimum damping length is
asymmetric with respect to the nominal plasma
discharges
by
circumference as the toroidal field strength is varied.
The abscissa is the major radius
resonance
length
between
The nominal
limiter
is
Fig.
5
Oscillogram in the middle shows the perpendicular charge exchange
signals at three different energies. Sharp spikes are observed on
the two high energy signals coincident with the appearance
toroidal
resonance.
The
lower
oscillogram
the
initiation
spikes. The top diagram
of
the
shows
rf
the
main
body
plasma
peak
at
3 ms
pulse is coincident with the
relative
locations
plasma, antenna and cyclotron resonance layer.
the
a
shows the loading
resistance in the expanded time scale. The second
after
of
of
the
The space between
and the antenna is filled with a tenuous
plasma.
Fig. 6
Oscillograms of the Rg7-computer output. The
cyclotron
resonance
surface is located near the plasma center in all cases. Variation
of the central electron density for each case is also shown.
Fig. 7
Theoretical loading resistance corresponding to the
conditions
strengths
obtained
of
the
based
by
center
upon
oscillogram
minority
and
of
second
Fig. 3.
experimental
(a)Damping
harmonic
damping
Stix. (b)Minority damping strength multiplied by an
enhancement factor of 15 and the second harmonic damping strength
by 100.
Fig. 8
46
Theoretical loading resistance for the experimental conditions of
the top-right oscillogram of Fig. 6.
upon
minority
and
second
harmonic
(b)Minority damping strength
(a)Damping strengths
obtained by Stix.
damping
by
multiplied
is
based
an
enhancement
fact'or of 1.8 and the second harmonic damping strength by 5.2.
Fig. 9
(a)Variation of the
across
the
slab
of
E.-component
for
three
the
different
wave
values of the confining
falls
B-field : one for which the proton cyclotron resonance
the
field
electric
at
plasma center (0.9 m) and two others for which the resonance
is displaced by 0.06 m on either side of
half-width
is
0.17 m.
The
hybrid
the
resonance
effects on the E., but they are asymmetric with
center.
(b)Variation
of
a
very
has
slab
only small
respect
to
the
of the E+-field for the same three cases.
The hybrid resonance has strong
appearance
The
center.
steep
effects
gradient
cyclotron resonance layers.
47
on
the
between
E+.
Note
the
the hybrid and
Table Captions
Table I
The theoretical wave damping length normalized
circumference
for
the
toroidal
to
the
toroidal
eigenmodes shown in Fig. 7(a).
The normalized damping length is the number of times the waves go
around the
torus
radial(l),
before
their
poloidal(m)
and
identifies the resonant mode.
based
upon
"pure"
fundamental
damping.
electron
The
negligible.
one-twentieth
the
e-fold.
toroidal(n)
damping
and
theories
deuteron
damping(E)
resistive
damping(W)
values
are
computed
set
for
of
numbers
computed
for the proton
second
Landau/TTMP
wall
A
eigenmode
The damping lengths
cyclotron
cyclotron(I)
The
amplitudes
harmonic(II)
is
completely
are
lengths
these
in
modes
cylindrical waveguide made of smooth stainless steel
walls.
a
The
right-most column(T) lists the total damping lengths.
Table II
The theoretical wave damping length normalized
circumference
for
the
toroidal
See captions to Table I for
symbol
to
the
toroidal
eigenmodes shown in Fig. 8(a).
explanations.
The
electron
Landau/TTMP damping is completely negligible except where entries
are made.
Table III
Influence
fundamental
of
the
two-ion
hybrid
resonance
on
the
proton
cyclotron (subscript I) and deuteron second harmonic
(subscript II). damping.
four
different
Some computed quantities are
of
for
locations of the cyclotron resonance layer (Re).
n,, is the parallel refractive index, P the power
slab
shown
absorbed
in
a
unit height and depth normalized in such a way that the
Poynting flux is 1 MW
.J-2 /(kAa)
the
per
unit
enhancement
height
factors,
of
the
slab,
J1
and
Q the quality factor, and
Lb/L the damping length normalized by the toroidal circumference.
Table IV
Influence
of
fundamental
the
two-ion
hybrid
resonance
the
proton
cyclotron and deuteron second harmonic damping. Some
computed quantities are shown for four different
on-axis
on
plasma
density
(neo).
symbol explanations.
49
See
values
of
the
captions to Table III for
Table I
(1,
m,
n)
I
II
E
W
T
(1,
(1,
1,
1,
3)
4)
170
56
230
100
-
13
11
12
-
(1,
(1,
2,
2,
3)
4)
240
79
410
150
-
17
13
16
11
(1,
3,
4)
95
220
-
17
14
50
9
a
Table II
(1, m,
n)
I
II
E
W
T
(1,-1,
(1,-1,
1)
2)
76
29
1.7
2.4
-
-
55
27
1.6
2.0
(1, 0, 6)
5.0
5.4
160
9.2
2.0
(1, 1,10)
(2, 1, 1)
(2, 1, 2)
3.1
81
33
24
1.7
2.7
80
-
17
54
23
2.3
1.7
2.3
(1, 2, 8)
6.5
20
140
8.6
3.1
(1, 3, 8)
7
20
160
9.6
3.3
51
d
Table III
0.98
0.94
0.90
0.86
28
29
30
31
25
1.6
78
55
2.0
6.0
7.2
64
2.8
1.8
Jl/(kAa)
13
5.2
1.2
2.1
QI
QJ
560
8500
180
3400
230
6500
2100
120000
(Lo/L)z
(LO/L) Z
14
220
4.5
6.4
89
180
58
m
ni
P1t
P1
JI
kW/m 2
kW/mz
4.0
52
0.11
3100
Table IV
10'/m3
neo
ni
kW/m
kW/m'
P,
P3E
J/k
)2
JE/ (kAa )
(Lo/L)
(LD/L)1L
0.8
1.5
2.5
3.5
14
22
30
36
300
95
55
12
2.0
40
0.55
15
590
3.1
54
5.2
1.3
1.1
570
3900
150
1200
230
6500
360
26000
1.2
7.1
3.7
6.4
180
8.8
640
44
30
53
1.8
0
U-
C
CL
CC
Uj
cnc
ANTI-RESONANCE
,F-R ESONANCE
Pt
I
I
I
1
10
pb
ms
I
Fig. 2
II
-
--
-
"Ali
S
CD
z
0
4
.
-
4
It
I.
i
15
0
0
(a) Damping Length
S
S
0
10
-j
0
S
-J
5
S
S
5
SF
0
I
110
I
-90
100
RD1=2] (cm)
HI-
I
I
80
70
-]
Plasma
1.0 r
.8 k
E
.6k
.4 k
(b) Bolometer Signal
.2
0
I
110
I
I
100
I
80
90
R [22
Fig.
(cm)
4
I
70
-
=2
Plasma (Nominal)
=>~ Tokamak Center
Antenna
± CX Signals
-
450 eV
-
900 eV
IZ7,
1800 eV
RF
~1~
I
10ms
Loading Resistance
-0
Fig. 5
4I
.
01--!:!7 1711 !t
I,
II A
ii-I4
3.01
E-a
NA~
-
ARI
4h
2.01
V.
I I
I
I
I
0
0
a)
1.0
20
Fig.
6
t ms
30
(a)
(1,1,4)
(1,1,3)
6- 6
(1,2,4)
To tal
4(,2,3)
m=+l
-
2
m=+3
m=+2
20
26
24
22
30
28
t [ms]
(b)
1.2
Toto
0.8
m +
+I
C3\
0. 4
--
\
m=+2
- ----- -
m =+ 2
0
20
22
26
24
t [ms]
Fig. 7
28
30
I
fK( 2,1,1 )
(a)
(1,-I, I)
4H-
1,0, 6)
(2,112)
2
(1,
0
)
+
V
M
,+
M=+2
2
2 57
22
I
I
I
24
26
28
30
t Ems]
31
(b)
2
Total
rI
crn
I
0
\+
-
-
- -_-=--
m=I
20
22
24
26
t [ms]
Fig. 8
28
30
LO
882H
0
LU
U,
0
i
Iscc
w-
0
I',
0
E
0
C%
~8H
tn
0:
2
0r
LLJ
/1
I
- U,
0
U,
2
I.-
U
'I
U
a:
0
LU
-
+
w
88H
U,
0