A REVIEW OF RF HEATING
by
Miklos Porkolab
Preprint PFC/JA-78-3
Plasma Research Report
PRR 78/19
May 1978
Invited lecture presented at ''International School of Plasma Physics,
Course on Theory of Magnetically Confined Plasmas", Varenna, Italy,
Sept. 1-10, 1977.
ABSTRACT
The funadmental aspects of RF heating theory is reviewed and
a summary of recent experimental results are given.
All frequency
regimes are covered, from very low frequency magnetic pumping to
electron cyclotron resonance heating.
I
A Review of RF Heating
P
+ P
n1
a
+ AP
$
- P
(1)
loss
Miklos Porkolab
where P
represents ohmic heating, P
represents a-particle heating,
APs
Department of Physics and
Plasma Fusion Center
Massachusetts Institute of Technology represents supplementary heating, and
Cambridge, Massachusetts 02139, USA
Ploss is due to synchrotron radiation,
We review briefly the present
bremsstrahlung, and heat transport to
status of the theory and the experi-
the walls and the limiter.
mental results of heating toroidal
that in order to contain a-particles,
plasmas (mainly tokamaks) by radio-
we need approximately 3.5 Mamperes of
frequency (rf) power.
ohmic-heating current.
We shall dis-
We note
Assuming that
cuss a '-lmber of different frequency
this is provided, in most devices
regimes beginning with the electron
with Z ff
cyclotron frequency (i.e., f- 100
tary heating will be necessary to
GHz) and ending with very-low
achieve ignition conditions.
frequencies (i.e. f w.1-10 KHz).
at present the forerunner is neutral
Before we discuss the details of rf
beam heating, in-reactor-size plasmas
heating, let us briefly point out
beam energies of the range c
the importance of supplementary
500 key will be necessary.
.heating of fusion plasmas in order
sent such beams do not exists, and
to reach ignition temperatures.
In
1 some form of supplemenWhile
300-
'At pre-
the low efficiency of neutralization
tokamaks initially ohmic heating will above 150 keV dictates the 'use (and
provide a relatively hot (T e Ti
2 keV) target plasma. However,- as
development) of negative ion beams.
An alternative approach would be to
the temperature rises, the resisti-3/2 In addivity decreases as T/.
learn how to use the large amounts of
tion, losses due to radiation in-
high temperatures.
crease as z n2T . Hence, at some
true due to the following additional
point radiative losses overcome
reasons:
RV power available to heat plasmas to
This is especially
(1) With the exception of
ohmic heating, and the attainment of a few clean tokamaks, in most devices
ignition cbnditions by ohmic heating
alone is doubtful.
Furthermore, the
Z
> 1.
However, since radiative
losses are especially sensitive to
ohmic current cannot be increased by Zeff, impurities will have to be
arbitrary amounts or MUD stability
will be violated.
I,
greatly reduced in reactor grade
A simple relation plasmas so that Zeff < 2,
(and
between typical losses and power
Z
input is given by the following
However, the effectiveness of ohmic
relationship:
heating in. "clean" plasmas
1 is highly desirable) [1).
2
above T w 2 keV is greatly reduced.
"conventional" large (low-density)
The effectiveness of supplementary
device,. an intermediate device, and a
heating can be domonstrated by the
high density, compact device ("igni-
simple formula (valid for At > TE)
tor").
The expected magnetic fields
and machine parameters are also shown;
AP-
V AKnT)
2
TE
(2) including the necessary temperature
where IE is the energy confinement
changes which the "supplementary"
time, AP is the supplementary heating heating system must provide in order
power, V is the plasma volume, and
to reach ignition temperatures.
Here
A(nT) is the additional rise Inplasmawe assumed that eventually the contri2
energy. If we take T - C 0na , where bution of ohmic heating becomes neglic0 is a constant (as is observed in
gible as compared with the supplemen-
present day. experiments) then
tary heating power (with the exception
of the last case
AP c (AT)R C ,
where C1
.
"ignitor" [3)).
While an intermediate
the
to l'arge system will require 55-125 MW
of net additional heating power, a
Thus, we see
small system can "ignite" with only a
2
37r /C0 , and R is
machine major radius.
(3)
which is called
that the smaller the major radius,
few MW of supplementary power.
Of
the less supplementary power one needs course, in a smaller device materials
to attain ignition temperatures.
damage due to neutron bombardment may
Implicit in these considerations is
be unacceptably high for reactor pur-
that
C 0 is independent of T. Although poses.
Nevertheless, "ignitors"
at present this appears to be consis- should be considered as relatively
tent with experimental observations,
"cheap" research devices where a-par-
in future, hotter plasmas where trap- ticle heating (for example) and
ped particles will be dominant this
reactor-grade plasmas could be studied.
may no longer be so, and a deteriora- It should be remembered that these
tion of TE .may be expected.
powers are the power absorbed by the
Let us now examine in a few
i
plasma, and thus the source power will
cases of interest how effective
likely be twice as much as that shown
supplementary heating is.
in Table 1.
In parti-
Since the total heating
cular,.we shall consider three dif-
pulse length is a.few seconds, for all
ferent sized machines and in each
practical purposes we must consider
case we shall assume that the energy CW power.
2
confinement time scales as TE - na ,
A typical rf heating system is
.with the constant C
obtained from
-represented schematically in Fig. 1.
the best present date results, namely It consists of an rf source, a transAlcator A [2].
Table 1 summarizes
mission line, a vacuum (and dc) break,
the expected confinement times for a and some sort of antenna structure
3
either inside or fiush mounted in a
port.
Finally, the radiated power
comments regarding their main features.
This is given in.Table 3, in order of
must be transported toward the plasma' decreasing frequencies, all the way
or electrostatic waves, and finally
from electron-cyclotron resonance
heating to very low frequency magnetic
this power must be dissipated in the
pumping.
interior by means of electromagnetic
form of thermal energy.
Of course,
if too much power is dissipated near
We should note that while
ECRH is the most recent experimental
entry into tokamak rf-heating (at
the plasma periphery, it will quickly B = 20 kG) there are some
be lost and the efficiency of bulk
new ideas for very low-frequency
heating will be low.
Thus, assuming
that the total efficiency of absorp-
heating [4).
Let us now discuss the
main theoretical aspects and the
tion is at. best 70% and that at least status of the experiments in each of
We note that
20% of the power is lost in the
these frequency regimes.
transmission line at the source we
in the theoretical discussions we
would need at least double the power
shall present only linear mechanisms.
shown in Table 1.
It should also be
In many cases nonlinear effects may
realized that we cannot expect more
become important at high pump powers,
than 35-40% electric efficiency for
the total rf source, and hence the
and nonlinear effects such as trap-
ping, stochasticity, and parametric
required "plug" power may be four or instabilities may have to be taken
A recent review of
-five times that shown in Table 1.
One important aspect of plasma
parametric phenomena has been given
rf interaction, which is often not
by the present author elsewhere and
mentioned, is that in addition to
we shall not discuss them here
into account.
[51.
actual bulk heating, rf power may be
used for controlling the plasma
properties.
Some of these roles are
listed in Table 2, and since most
are self-explanatory (and some are
speculative), we shall not discuss
them here any further.
Clearly,
3)
Electron Cyclotron Resonance
Heating
A.
Theoretical Considerations
The linear theory bf electron-
cyclotron resonance heating applicable
to tokamaks has been examined recently
The
considerably more research work is
by a number of authors [6-8].
needed to determine which of these
renewed interest in this high fre-
techniques hnvc potential for prac-
quency tegime stems from new developments in the field of high-frequency
tical applications.
technology,namely the development of
nterest in RF Heating gyrotrons [9,10). Thus, it appears
Let us now list various frequen- that cw sources with power levels of
cies of interest, and also make some the order of P = 0.1-0.2 MW, f - 28
Megimes of
4
GHz, and pulsed sources with P v 70
kW, f = 65 Gilz are available now [11]
mode of propagation is useful
and cw sources with frequencies of
absorption coefficient at wu
f = 100 CHz and P = 100-200 kW expec-
is given by (using v2
ted to be available within three
R 2
B W
only when w
< w
pe
years [12].
ce
.
The
T/m)
v2
2
.
(6)
The linear absorption mechanisms
in this frequency regime are due to
For example, for f - fce
either finite values of k,, and cyclo-
GHz one finds that 2nn > 1 if
tron damping, or due to the inhomo-
Te > 0.4 keV.
geneous magnetic field.
is efficient only in hot plasmas.
summarize
Let us now
theoretical. predictions obtained by
this mode of propagation is not
Defining
effective in heating the plasma.
Extra-Ordinary Mode of Propaga-
b.
T - exp[-2inr)
Thus, absorption
It is also found that at w = 2wce
the main results of the
Antonsen and Mnnheimer [6].
100
tion
(3)
In this case E0 B, k0 B, and we
as the transmission coefficient,
are interested in absorption
A = (1-T) is the absorption coeffi-
near
cient.
Furthermore, the magnetic
= we
2
2
w
ce' WUH (where
,
+w2
is the upper-
field gradient scale length is de-
hybrid frequency).
fined as
accessibility is prevented by a
Direct
cut-off layer in front of -the
VR
(4)
B/RD
upper-hybrid layer when the wave
propagates from the low magnetic
where the spatial derivative is
field side.
taken along the major radius, R.
is given by
The cut-off layer
These relationships predict efficient
absorption of the microwave beam in
one passage when
.
W
w
W+
27m > 1.
(5)
ce- 1
2
+ +7
4W2
211
C
m2
ce
The wave may propagate to the
upper-hybrid layer by tunneling
The results of the theoretical
predictions are as follows;
a. Ordinary mode of propagation
In this case E0 |
.
IB,
k2.tB, where
IR the external magnetic
field.
Accessibility requires
that b0
>
W , and hence this
from- the low magnetic field side
*2 U W
2
.
Since in toka-
maks we = wpC. accessibility
exists when the wave propagates
from the high magnetic field
side [13] or when w = 2wce
Thus, we see that even if the
5
wave is launched from the outside in use w = 2w . In addition, one may
ce
the equitorial plane, accessibility
want to beam the microwaves at some
may be achieved by scattering around critical angle to optimize the k,,
the torus (between the chamber wall
spectrum for the 0-mode of propagation
and the plasma column) and penetra-
(although a spread in the k,, spectrum
ting from the high magnetic field
will automatically be introduced due
side.
to the finite launching horn-size).
i.
W- W
C~e
One finds then that if k,,
strong absorption by cyclotron
damping takes place if w
On
date on ECR! heating of tokamaks have
been obtained in the TM-3 tokamak
In these experiments, the
[11).
freqincies used were in the range
+ 0.
f
w - 2wce
35-65 CHz, and the power was
injected by open ended waveguides.
Absorption can be efficient even
if k,
Experimental Results
The only experimental results to
Wce does not take
place since IE
ii.
wc .
B.
0,
0 efficient
the other hand, if k,,
absorption at w
#
0 and 21m > 1 where
T1 = Rg W
At low densities (n < 10
13
-3
cm
) most
of the energy ended up in a high
energy tail.
vte
(8) (n
c c2
C
At high densities
1013 cm
) efficient bulk
electron heating was observed at both
Again, this absorption mechanism is
efficient if the plasma is relatively
hot (i.e., T ; 0.5 keV).
iii.
W
w - w
and
at w m w
w = 2w
.
To date heating
has not been observed.
Typical pulse lengths were At
Wu
= 1 msec
and maximum power levels at 70 GHz
uh
In this case heating occurs by
were P = 60 kW, which raised the
absorption of the Bernstein wave
peak electron temperature in the cen-
which is a consequence of linear
ter of the discharge by as much as
mode-conversion of the extraordinary
ATe = 150 eV.
mode at a) Wuh [14).
The maximum absorption
Alternatively, efficiency for bulk electron heating
Bernstein waves may be generated by
in these experiments was estimated to
parametric decay processes also [15). be 30%.
There has been experimental
Accessibility is best by propagation
evidence that the electron energy
from the high magnetic field side,
confinement increased with electron
or by tunneling if W2 <
temperature, at least in the range
pe
, or by
ceo
the so-called z-hole effect [16].
From these considerations one
T
u
100 to 500 eV.
Whether these
trends would remain at temperatures
may conclude that the best strategy
above a few keV, remains to be seen.
is to use the 0-mode in tokamaks
In summary, while the mechanism of
with WPC < Wce, and when w
absorption was not determined, the
> we
6
initial. ECRII results are encouraging, W - th (0) and k_ +
and we expect to see a big increase
effects are included,
in the research activities in this
regime.
Whcthcr enough CW power can
If hot plasma
the slow-wave
.converts into an ion plasma wave
before the resonance layer, i.e. where
be developed to heat large tokamaks
to ignition by the use of ECRH alone
remains to be seen.
(11)
w = 3k Lyti'
Nevertheless,
one can consider ECRH heating to
and this ion wave then propagates
provide power for radialprofile
outward.
modification in order to improve MID
absorbed by "ion-landau" damping in
stability.
the inhomogeneous magnetic field
The ion wave is either
[24,25) or is soon converted once
4. Lower-II brid Range of Frequencies more into an ion-Bernstein wave which
A. Theory
then propagates back toward the center
in the lower-hybrid range of
and is absorbed by harmonic-ion-cy-
frequencies we may consider heating
clotron damping at the next exact
at the lower-hybrid resonance fre-
cyclotron harmonic layer [13).
The
quency (or slightly above if tempera- accessibility conditions is roughly
ture effects are included),
Wh
mwh pg(l+w2pe 1w2ce)-1
[17)
2
n i> 1 +
(9)
22 2
W/ W
pe ce 'Res
(12)
ck,,/w is the index of
by the process of mode-conversion
where n,,
into ion plasma (Bernstein) waves
refraction, and the right-hand-side
[13,17] or we may consider heating by is to be evaluated near the resonant
electron Landau damping if w > wth
[18,19,20). In both cases one
layer.
assumes that the so-called slow-wave
accessibility condition must be
When such a layer is not
present in the plasma, a more exact
satisfied, namely
( ++ k-2
I
k
(10)
cJ
W/w
=wn,,y
2 2
/l +nil(y
- 1)
(13)
is excited at the surface by a slowwave structure, and the wave propa-
where y 2
gates inward along "resonance cones"
(which are trajectories of the group
is especially important when one
velocity in the
2, r plane) [21-23).
w /wcewci. Equation (13)
considers an inhomogeneous plasma,
or when one desires to heat by elec-
Since the k., spectrum is fixed by
tron Landau damping.
the slow-wave structure, kL = k
r
increases as the wave propagates
case, one relies on the fact that if
radially inward until at r - 0,
in the outer layers of the plasma
In the latter
the kl, spectrum is "tailored" so that
7
(14a)
c/vte >> 3p,,
B.
Summary of Lower-Hybrid
Experimental Results
and near the center
In recent ypars there have been
several large-scale lower hybrid
(14b)
C/vte
experiments performed in tokamaks,
and some of these investigations -are
then absorption occurs near the cen2
ter (where
v2
still going on at the multi - 100 KW
(T /m )). I* fact, level [30-32].
in a hot plasma (i.e. Te
The great interest
10 keV) it in this regime arises from the parti-
may be difficult to satisfy Eqs. 13
cularly advantageous technological
and 14a simultaneously, and at these
aspects of LURF heating, namely that
temperatures we may have to abandon
one can use phased-arrays of wave-
the slow wave as a useful mean s to
guides mounted in ports as sources of
deliver the rf energy to the core of
the plasma column.
However, one
the microwave power [33,34).
It has
been shown recently that such a
could then use the whistler mode
"grill" structure behaves as a slow-
(fast wave) and rely on electr on
wave structure [35], and the theore-
Landau damping for absorption [25).
tical work of Brambilla [34) allows
Alternatively, sufficiently strong
one to design grills for different
rf fields could flatten the distri-
machine parameters (the "Brambilla
bution function and allow pene tration code").
[19,20).
sources are available, or can be
Because the path of wave- pene-
tration is long, (typically
L
= (mi/m e) a, where a is t
In addition, ample power
built upon demand, with frequencies
up to 10 GEz.
e
In most cases one would
use f v 1-5 GHz for the frequency.
plasma radius and Lz is the path
The most recent experiments at the
length along the magnetic fIel d)
lower hybrid frequency were carried
nonlinear effects in the outer plasma out on the Princeton ATC tokamak at
lyers are particularly dangerc us,
f
especially parametric instabil ities,
0,11, were used as a phase-array
solition formation, and scattc ring
system [30,31].
by drift wave fluctuations [2 F,28].
Wega experiment loops phased appropriately are being used to inject
Again, some of these phenomen i were
= 800
MHz where 2 waveguides phased
At Grenoble in the
paper and we shall not discus S them
the rf power at 500 MHz [32). In
preparation at present is the general
here. We note, however, that in
Atomic Doublet 11 experiment where a
addition to parametric effect s
slotted slow wave structure will be
[27,28) stochastic rf fields could
used to launch waves at w > w
also play an important role in non-
heating by Landau damping will be
linear ion.heating [29).
attempted. At MIT, preparations are
discussed in our recent reviec
, and
8
under way for a 100 kW split wave-
represent T
guide experiment on Alcator A.
From the past experiments the
showed little or no heating,
the OVII on CV lines showed
following conclusions may be drawn:
good heating, although typi-
a. -The phased array grill works
cally less than the perpen-
as predicted by the Bram-
dicular charge-exchange
billa theory; namely, while
measurements indicated (i.e.
AT w 50-100 eV).
in a single waveguide most
of the power is reflected,
in at least 2 waveguides
driven out of phase (0,180)
in Wega in some cases a
the reflection was reduced
density increase of 50% (or
to about ten percent, as
more) was observed.
In
f.
The total energy absorbed
particular, no tuning elements were needed, which is
by the bulk of ions was
difficult to estimate; how-
an important technological
ever under reasonable
advantage of this type of
assumptions 15-20% could be
coupling system.
accounted for by bulk-ion
In the
future four (or more) waveguide grills will be used.
heating.
g.
In ATC a definite threshold
Perpendicular charge-ex-
power and density for heating
change results are similar
was observed [31].
in both Wega and ATC; namely,
upon application of
was decreased, the threshold
P
power increased from P = 10
100 kW, typically
ATi
c.
e.
While in ATC the density
increase did not exceed 10%,
predicted by theory.
b.
near the edge)
100-150 eV increase
In
particular, as the density
kW to P > 100 kW, until
was observed.
neither parametric decay
While in ATC parallel
(monitored by a probe at
charge-exchange measurements
the plasma edge) nor plasma
were inconclusive, in Wega
heating were observed.
gqod agreement with the
a theoretical analysis it
perpendicular measurements
was concluded that in ATC
were obtained.
the heating took place at
d. Doppler-broadening measure-
(r/a) v 0.3-0.5 when
ments of OVIT, CV and CIV
1 Z wo/w
impurities showed similar
heating was studied only
when 1 Z Wo/wlh Z 1.3.
results in both ATC and
Wega.
In particular, while
the CIV measurements (which
< 2.
In Wega
h. More recently, in Wega
heating of electrons was
From
9
also observed [36].
It is
2
not clear, however, whether
such an electron heating
was due to impurities or
2A
2
N
(M -
(15)
cil
where
direct heating by the pump
wave, or by the parametric
VA - AB/[47rn(m e4n ici
/W p)c
p1
(W
decay waves.
We also note that in addition to bulk is the Alfven speed, R is the major
ion heating an energetic perpendicu- radius, a is the minor radius, and
lar ion tail was also obsekved, with
a life-time of 50-100 psec.
It was
V a P + jM/21
concluded that this ion tail was
produced on the surface of the plasma and where p, M, and N are integers.
Similarly, an- increase in soft and
In paz,.icular, v is the "perpendi-
hard x-rays were also observed.
cular" wave number consisting of
In summary, while the initial
the .radial mode number p and the
results are encouraging a clear
poloidal mode-number M, and N/R = k,,
understanding of the physics of LHRH
is the toroidal mode number.
The
is not yet available.
In particular, important prediction of Eq. 15 is
the fate of the rf power-flow is not that fast-wave propagation is
understood, and clearly more experi-
possible only if the density and
mental work is required.
radius is such that V i
,
N
0, and
>
Eq. 15 becomes
5. Ion Cyclotron Range of Frequencies (CRF)
na2>
2
2 2
B
F
A. Theory
5xl
m
2
si
e
2(16)
Heating near the ion-cyclotron
frequency may be done via the slow
wave near w L wei or the fast wave
near w V tci [37-40).
Thus, fast wave propagation demands
sufficiently large and dense devices,
In tokamaks
especially if w < w i is desirable.
the most important regime is w = 2 Wei
The heating rates are given by the
and heating is achieved by the small following
expressions [42):
left-hand component of the fast wave a.
W
(magneio-acoustic wave).
The slow-
wave is expected to mode-convert
near the tokamak surface, and it may
W0 i
1
4w
2
i
0 cc2 a
E+ 2
(17a)
be used mainly in heating stellara-
tors. The dispersion relation for
fast waves is given by the following
expreenion [40,41):
We notice that since in a single ionspecies plasma E + 0 at w - ci , the
power absorbed, P
0. However, in
a two-ion species plasma this is no
*
ii
10
longer so.
the RF voltage (namely, RM depends on
b. w w 2w
Q, and Imax is limited by voltage
breakdown).
P i2 R I1+1
2 ..R+E |2
16iw r
where E+
Vr
0
are excited it is desirable to "lock
onto" one mode. and "stay with it"
E+ is'the gradient
during the rf pulse.
Ilowever,
since
81nT /B2 is the ion
of E+, and 0
"beta".
(17b)
Thus, if cavity modes
(19)
Q - (wo/Aw) - 2no/An
We note that while Eq. 17b
predicts weak absorption in present
tokamaks,
in future high density, hot
where (An/no) is the fractional den-
tokamaks the absorption is expected
sity change, we see that as the den-
to be efficient.
sity changes in time one jumps from
c.
one mod(. to the next, and a feedback
Ileatin1,
jylectron-Landau Dapng
system may be necessary to track the
This has been calculated by
This is a rather difficult
Stix [40,41), and again, in hot toka-
mode.
maks it is expected to be an effi-
technical task if many modes are
cient absorption process, while in
present in the system.
machines with T Z 1 kev it is a
a reactor size plasma hundreds of
An advantage of
rather weak process.
In fact, in
modes may be excited as it will no
this process is that it allows opera-
longer be feasible to track modes
tion at w < w i, where it can be
[41).
cofisidered a type of "Alfven wave
the effects of the rotational trans-
One may have to consider then
form also [43].
heating" (rather than ICRF).
e. Impurities
d. Cavity Modes
Recently a number of authors
Once the fast wave propagates,
It is possible to set-up standing
discussed the effects of even a few
waves around the major circumference
percent of impurities present in the
of the torus, the condition being
plasma upon the fast wave propagation
kJZ/271
integer.
This allows set-
[38,44,45].
For example, a few per-
ting up modes with relatively large
cent of hydrogen present in a deu-
values of the "quality factor", Q.
terium plasma presents an 11
Since the power absorbed depends on
the coil impedance 'Ri, and is given
funda-
mental resonance (w = wci(H+)) when
w
2w ciD +) which may provide a
mechanism of efficient power absorp-
by In terms of the current I, by
tion by the H+ minority rather than
SRH
(18)
the D majority species. Absorption
by H+ leads to a large and fast H
it turns out that relatively large
-tail which will have to be confined
values of Q allow effipient power
and its energy transmitted to D+ by
absorption for acceptable values of
slow collisions.
Additional problems
11
may be created by setting up an ion-
w L 2wci(D ) heating experi-
ion hybrid resonance layer near the
plasma edge which may lead to mode
ment was carried out [47].
Coupling experiments carried
c.
conversion and large damping near the
out ot low powers in the
edge.
French TFR device showed
Additional enhanced absorption
near the ion-ion hybrid layer may
lower-Q values (by one to
also be due to large gradients in E
two orders of magnitude)
between the cyclotron and the hybrid
than predicted by theories
layers [45).
which ignored impurity
effects [48]. Again, the
importance of small amounts
B.
Ex erimental Results
Experimental results on ICRF
of impurities were blamed
heating of tokamaks have been
for the results, especially
obtained .n the past in a number of
in D2 plasmas.
devices, and further experiments are
d.
The Princeton ATC experi-
planned on the TFR device in France
ments showed the best ion
and the PLT device in Princeton, USA.
heating to date using ICRF
In particular, the following is a
power near w
brief summary of the main features
In this experiment good bulk
of these experiments.
ion heating was obtained.
a. On the Princeton ST device
In addition, a significant
the ICRF experiment showed
ion tail was also produced.
ion heating, but there were
However, it was not deter-
also impurity problems
mined experimentally whether
present [46].
this ion tail consisted of
In particu-
lar, the impurity level
deuterium or hydrogen.
increased with injected
maximum input energy was
P Z 2 kJ which was compar-
energy.
When the total
external rf energy exceeded
the plasma thermal energy,
namely
b.
2w ci(D+) [49].
tPRF > nkT : 200
The
able to the thermal energy
content of the plasma. At
higher energies the m = 2
Joules, disruption occurred
MHD mode was intensified and
[46J.
no experiments were carried
These results were
blamed on bad banana orbits
out.
and fast particles hitting
efficiency was about 40%,
the wall..
and the energy absorbed by
On the Soviet TM-l-VCHI device efficient heating of an
ions was comparable to that
energetic H tail wan detec-
heating.
red (rather than heating of
anisms in these experiments
D ) when an
The 'maximum heating
obtained by neutral-beam
The heating mech-
2
were not determined, and Q-measure-
where 0
=
C
8'n T /B is the total
ee
±
ments and their comparison with
beta of electrons,
theory have not yet been published.
E
In summary, the ATC ICRF experiments
(We note that Eq. (21) may be used
are most encouraging and further
both at w < wci and wi<
improvements can be expected in the
Since the damping is weak, high Q-
forthcoming PLT experiments.
values may be established for one or
~0
2T /m and
is the electric field intensity.
< 2 wcV)
two cavity modes which may then be
6.
Alfven Wavelleating
tracked by a feedback system.
A.
particular, by lowering the frequency
Theory
This is the regime of low
the number of modes in reactor-size
frequencies,
such that w < w , typically 300 KIz - 10 MHz in toka-
maks.
In
plasmas can be lowered by orders of
magnitude from the ICRF regime.
Two types of modes are of
In the case of the shear Alfven
interest, namely the compressional
Alfven wave (e
k vA) and the shear
Alfven wave (w
k,.vA).
wave one would use a coil structuresuch that the resonance
The com-
pressional Alfven wave with a small
w
= k.,(r )v
!r B +kZB
/4np
(22)
component of k,, may be described by
and it
Eq. .(15),
may be used to heat
would be achieved at some radial
electrons by Landau and transit-time
position, r. Then a mode conversion
damping in reactor size plasmas [41).
into short wavelength electroqtatic
The frequency to be used is expected
waves may take place [50) or MHD
to be w rw ci/10, so that direct
multi-spectral absorption may take
acceleration of 08+ impurities may
place
The critical density
be avoided.
[511, both leading to absorption
of the energy.
In particular, in the
mode-conversion model [50) the con-
may be set at
verted wave would have to travel
na2 ; 5x10
7
cm-.
(20)
around the torus once or twice
(similarly to the lower-hybrid wave)
For example, a plasma column with
and absorption by collisional,
3
n ; 5x10. , a = 100 cm would satisfy Landau, or parametric effects may
this condition (such as the forth-
coming TFTR device at PPL).
The
damping rate and power absorption is
given by Stix as follows [40,41'):
16F [
J
I
2 '60c
2 2
xe
;-1
take place.
k.tej
(21)
B. Experimental Results
There are to date two experiments where shear-Alfven wave heating
have been studied: (a) In proto-cleo
at Wisconsin a doubling of T
and T
was observed when a power of
P m 10 kW was applied in an initial
13
low density plasma with temperature
T
v10 eV [52].
of f
100 Klz are contemplated.
However, after the
heating strong "pump-out" was. obser-
ved in this device.
B.
(b) In the
Experiment
The only results to date have
Heliotron-D torsatron at Kyoto the
*been obtained by using ion TTMP on
following results were obtained [53]: the Petula tokamak at Grenoble [55].
applying a power of P = 400 kW,
This experiment confirms the preTe
changed from 150 eV to 300 eV and T dictions of theory, namely good bulk
changed from 20 eV to 5Q eV. A maxi- ion heating has been obtained
mum heating efficiency of about 30%
was observed. The physical mechanisi
responsible for the heating remains
unclear.
(AT
- AT,,
-
T /2) with
(AE/B)
0.015 without major disruption or
impurity problems.
In particular,
the results were in agreement with
theoretical predictions, and scaling
7. Transit Time Magnetic Pumping
A.
Theory
should improve with larger machine
size.
This technique has been studied
years ago extensively by Canobbio at 8. Very Low Frequency Heating
Grenoble [54].
The electron TTMP is
similar to the compressional fast
This case was studied originally
by .Schlhter in the 1950's and the
wave heating proposed by Stix more
main absorption mechanism considered
recently, namely toroidal eigen-modes was due to ion-ion collisions [56].
are set up and absorption takes placc The absorption-rate obtained is given
by modulation of the magnetic field.
by
The power absorption is given by
v
P/vol = 3
2
V n-It Q
Vh
-8B J
-
nT .
(23)
W
(24)
22
VIi
where (At/B) is the modulation depth.
where R is the major radius, and B
The advantage of this heating
is the modulated magnetic field
technique is due to the possibility
intensity.
of using a few kHz for the frequency
Electrons are preferen-
tially heated since w/k, = vte is
so that the rf coils could be placed
set up by the coils.
outside the vacuum chamber.
Frequencies
However,
contemplated are typically f = 1-10
since we expect that w Z Vi, we see
Mlz.
that in order to obtain absorption
On the other hand, ion TTMP
relies on forced (evanescent) rf
rates comparable to the ion collision
fields such that w/k, v vti, and
frequency one needs AB/B * 1, which
since vti
is energetically not practical.
scent.
heated,
VA the mode is evane-
Here ions are preferentially
More recently Canobbio proposed
and frequencies of the order that the efficiency of this type of
14
magnetic pumping could be improved by pursued.
coupling to a low-frequency resonance, for example that due to the
Acknowledftements
VB drift, and in that case consider-
This research was supported by
able improvement over Eq. 24 could be the US Department of Energy.
obtained [4].
This work is still in
progress and preliminary indications
are that significant heating (i.e.
References
AT/T = 1) may be obtained by modula-
1. D. Meade, Nucl. Fusion 14, 289
tion depths of AB/B =
.1-0.2.
(1974).
2.
9. Summnry and Comments
M. Gaudreau et al., Phys. Rev.
Lett. 39, 1266 (1977)
It is clear that the physics
of rf heating is complicated, and
3.
B.
Coppi, Comments Plasma Phys.
Cont. Fusion, 3, 47 (1977)
there are still too few experimental
results available.
However, it
appears that in the available experi-
ments efficiencies of 20-40% have
4. E. Canobbio, Proc. Conf. on
been obtained (where by efficiency
Plasma Physics and Controlled
we mean ri = A(nT)/P(incident).
Nuclear Fusion Research, Berch-
Furthermore, in many cases improved
tesgaden,
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Vol. 3, p. 19
W. Germany, Oct. 1976,
5. M. Porkolab, Nucl. Fusion 18, 367
machine size and hotter target
plasma, and theoretically efficien-
(1978)
cies of n = 60-70% appear achievable. 6. T.M. Antonsen, Jr. and W.M. ManThe efficiency of rf power generation
heimer, "Absorption and Trans-
is good, typically 30-70%, and thus
formation of Electromagnetic Waves
technologically speaking rf heating
in the Vicinity of Electron Cyclotron Harmonics", NRL preprint, to'
is attractive for the purposes of
reactor-grade plasma heating.
More
experiments are clearly needed in
be published in Phys. Fluids
7. 0. Eldridge, W. Namkung and
order to learn more about the physics
A.C. England, Oak Ridge National
of wave penetration, heating and
Laboratory, ORNL/TM-6052, Novem-
impurity problems.
ber, 1977
Finally, more
materials research will be necessary
8. I. Fidone, G. Granata, R.L.
to solve the problem of coil
Meyer and C. Ramponi, Association
shielding at low frequencies. Since
Euratom Report EUR-CEA-FC-912
at present no clear "winner" is
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9. N.I. Zaytsev, T.B. Pankratova,
M.I. Petelin, and V.A. Flyagin,
Radio Eng. Electron Phys. 19
--
15
103 (1974)
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21.
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R.L. Berger, F.W. Perkins,
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V. Cinzburg, The Propagation of
M. Porkolab, Phys. Fluids 20,
2058 (1977)
28.
R.L. Berger, L. Chen, P.K. Kaw,
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Also, C.F.F. Karney, Stochastic
Phys. Fluids 19, 995 (1976)
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Also, A. Bers in
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[Editrice Compositori, Bologna,
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S. Bernabei and I. Fidone, in
Plasma Phys. 2, 212 (1976)
klectromagnetic Waves in Plas-
17.
(1976)
'Third
329 (1972)
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I. Fidone, Phys. Fluids 19, 334
Also, Alikaev et al., Sov. Phys.
78 (1965)
14.
P.M. Bellan and M. Porkolab,
Phys. Fluids 17, 1592 (1974).
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12.
R.J. Briggs and R.R. Parker,
Phys. Rev. Lett. 29, 852 (1972).
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R.K. Fisher and R.W. Could,
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M. Porkolab, S. Bernabei, W.M.
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F.J. Paoloni, R.W. Motley, W.M.
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Hooke and S. Bernabei, Phys.
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32.
P. Blanc, et al., Proc. of Conf.
16
on Plasma Physics and Controlled
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Nuclear Fusion Research, Berch-
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tesgaden, W. Germany, Oct. 1976,
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Vol. III, p. 59.
43. C. Cattanci and R. Croci, Nucl.
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P. Lallia, in Proceedings of the
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35.
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Physics Laboratory Report PPPL
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38.
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5th Conf. on Plasma Physics and
H. Pacher and the WEGA team,
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T.H. Stix, The Theory of Plasma
42.
p. 65
V.L. Vdovin, et al., in Third
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J. Adam, et al., Euratom Reports
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EUR-CEA-FC 579 (1971 and EUR-
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T.F.R. Group, in Third
p.
349
Int.
F.W. Perkins, Symposium on
meeting on "Theoretical and
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July, 1976, p. 87.
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T.H. Stix, Nuci. Fusion 15, 737
(1975)
41.
1974, Vol. II,
47.
Waves (McGraw-Hill, New York,
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Rev. Lett. 34, 866 (1975)
CEA-FC 711 (1973)
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47 (1976)
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37.
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Takahashi,
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et al., Phys.
39, 31 (1977).
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ibid, Phys. Rev.
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(1975);
(Editrice Compositori, Bologna,
ibid, Phys. Fluids 19,
Italy; 1976, p. 156)
(1976);
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ibid, Phys.. Rev. Lett.
Heating and Injection, Varenna,
(1976).
Italy, 1972 (Editrice Composi-
51.
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370
1924
36,
1362
J.A. Tataronis and W. Grossmann,
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Z. Phys. 261, 203 (1973);
p. 83)
ibid, 217
Also, in "Third Symposium on
Also in Proc. Sec. Conf. on rf
Planma heating in Toroidal
Heating of Plasmas, Lubbock,
17
Texas, 1974;
55.
also, Nucl. Fusion 1_6, 4 (1976)
52.
S.N. Colovato, J.L. Shohet and
J.A. Tataronis, Phys. Rev. Lett.
53.
published in 1978
56.
A. Schl'tter, Z. Naturforschung
12a, 822 (1957).
37, 1272 (1976)
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54.
Petula Group, Grenoble, to be
FIGURE CAPTION
E. Canobbio, Nuclear Fusion 12,
561 (1972) and Course on the
Fig. 1) A schematic representa-
Toroidal Reactors, Erice, Italy, tion of an rf heating system.
p. 71, (1972)
Also, in "Symposium on Plasma
Heating and Injection", Varenna,
Italy, P. 14 (1972).
18
ANT ENNA
R.F
TRNSsMISSION
SOURCC NETWOXtK
MATCHWu
NETWOR11
WAVES
VACUUM
WIN DOW
Fig. 1
19
h (0)
(Te;n)
n
a
R
xE
(cm)
(i)
(W)
(sec)
8
2
4
1xIO'4
6
4x10
14
15
1xlO15
2
AT
(keV)
10
V
AP
cm
OVW)
6.4x10
125
&6
1
4
2
8
8x107
54
0.2
0.5
0.2
5
4x105
4
Table 1
Examples of Ilypothetical Tokamak Parameters Near Ignition Conditions
B - toroidal magnetic field; n - average density; a = minor radius;
R - major radius; xE . energy confinement time; AT
AT
AT
tempera-
ture increase needed to achieve ignition (assuming that the ohmic heating
contribution will be negligible); V - plasma volume; All
power absorbed to
achieve the AT indicated.
1.
Bulk plasma heating
2.
Tail-Heating for a 2-component tokawak
3. Radial temperature profile modification
4.
Measuresient of the transport coefficients by local heating
5.
Feedback control
6.
De-trapping of trapped particles
7.
Enhanced dc resistivity
8.
DC currents due to quasi-linear effects
9.
RF plugging of mirrors
10. Density build-up
1.
RF confinement.
Table 2
Possible Roles for RF lcating
20
Nature (Type)
Electron Cyclotron Resonance
(ECIU)
Coupler
Sburces
Waveguide
or horn
Gyrotron
Characteristic
Frequency
f e,2fc' fuh
-100 Gilz
Comments
(CW sources)
Presently
available:
P ' 200 kW,
f
28-65
CII z
Next 3 years
P
200 IJI,
f ~ 100 GIIz
Lower Hybrid
Range of Frequencier, (LIURF)
fLh
pi
Waveguide
Array
(grill)
1-8 Glz
Klystrons
Presently
available,
0.25-0.5 M%'/
tube,
1 1M possi-
ble, CW
Coils
(Ridged
waveguides
may be
possible)
Tubes
Presently
available,
0.5-1 11
f < fci
1-10 MIZ
Coi]p
Tubes
W-S
Transit time
mapnetic pnrnpinp (TTIP) electron or ion
0.1-1 MHz
Coils
Tubes
MW-S
Very low frequency (VLF)
1-10 klz
Coils
outside
chamber
Generator
MW-S
Ion-Cyclotron
Range of frequencies (ICRF)
f ,i2f
5-100clIz
Alfven wave:
shear or com-'
pressional
.
Table 3