PFC/JA-86-5
Simulation Model
for Lower Hybrid Current Drive
Paul. T. Bonoli
Ronald C. Englade
February 1986
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
Submitted for publication in:
The Physics of Fluids
This work was supported by the U.S. Department of Energy Contract
No. DE-AC02-78ET51013. Reproduction, translation, publication, use
and disposal, in whole or in part by or for the United States government is permitted.
By acceptance of this article, the publisher and/or recipient acknowledges the U.S. Government's right to retain a non-exclusive,
royalty-free license in and to any copyright covering this paper.
-1-
Simulation Model for Lower Hybrid Current Drive
Paul T. Bonolia
Ronald C. Engladeb
Plasma Fusion
and
Centera
of
Laboratory
Research
Electronicsb
Massachusetts Institute of Technology
Cambridge, MA 02139
(Received
ABSTRACT
A simulation model
for
steady state,
lower hybrid current drive is
described which incorporates a relativistic, one-dimensional Fokker Planck
calculation and a toroidal ray tracing code.
Two-dimensional (vj) effects
are included in the Fokker Planck analysis in the form of a large perpendicular electron temperature due to pitch angle scattering.
in the parallel
refractive index of the lower hybrid waves,
toroidal geometry
effects,
injected RF waves
at
is
proposed
high phase
as a physical
velocity
(ve
An increase
arising from
mechanism
<< v 0
c)
whereby
can interact
via the Landau resonance with -electrons at low phase velocity (vp$
Numerical results
relevant
presented which demonstrate
to
the
Alcator
the dependency
ciency on various plasma parameters.
PACS Numbers 52.40.Db, 52.65.+z, 52.35.Hr
C and
of
the
PLT
experiments
current
drive
3
ve).
are
effi-
-2-
SIMULATION MODEL FOR LOWER HYBRID CURRENT DRIVE
I.
In recent years,
Introduction
toroidal current generation in tokamaks using driven
2
lower hybrid
wavesl,
devices..3-7
In experiments performed on PLT 5 and Alcator C6 , the primary
has
been demonstrated
circuit of the ohmic heating
current decayed
inductively
(OH)
on
a
variety
of toroidal
transformer was opened and the plasma
with
a
typical
time
scale
L/R -
t
Lp/Rp,
where Lp is the plasma inductance and Rp is the plasma resistance.
injection of a nearly unidirectional
the plasma,
the current decay was
maintained at
a constant
period of time
value
T L/R (thus
stimulated
spectrum of lower hybrid waves into
stopped and the toroidal
with zero
loop voltage
for
current was
at least
a
insuring that the plasma internal inductance
and vertical equilibrium magnetic
experiments have
Upon
field were no longer varying).
These
considerable interest in the development
of
realistic models for lower hybrid wave propagation and quasi-linear wave
absorption as a framework within which to understand
the present steady
state current drive results and extrapolate them to future devices.
The
model proposed in this paper is a response to that interest.
If the
total
maintain steady
current was
injected
state
RF
power was greater
current drive
observed to increase
fraction of ( TL/R)-'
in PLT
at a
rate
than
that
and Alcator
C,
required
to
the plasma
which was a power-dependent
and the loop voltage became negative.
A treatment
of lower hybrid current ramp-up is beyond the scope of the present work,
but we note that a generalization
of the
steady
state
simulation model
described below can account for several important features of the observations.
41
The major code modifications
include linkage with a transport
-3calculation for the time evolution of the poloidal magnetic field,
consistent calculation
of
the
DC electric
parallel electron distribution
function,
field
and
its effect
the explicit
and
on
inclusion
spatial diffusion as a loss mechanism for suprathermal electrons.
treatments of current ramp-up can be found in Refs.
selfthe
of
Other
42 and 43 and a numer-
ical model that is similar in some aspects to that of Ref.
41 has been
developed by Valeo and Eder.44
A puzzling result of the PLT 5 and Alcator C6 experiments is that the
parallel phase velocity (along the applied magnetic
jected lower hybrid waves
is much greater
than
field B0 ) of the in-
the
phase velocity
re-
quired for electron Landau damping v1 (ELD), and the subsequent generation
of RF
In
current.
fact,
the
v 0 $ c, whereas
v n(ELD)
temperatures of
(1.0-1.5)
(2 Te/me)1/
2
injected
phase
velocities
Here
key.
is the electron thermal
c
is
the
speed
of
light
-
speed, Te the electron temperature,
are able to interact
with plasma elec-
trons and thus significantly transfer their momentum and energy.
anomalous Doppler
ve
The question immediately arises as to how
these high phase velocity waves
posed explanation 8
satisfy
<< c for the observed central electron
< 3 ve
and me is the electron mass.
wave
for filling the spectral
instability.
9
P1 0
One pro-
gap in velocity space is the
Another mechanismIl
comes
from con-
sidering the injection of RF waves into a plasma where the electron distribution includes
as well
the effects
of magnetically
trapped
as an electron tail which may or may not be
electron orbits
"sliding away"
12
under the influence of an electric field which is not too large relative
to the
critical
runaway
scattering by density
proposed 16 as a means
or
Dreicerl
3
fluctuations1 4 ,1
of
injected lower hybrid waves.
lowering
electric
5
the
(at
the
field.
More
tokamak edge)
recently,
has been
parallel phase velocity
of the
-4-
paper a simple linear mechanism
In this
varia-
which incorporates
tions in the parallel wavenumber k11 - k . B0 / 1B0 1 of the injected waves
in the
due to toroidal geometry effects is utilized to effectively fill
gap in velocity space.
Recall
that the poloidal mode number m - rke is
a constant of the wave motion in cylindrical geometry.
longer conserved
to vary due to the poloidal
eg (n/R) and B - e Br + e
0
+ (n/R)
dicity (m
B(b
Taking k -
gradients.)
erkr
+ ee (m/r)
magnetic
with r
$ the toroidal
shear.
Here
the minor radial
angle.
The
(r, 9, h) are
e
position,
canonical
(kr,m,n) with kr the radial wavenumber, n - Rk
the
usual
the poloidal
momenta are
given by
the toroidal mode number,
(r/a)cos e I the major radial position, R0 the torus major
R - Ro 11 + .
radius,
a the torus minor radius,
of the
torus.
effect
This
tracing technique.
The
has
and
been
numerical
e - a/R0 the inverse aspect ratio
using
calculated
results1 7 ,18
a
indicate
decreases
in significant
in
the
parallel
phase
velocity
increases
v
In the present work it is found that the toroidal variation in
with a one-dimensional
5 6
,
ray
toroidal
kq by as much as factors of three or more over initial values,
ally observed
+
fkr Br +
BA + et B$, it can be seen that kp -
and
toroidal coordinates
conservation requires m
(Momentum
will vary due to the combined effe cts of toroi-
/BI
variation)
angle, and
m is no
in a tokamak equilibrium due to the poloidal inhomoge-
neity of the magnetic field B (r,e).
(m/r)B
However,
in
resulting
-
W /
k .
k g coupled
Fokker Planck analysis reproduces the experiment-
levels of RF current generation.
The model also repro-
duces the correct qualitative dependencies of the steady state RF current
drive efficiency
netic field
R e(10
Be and relative
average plasma density,
4
cm-3 )I,(MA)R (m)/P(MW)
waveguide phasing
Ip is
the total
Ad*
toroidal
on
toroidal
mg-
Here ne is the line
plasma
current
main-
-5-
tained by the RF,
of
and Pin is the injected RF power.
The highest values
are obtained numerically at the largest values of B
T
lest values of A.
and the smal-
Larger toroidal magnetic fields allow higher parallel
phase velocity waves to propagate
proved accessibility19,
20
to the
central plasma
and smaller waveguide
because
of
im-
phasings produce "Grill"
spectra 2 l which are richer in high phase velocity components.
The pro-
posed model does not reproduce the experimentally observed density limit
for lower hybrid current drive. 3 - 7
other wave processes
hybrid pump wave.
However this phenomenon may be due to
such as the onset of parametric decay of the lower
Parametric decay
is
found both
experimentally
22
and
theoretically 2 3 to have a density threshold at fixed pump wave frequency
of
W /
W of~~ 2, where wb
frequency,
i-
(4
n e2
+/ 4e
/
-
)1/2 is the lower hybrid
ci(Ie
mj)1/ 2 is the ion plasma frequency,
(4n ne e2 /me)1/ 2 is the electron plasma frequency, and Q ce
-
)pe-
e B / (mec)
is the electron cyclotron frequency.
This paper is organized as follows.
The Fokker Planck analysis is
described in Sec. II, including a discussion of how perpendicular velocity
(vi) effects are included by retaining an effective perpendicular electron
temperature "T1 ".
The
toroidal
ray tracing
calculations are described in Sec.
lation of
cient.
the flux surface
III.
averaged,
and
Shafranov
equilibrium
Section IV describes the calcu-
quasi-linear
RF diffusion
coeffi-
Numerical results are presented in Sec. V, including a comparison
of current drive results with and without the toroidal upshift phenomenon
and the dependence
field,
of the current drive efficiency on toroidal magnetic
relative waveguide
phase,
RF power,
perature, and electron tail confinement.
plasma density,
electron tem-
-6-
Model Fokker Planck Equation
II.
A.
Derivation
The starting
point
for
the
Fokker
Planck
is
analysis
the
two-
dimensional equation,
af e
--
af e
-
-eE
+ -
afe
Fj
Drf (PH) -
fe
+ C (fe)
(pj,pP) is the electron distribution function,
where fe
field and
Ep is the DC
24
diffusion coefficient
is
a
electric
field.
Drf(Pn) is
time
for tail
electrons,
source of particles at low parallel momentum.
of the
Balescu-Lenard
collision
operator.
pH / p, P2 .
coordinates where p -
2 +
(1)
is the
quasi-linear
in the plasma,
and rs 8(pp)
is
a
C (fe) is the Landau limit
25
2
pi (pl)
the
due to the presence of RF waves
confinement
6 (ps),
to the applied magnetic
(parallel)
component of momentum perpendicular
rt (p p)
rs
+
-
Using
cylindrical
and considering
(4,p)
fast test
electrons scattering from a background of slow field particles (ions and
the relativistically
electrons),
can be
correct expression for C (fe)
written as 26,27,
-(1-
+
-1
1
C (fe)
- 2
v
p2)
--
-- p2 6p
f
(
2
e +
p
}
a
y
- fe) + p3
6p
Z
+ 1
X
2
,
vo - (e4log A / (2n ne vn)
where Pe m meve,
Zi is the ion charge state,
(2a)
(2b)
log \ is the Coulomb loga-
-7-
rithm, p = Ymev,
This corresponds
neglected.
=
1 + p2 / (mec)2 , and me is the electron
terms
(2a),
Eq.
obtaining
In
rest mass.
Y
ymev,,,
POl
fe is a Maxwellian
with
mentum variable
Fe (P )
fe (p ,p)
p2 ) were
/
>> y2
can be integrated over perpendicular momentum
and (2)
by assuming that
2
to the high velocity limit of the collision
operator in the region of RF waves (p
Equations (1)
(y2
order
to
effective
an
function
in
the perpendicular
spread
momentum
of
(2
me
mo-
T,)1/2,
2/2 meT
exp (-p 1
e
-
2 7=e TI
(3)
Fe (pl) is a function of parallel momentum only and is normalized so that,
co dp 2 fdpI fe(pj,p)
-
(4)
1
We further approximate y to be independent of p1 as,
2
y2
+
Pff
(mec)2
dp2
Applying n f
(5)
(m c)2
2meT.
e
where p2
Pjeff
2
p2
to
This approximation is valid provided
Eqs.
and
(1)
(2)
and
utilizing
Eq.
2T
(3)
<< mec 2.
yields
after some lengthy but straightforward algebra
-
a
Me
+ -eE 1 -
at
bp I
We
Drf(p H)
apgIp
6
1
C (Fe)
= -
voPe -
2
op 0
I
Te
Tj
6
Fe + C (Fe)
3P0
poy [YI1 (Pit)
Fe
-
(6)
+ rs6(p 1)
-
(p 3)
+ (1
+ Zi)
12 (pit)]
Fe (Pit)
-8-
Te
I + Zi
1
13 (P ) + -
[
+ y -
e
1 4 (P
In obtaining (7), terms to order (Y2 Pe 2 /
The derivatives
glected.
space by using p, -
a
2p2
ap
p
ap
- -2p p
--
pp, p
p
8
P
6p
in (4,p)
-
space
were
2) have again been netransformed
to
(p,,
p2)
p2 (1-42 ), and
8
8
-
(7)
2
2
T
2
2
pe 2 P 2
+ p
-
8p
The integrals Il to 14 are given by
dp2
1-
12
13
(8a)
exp (-p /2meT )
2c
2
dpi
3
P3
dp2
d
3
(-) 2
P
exp
2.
p1
p2
-p2/2m
exp (-p2 /2m Tj)
(8b)
(8c)
T,)
dp2
14 -
(8d)
exp (-p /2meT)
d
5.
p
These expressions
arbitrary values of T1
the limit Tj
=
can be evaluated
in terms of error
functions
both within and outside the region of waves.
for
In
Te and y + 1, the collision operator given by Eqs. (7) and
2
(8) reduces to the familiar nonrelativistic, one-dimensional form
-9-
V
2
0
2
B.
1
p3
p
2+Z
-
C(Fe)
e_
-
e
-p2
-Pe
2e
+
+
fp g F,
3
--
2
a
Fe
(9)
}
Moment Calculation
Once fe is known at some position x in the plasma,
the RF
current
density Jrf, the RF power density Srf due to quasi-linear electron Landau
damping,
the power density
Sd,
due to
electrons on bulk plasma electrons,
collisional
and
slowing down of tail
the power
density
S
due
to
electron tail losses can be calculated from
Jrf (x)
Srf()
Sd (x)
S
(x)
f7
-
f*
-
rf
r
-
f
-
1dp
ndp2
o
dp%
dp (nev)fe
dp
(10a)
nemc 2 (T1)()
eme
t
(10b).
rf
dpA nemec2(y-l)(-)
at
n dp
dp
nemec 2 (-
,
(10c)
,
(10d)
d
-afe
J)(-)
at
has been used.
where the normalization of Eq.
(4)
be rewritten by taking account
of Eqs.
Jrfi
-
(3),
fdp , (neev ) Fe(p,)
srf("-) - SdpR nemec 2 (y-f)
--
Drf((i)
W
(6),
These expressions
and (7)
and we
can
obtain
,
(Ila)
,
(11b)
-10-
dpo nemec 2 (y-,)
Sd (x)
(11c)
C(Fe)
Fe (PH)
(x)
S
=
dp I
f
nemec 2 (y1)
,
(lid)
2g (p,)
Numerical Solution
C.
Equations (6) and (7) are solved numerically by taking aFe /
3t
0
-
and ER - 0, corresponding to the steady state current drive experiments
Defining Fep (pjl) as the solution for
that are being modelled.
Fem (pg) as the solution for p,, <0, Eq. (6) is integrated from
to pH and from pj
to p1 , assuming Fe (+
=
<
-00)and Drf (+ )
pq >0
PH - -<
0.
The result is
[Drf (Pp) + D1 (P)I
-
Fep (pl) + D 2 (pl) Fep (PH)
SFep (p'ip)
dp/H
F
+
PH
0
--(P/ 0)
,
(12a)
,
(12b)
,
(12c)
*
(12d)
Fem (P#) + D 2 (p") Fem (Pl)
[Drf (Pp) + D1 (pp)}5P P
P H
Fem (P /u)
dp 1 1
(
-
'g
--
D1
(Pu)
=
-
p
1.
Te
2
VoPe - Y [T1
I
D2 (PH)
0
Te
vope -
2
T
1+
2
1
Zi
13 (PH) + -
POY [Y1 1 (PH)
+ (1
2
p2 p
2
4
+ Zi) 12 (P)]
(
1
-11-
and (12b)
The system of equations (12a)
for Fep (pH)
is then solved numerically
The solu-
and Fem (PH) using standard shooting techniques.
tions are normalized subject to the constraints,
Fep (0)
=
Fem (0)
P
= dp, Fep(Pd) +
(13a)
dp
Fem(P)
III.
A.
=
1
(13b)
Wave Propagation Model
Toroidal Ray Tracing
An extensive literature exists
on the application of the ray equa-
tions and eikonal method 28 to the study of lower hybrid wave propagation
in tokamak plasmas.17,18,
a wave
packet
which
where we take w, k,
29
satisfies
The ray equations
Do
(x,
k,
)
-
0,
If one utilizes the Hamiltonian
30 they take on a particularly simple form in
Recall
from
the
Introduction
defined as the usual toroidal coordinates
cally conjugate momenta.
give the trajectory of
the local dispersion
and Do to be real.
nature of these equations
toroidal geometry.
-31
that
and (kr, m,
n)
(r,
0, 4) were
as the canoni-
Using these variables, the ray equations can be
written as 30
dr
--dt
= -
d9
--dt
-
d6
--dt
= -
6 Do /kr
(14a)
6 Do /w
6D
/
am
(14b)
-
b Do /w
6D 0
/
bn
(14c)
6 Do /w
-12-
/ br
dkr
---dt
3D 0
6 Do
aw
dm
Do
e8
(14d)
(14e)
dt
8 Do /w
dn
-dt
Do /
b
0
o D/
(14f)
.
o
Equations (14e) and (14f) clearly display the fact that for a tokamak
equilibrium with poloidal
poloidal mode
number
( 9)
will
inhomogeneity and axial
not
be
conserved
( )
whereas
symmetry,. the
the toroidal
mode
number is a constant of the wave motion.
B.
Dispersion Relation and Wave Damping
The dispersion
relation
used
for
lower
hybrid
waves,
including
electromagnetic and warm plasma effects is 17,31-33
D0 (x, k, w)
P6 n6 + Pn4 + P2 n2 + P0
-
-
Po - F
[ (n
P 2 - (e
+ e ) (n
P4 -
-
(15)
0
2
e)2
-)
+
2
E
3 -2
--
P6 - - I ---
3
v2
-
- --
c2
2
2
--
+
8
v
---
c
g2
Oe
Here ni - ki c/w is the perpendicular refractive index, nj
is the parallel
k
k . B /
IB
refractive index,
.
vi -
(2Ti
/
k -
k1 m,)1/
2
qBI,/
kRB
ki
I
=
kp c/w
kl|,
and
is the ion thermal speed, Ti is
-13-
the ion temperature,
assumed that
B -
and mi
line ion
Larmor
electrons were
Pe -
ve / Oce
where
ez B0
local approximation).
is the
ion mass.
Bo is
In deriving (15)
constant
and
The ions were taken to
orbit
relative
treated
as
to
a
strongly
perpendicular
radius
.
E_,
Pe)2
el,
<<
or
(straight
wavelength).
[(k
was
16/ a I(WKB
be unmagnetized
magnetized
is the electron Larmor
ki >>
it
The
where
1],
are the
and ey
elements of the cold plasma dielectric tensor of Stix 19 evaluated in the
limitQ 2 >G
2
>
ce
E
Exy
ce)
1 + ( e
S-
-
They are given by
ci*
/ w)2
(
2
(u
w)2
,
(16a)
w 2
-
(16b)
(060)
e
'$Se
An evolution equation
for the power (erg
ray path is integrated simultaneously
/ sec)
flowing along the
with the ray equations and has the
form
dP
-
- - 2y P
Y-
(17a)
T
dt
Ye + yi + Yc
Note that
(17b)
focussing terms in (17)
a power density.
have
been neglected
The damping decrements
electron and ion Landau damping and
electron-ion Coulomb collisions.
ye and
so that P is not
yi are due to
resonant
Yc is the nonresonant damping due to
yT is calculated in the usual
way by
writing
D (x,
k,
w -
iYT)
-
Do
(x,
k,
w -
i yT)
+
i
D(0 (x, k,
a,= e,i,c
w -
i yT)
-14-
where Do
and
D(a)
for YT
real
are
0,
=
and
Expanding D for small yT and D(a) with D0 (x,kw)
-
SD(a) (x,
e,i,c
T =
T
k,
the
are
imaginary
relation due to the various damping mechanisms.
parts of the dispersion
a
D(a)
0 yields,
w)
(18)
o Do /
bw
The expressions for D(a) are given by 31,32
2
D(e) = -
(mec)2
(n
2
D(i)
-
+ n2)
(19a)
43
2nl/2
(-
n
)
exp (-
-
)
(19b)
kvi
22
2
Dc
-
n2 +
[
4
n2
n2
(19c)
ce
plasma
For a
with a
single
impurity
species,
vei is
by
given
34
2
vei - -
(20)
vo Zeff
where vo was defined
in Eq.
(2b)
and Zeff
(14)
and
dispersion
ion charge
is the effective
state of the plasma.
The ray
equations
relation
integrated numerically using the predictor-corrector
(15)
-
(16)
were
algorithm described
in Ref. (35).
The accuracy of the integration was checked by calculating
the variation
of
component of DO*
Do
(x,
k,
w )
from
zero,
normalized
to
the
largest
The maximum deviation was found to be typically < 10-4.
-15-
C.
Model Toroidal Equilibrium
expansion of the magnetic field.
(e - a /
0 <
Shafranov's
aspect ratio
36
A straight
cylindrical
plasma model
is used for the lowest order configuration.
Ro+ 0)
into
is divided
from
equilibrium is obtained
The toroidal
current
a
r < a and a region
region
carrying
inside
the
The plasma
limiter
radius
where the current density is assumed to vanish
extending from the limiter radius to the ideally conducting chamber wall
a
< r < b.
The
cylindrical
equilibrium
quantities
are taken
to
be:
(i) For 0 < r < a,
exp (
ne(o)(r) - (neo - nea)
)
exp (En)
2
exp (-Er
Te(O)(r) - (T
exp (%r
-
2
/a2 )
- nea
(21a)
+ Tea
(21b)
1
-
/a 2 )
exp (-Fe)
- Tea )
I - exp (-e)
2
exp (-Fr
/a2 )
-
exp (-E )
+ Ti
T1 (o)(r) - (Ti0 - Tia)
1
b (r/a)
1 +
P(O)(r)
ne()(r)
-
(ii) For a
<
,
(21c)
exp (-r,)
/ qo)(r/a) Bo
(
B O)(r)
-
2
ITe(o)(r) + Ti(o)(r)1
(21d)
(21e)
r < b,
b- r
ne(0)(r) - n
-
b -a
(22a)
-16-
b-r
b-a
Te(o)(r) - (Tea - Teb)
+ Teb
(22b)
+ Tib
(22c)
b -a
b -r
- Tib)
T i(0)(r) - (T
b -a
(22d)
B@(o)(r) - Be. (a/r)
(22e)
0
P(O)(r)
Teo, Tjo, and neo are the central (r
of
values
a)
-
electron
q(r) -
zero.
be
plasma
volume average
is
qo
current
Fe,
Finally,
2 Ip/(ca).
and nea are the
and
The electron density at r - b
of
value
the central
the
safety
factor
)(r)) and the constant Eb is determined by specifying
rB0 / (R0 B
the total
Tia,
ion temperature,
temperature,
ature at the chamber wall radius (r - b).
to
Tea,
Tib and Teb are the values of electron and ion temper-
electron density.
is taken
0) values of electron tempera-
and electron density.
ture, ion temperature,
limiter (r
-
values
using
I
and
,,
and Fn are
<n o)
(r)>,
the
relation
determined
<TeO)(r)>,
and
Be
by
-
B ()(a)
specifying
<T (0)(r)>,
the
where
2
(F(r)> 2---
Fa rF (r) dr.
a2
The formulation of B(r,e) in the region 0 < r
<
a proceeds as follows.
Using Eqs. (6.21), (6.24), and (6.29) of Ref. (36)
er Br (r,9) +. e
Be (r,9) + e 0 B$ (r,$)
t(r)
Br (r,e)
-
B(0)(r) sine
-
r
,
(23a)
-17-
d
r
B
(r,0) -
(r)
B(o)
-
+
(r)
B50)
A (r)
+ A(r)
-
Ro
r
B
(r,e)
=
B(o) (r) +
4Ro
'
The asymmetry
-
Bio)(r)]cos
9
, (23b)
dr
d
(r) + A (r) - B(o) (r)] cosE
dr
B(O)
parameter A(r)
and
the
flux
surface
, (23c)
shift A
(r)
are
given by Eqs. (6.11) and (6.32) of Shafranov 36,
<p(O)(r)> - p( )(r)
A(r)
-
8
[Bo) (r)12
<[B
2
[B
)(r)]2>
f
0
+
(r)]
(24)
(21
-*
2
r
r
A(r) -
1
+--
[A (r') + 1] dr/
-
. (25)
Ro
and the volume average <F (r)> has been defined above.
The lowest
order
toroidal
B(O)
field
(r)
is obtained
by
considering
the radial component of equilibrium pressure balance,
1
d
-
p(0)(r) -
-
[J 0) BCO)
-
Law for JZO),
Using Ampere's
to the boundary condition B o) (r
p(0)(0)
B(o)
o)]
j(o) B
c
dr
(r) - B
-
1 +
B(0)
-
0)
-
p(o)(r)
(r)
B2 /8,n
0
be obtained
subject
B0 ,
F2
-
0
can then
q2
0
1
--
x
l
1
i+ F
[ 1 -()2]1/2
/
r 2 /a2
(26)
-18-
Here p(o)(0) - neo (Teo + Ti ) from Eq. (21e).
B (r,9) in the region outside the limiter radius
The solution for
V. B
a < r < b is found from V x B - 0,
conditions on B at the plasma surface r
=
a that the normal
component of
The solutions
- 0.
(r - a)
surface current J
B is continuous and the
together with the boundary
0,
=
correct to O(E) can be written as 36
1
Br (r,B)
a
(1-a
/r
AS
a
1
a
-
+ -
Be (r,e) - Bpa (-)
-a
+
2
Cos
a),
(24-26).
Equation
0
Aa
placement of
the
above and Eq.
1
A(r) - Aa
+ -
(1
a (21d)
flux
+ B /ea / Bea)1
A (r
is
cos 9
=
surfaces
to
is
for
I x [
I - (r/Ro) cos91
(29)
found
< r
Bp.,?Ba
using
<
using Eqs.
a)
- B,(o) (r
calculate
a
(28)
B
a
and Bt
a),
used
6 (r/a)]a
(6.31)
r2
f ln
-
2
B /b
-
(1 - a 2 / r 2 )}
a
a
(-)
[b-BC0 )(r)/
and B
/ B
a
1 - (a/Ro)
(28)
B
-
Ba { 1 +
A(r
A
) + - -
a
Aa
1
+ (1 + a 2 /r 2 ) (Aa + -
Here A-
1
In (r/a) l
Bpa cos
2
r
B6 (r,@)
(27)
)}
a
a
2
+
/
Ba / B
-
+ -
) (Aa + -
(1+a2/r)
a
a
1
2
ln(r/a) +
R
2
2
R
{
sine
Bp
-
-
o)(r)/ (r/a)1a,
Eq.
The
dis-
using
Eq.
Aa
-
Ro
-
x
a
a
(26).
computed
b is
of Shafranov 36
(r/a)
+
(1-a
2
/r 2 )
1
[A, +
-
+
2
.
(30)
-19-
where p
Ti - Ti (p),
we
ko)
(xedge,
ko and
wavenumber between
calculating
the
flux
(in velocity space)
ko + &0k (k 1 -
power
This
0).
(14)
with parallel
edge
the plasma
near
launched
density &
energy
propagating
(x,
k)
assumed
is
in a tube of
and (17)
section a with a varying group velocity Vg.
total
uniquely
labelled
can be
We consider an amount of energy per
according to the ray equations
constant cross
tube, the
and
coefficient
surfaces
flux
that magnetic
with the toroidal radial variable r.
to propagate
( p),
For simplicity, we neglect the Shafranov
due to the RF electric fields.
unit time AP
for
a procedure
outline
quasi-linear diffusion
shift and assume
Te
-
Te
ne( p),
r + A(r) cose.
-
section
surface averaged
-
Formulation of RF Diffusion Coefficient
IV.
In this
ne
Thus
p .
radius
constant
surface of
on a magnetic
are taken to be constant
The density and temperature
Inside the
(x,
=P
/
k)
[a Vg (x,k)] can be expressed as the sum of the electrostatic energy density I E
(x, k,
t)1 2
/
8-T and
kinetic
the
energy
density
associated
with the coherent motion of oscillating plasma particles. The result is19
&Z (x,
k) - -
where do
point x
|
-)
8n
(x, k, w) /
a D
inside
E (x, k,
t) 12
(31)
,
aw
tube,
the
diffusion coefficient
(n2 + n2)
an
2
is the dielectric
expression
in the presence
of
24
for
resonant
the
a strong
constant.
magnetic
At a
parallel
field
can
then be rewritten as
8,e
Drf (x, v
ko)
2
-i-rk
me
k2
N (x,
k)
8
k0___)
aw
v )
(32)
-20-
where k
is
time t
volume
the
AV(r)
ray
(14).
equations
surrounding
magnetic
a
Suppose a packet of wave energy flowing along the
surface at radius r.
this
enters
k0 through
and
a differential
Next consider
ray tube
from x
determined
at
volume
+ Att and location x1.
at
/ and leaves
location
te and
time
If several such transits occur, the
ef-
fective incremental quasi-linear diffusion coefficient acting on a typical
electron within AV(r) is easily seen to be
(rf,
tn + Atn
8r2 e2
1
me
AV(r)
Z f 6(u - kn
n
tn
kO) -
)(
) n
dt
2
a
The integral in (33)
is over transits.
where the summation
k2
)
is readily
evaluated using the numerical solutions of the ray equations (14)
Finally we must
lrf(r,
sum
k, representing the
assumed to
(k)
k0.
full
be launched
The
that possible
electric field
result
ko) over
v
range
of
the
sufficiently many wavenumbers
Brambilla
at the plasma edge,
of
this
and (17).
with
summation process
power
aP
(xE
is Drf
spectrum
edge,
P(kO)
k1)
(r, v. )
P
Note
interference effects due to the addition or subtraction of
vectors
tE
associated
with different
values
of
k0 have
been ignored.
V.
A.
Numerical Results
Numerical Details of Model
The plasma cross section is divided into forty equally spaced radial
zones and the Fokker
radial locations.
(shown in Fig.
1)
Planck system Eqs.
The
Brambilla
power
(12)
is solved at each of these
spectra
for
Alcator
C and
PLT
are divided into approximately fifty intervals or bars
-21-
< Inid' 8.37.
of power between 1
A grid spacing AnN
1<In iI < 1.99 and Ang - 0.22 is used
< InIg<
for 1.99
for each tng interval,
ray trajectory is launched
0.044 is used for
-
16.37.
A single
weighted to the approThe quasi-linear
priate power according to the input Brambilla spectrum.
RF diffusion coefficient is calculated on this no grid following Sec. IV.
solved in the region of RF waves using a
Planck equation is
The Fokker
parallel velocity (vp) grid with nonuniform spacing Avj
-
vp &Ag / ng.
parallel velocity grid is used in the region below the RF waves 0 < v p
(c/16.37) with
spacing Avg -
uniform
0.05
(c/16.37)
(i.e.
twenty
A
<
grid
intervals).
Based on the numerical 2-D Fokker Planck solutions reported in Refs.
2,37,
and
38,
and on those obtained
provided them by D.
by the authors
the effective
Hewitt,
from a code kindly
electron temp-
perpendicular
erature is modelled by the following prescription for vl>O:
O < v, < v
T1 (keV) - Te
v 22
-v1
2
1
Ti (keV) - min [(1 + 2
Ti (keV)
) Te, 50
x
Tel,
V1
v I < Vh
ve
- 50 x Te
,
vh < vi < v2
Here vl and v2 refer respectively to the minimum and maximum values
of parallel phase
Do -
Drf
(v1 )
/
velocity for
(vo
/
v
typically taken to be 3.0.
2)
which D(v)
and er is
Do (v
a
threshold
is reached then vh
-
<
ve) 3
>
where
value
which is
Vh corresponds to the first velocity location
below v2 for which D(v) N) < eT and vj indicates the first
above vI for which D(v1 )
/
eT.
If f(vN) does not fall below er until vl
v1 and vj - v 2 .
the negative velocity region.
velocity location
Similar expressions for T, apply in
-22-
The choice of Ti - Te below the resonant region requires some discusIt
sion.
is well-known 2
concert with
that one effect
angle
of pitch
scattering
in
strong quasi-linear diffusion in a region O<va<v R<vb is to
enhance the perpendicular spread of nonresonant particles with O<v
This enhancement
is especially pronounced
<va.
"boundary layer"
in a narrow
close to. va and results in a greater magnitude of the parallel distribu-
We
enhancement.
attempted
have
than would
region
resonant
in the
tion function
determine
to
the
be
the
importance
with
several
below the resonant
reasonable
regions,
estimates obtained by Fuchs,
current exceed that
than 15%.
prescriptions
including an
et al.
45
for
an
Secs.
effective
adaptation of recent
no
this
of
effect by redoing the model examples discussed in the following
and VC
with
case
VB
TL>Te
analytic
In no case did the calculated RF
obtained in the corresponding model example by more
Careful examination of the code results reveals that changes
in the parallel distribution function at various radii directly attributable to an enhanced nonresonant T1 alter the wave damping and hence the
self-consistent quasi-linear
coefficient
diffusion
manner as to produce nearly counterbalancing
Drf(r,vo)
in
such a
changes in terms of contri-
butions to the RF current.
The form
qt-
roP where Tois
electron energy
the suprathermal
a
constant
confinement
current
time
carrying
whose
the
of
in
Ref.
39
for
the
is
value
on
plasma
bulk
electrons
particular velocity dependence represents
results obtained
was
confinement
electron' tail
the
for
are
the
and
of
to
be
order
of
the
p-3. 3 9 . Thus
well-confined.
a numerical fit
ratio
chosen
to theoretical
the path lengths
stochastic field lines of energetic and thermal electrons.
This
along
To be fully
consistent with the concept of electron propagation along lines of force,
-23-
the expression
be multiplied
for -rl should
and
by (ve/vN)
-TO
should
be
Qualitatively these
taken to represent a bulk particle confinement time.
changes tend to compensate each other.
The Brambilla spectrum of
denoted by Feo (corresponding to Drf - 0).
(12)
Equation (12)
waves is then damped and Drf is calculated based on Feo.
is then
The spectrum of waves is then damped
the quasi-linear distribution Fe.
This process
now based on Fe.
again,
and Drf is calculated
approximation to
This gives the first
solved using Drf (Feo).
of Eq.
solution
initiated by using the Maxwellian distribution function
is
equation
Planck
Fokker
the
solving
for
process
The numerical
is repeated
until the radially integrated RF current moment does not change.
B.
Model Example for Alcator C
In Figs. 2 and 3 we show model results for an Alcator C steady state
current drive
neo
=
-
used were n
The parameters
scenario.
5
x 1i13 cm- 3 ,
l.ne, nea " 0.15 neo, Teo - 1.5 key, Tea - 0.03 keV, Teb - 0.005
keV, Tio - 0.7 keV,
R0 - 64 cm,
17.8 cm,
Tib
Tia - 0.03 keV,
relative
parameter 1, were qo
-
1.0,
phase
waveguide
5 x 10-3
&n -
0.005 keV,
a -
b -
16.5 cm,
total plasma current Ip - 170 kA, deuterium plasma,
toroidal magnetic field B, - 100 kG,
450 kW,
-
sec.,
0.90,
E,
and
-
Zeff b
-
r/2,
,
-
roY 3 .
electron
The
and Ei -
3.99,
<n 0)(r) / ne> - 0.51, <T o)(r) / Teo>
2.0, injected RF power Pin
=
tail
profile
3.29,
=
confinement
form factors
corresponding
0.25, and <Tjo)(r) / Ti >
=
to
0.30.
The resulting RF power density and RF current density profiles are shown
in Figs. 2(a) and 2(b).
The profiles are peaked off-axis at r
-
2.48 cm.
Approximately 22 kW of the injected RF power was damped due to electronion Coulomb collisions
at
the plasma
periphery
(13
cm
<r < 17.8
cm).
-24The remaining 428 kW of RF power was absorbed due to quasi-linear electron
Thus
Landau damping with 78 kW eventually going to electron tail losses.
for this case Pcoll - 22 kW, Pr
=
78 kW, and Prf = 428 kW.
integrated RF current in Fig. 2(b) is Irf
n .
171 kA, with Irfp - 215 kA due
=
with positive n q and Irfm - -44
to waves
The radially
kA due to waves with negative
Since the total plasma current is I , M 170 kA this then corresponds
tion is ;
-
the Alcator
case.
0.121 for this
C experiment
efficiency
drive
current
state
The steady
" Ip and E1 (r)
Irf
to a case of steady state current drive [i.e.,
in
; defined
Introduc-
the
This value is in good agreement
with
by molyb-
constrained
for low Zeff discharges
01.
-
6
denum limiters.
parallel velocity electron distribution function for this
A typical
Fe is plotted as a function of parallel ki-
2(c).
case is shown in Fig.
on the lower
netic energy (E.)
tion in
Fig.
maximum of
corresponds to a
2(c)
the
RF
power
radial location
deposition
index (no)
The distribution
2/n1/2-1.
on the upper axis, where Eymmec2
refractive
and parallel
axis
The
profile.
func-
2.48 cm at the
r long,
flat
plateau
extends from no - 1.287 (near the critical value of nog for accessibility)
up to no - 5.51.
For the chosen electron temperature profile [Eq. 21(b)],
the plateau extends from vN / ve
10.54.
to raise
function for
The distribution
to that for n q >0.
a
sloping
vl
/ ve - 2.46 to v
no <0 is an
interesting
Although the RF power at negative n
plateau,
electric fields is not
the
quasilinear
diffusion
sufficient to overcome
/ ve av
2
/ ye
contrast
is strong enough
due
to
the
RF
the collisional diffusion
and completely flatten Fe.
In Fig. 3(d) we have plotted the RF power density Srf due to electron
Landau damping as a function of EY and n11 , corresponding to the distribution function at r=2.48 cm,
shown in Fig.
2(c).
The self-consistent
RF
-25-
About one-fourth
is due
to
waves
[Fig.
Brambilla spectrum
clear that
at n p>
3
1(a)]
has very little
increases
toroidal
being carried to higher values
elevated quasi-linear
plateau
p11.
the
in significant
in nig have resulted
of parallel refractive index,
to
be maintained.
injected
at nil > 3,
power
is
power
enabling an
existence
The
it
of
an
of waves at nil < 2 on the
elevated plateau then results in the damping
suprathermal electron tail (E
particular
Since
35 keV.
and E Y <
the
radial location
at this
RF power density
of the total
damping
cm at a
r=2.48
at
damping
rays
all
contributions from
includes
11(b)
Eq.
in
Srf
evaluate
to
used
diffusion coefficient
Approximately one-half of the
> 100 keV).
RF power density in Fig. 2(d) is due to this tail damping of waves at low
nil.
Very little
waves at n q < 0.
negative parallel
is dissipated at this
RF power
This is
refractive
since
to be expected
index is
radial location due to
the launched
characterized
tends to damp at a lower electron temperature
by -6< n,
power
-4
at
and
after toroidal
(r 14 cm)
upshifts in nil.
The damping and nil behavior for a particular ray trajectory from the
case discussed above is shown in Fig.
Figure 3(a)
3.
is the projection
of the ray trajectory in the poloidal plane of the tokamak,
the variation in p(r,e)
the poloidal mode number
geometry),
of n
and Fig. 3(d)
along the ray path,
(m)
Fig.
Fig.
3(b) is
is the change in
3(c)
this trajectory (due to the toroidal
along
The initial
is the resulting nil variation.
= nO - 1.418 is upshifted to n
tions of the ray from the plasma edge.
-3.8,
value
after several radial reflec-
Fig. 3(e) shows the wave amplitude
due to quasi-linear electron Landau damping using the self-consistent distribution function solution for nil >0.
There is strong
tail damping of
the wave on the initial pass into the plasma when n; is still low, (i.e.,
-26-
occur
in n 1 which
upshifts
the
However,
and 0< s (rad)i 4).
1.6
n,
for this ray and other rays on later passes, are still necessary to maintain the elevated quasi-linear plateau [see discussion above with Fig.
2
(d)].
of multiple radial
The importance
illustrated by
out
plasma and
to
to the edge,
single
a
-
was Irf- -35 kA with Irfp = 2 kA
and Irfm = -37 kA.
Fig.
was
The net RF current generation
absorbed with Pcoll - 8 kW and P.,
<-4 [see
variation in
184 kW of the injected 450 kW of RF power
this case only
current is not
with
into the
pass
radial
only a modest
thus allowing
can be
rays
example
model
our
for
calculation
restricted
ray trajectories
all the
n j. In
the
repeating
of the
reflections
because of the incident
surprising
average
The
1(a)].
32 kW.
plateau
value
The larger negative
n
RF power at -6
n, >0
for
was about
a factor of ten to one hundred less than for the case shown in Fig. 2(c).
it
As a final point,
assuming Ti
>>
with Ti - Te.
Te
in
the
is interesting
resonant
In this case,
to examine the consequence
region by redoing
the RF current
the model
is reduced
of
example
by more than a
factor of two to 70 kA and 406 kW of RF power is absorbed due to electron
Landau damping and 49 kW is absorbed due to electron-ion Coulomb collisions.
of RF current
The enhancement
by a large Ti can best be under-
stood by considering that the steady state parallel distribution function
Fe is determined by a relation of the form
Fe
Drf (P 0)
-
()pI
Fe
+
pg A
(p l,Tj)
Fe
+
B
(p p,Ti)
-
bp q
.
0,
-27-
ignore
A and
B involve
of
integrals
the
Eq.
and
(8)
be
can
it
6
Eqs.
(see
confinement
of electron
the effects
if we
7).
and
that
shown
p IIFe/(Drf/A + B/A)|
aFe/ aP 0
In regions of phase space where the self-
is a decreasing function of Tj.
consistent Drf is not large, Ti can play an important role in flattening
C.
the
thus increasing
density.
current
RF
Model Example for PLT
Model results
-
a PLT
for
state
steady
3.75
1.5 ne, nea - 0.1 neo, Teo - 1.5 keV, Tea - 0.03 keV,
Teb
x
1012 cm
-
0.005 keV,
Tio - 0.5 keV, Tia - 0.03 key, Tib - 0.005 keV, a - 40 cm, b -46
- 132 cm,
Ip - 200 kA,
100 kW, A4 -
-to - 7.5 x 10-3 sec.,
n/3,
factors were qo -
<ne(o)
to
(r)
are
En - -0.571,
1.0,
/ neo > - 0.51,
shown
in
Figs.
peaked off-axis at r - (4-9)
- 96 kW.
and
Fe -
< Te(o)(r)
, -
Zeff - 4.0,
rOY4.
i
3.99
/ Teo >
-
Pin
-
The profile form
2.90, corresponding
and
0.25,
< Ti(o)(r)
4(a)
Again the
and 4(b).
cm with Pcoll -
4 kW, PV -
profiles
are
53 kW, and Prf
The radially integrated RF current is Irf - 196 kA with Irfp "
217 kA and Irfm
is smaller
Bb - 31 kG,
cm, Ro
The RF power density and RF current density profiles for
/ Tio > - 0.34.
this case
deuterium plasma,
are
scenario
drive
current
The parameters used were ne -
4 and 5.
shown in Figs.
neo
and
function
the distribution
-
-21
kA.
The negative contribution to the net current
for this PLT case than in Alcator C.
This is not surprising
since the RF power propagating at negative ng for the PLT, rT/3 phasing is
only 23% of the total
injected power as compared with 36% for Alcator C
[see Figs. 1(a) and 1(b)].
-
The current drive efficiency for this case is
0.099 which is in good agreement
obtained on PLT (6
0.10-'0.13).5
with the
best experimental
values
-28-
A typical parallel velocity electron distribution function for this
case is shown in Fig.
r
-
to a radial location
9 cm, at the maximum of the RF power deposition profile.
from ng
extent is
v2 /
This plot corresponds
4(c).
5.74
n
or
/
from vi
ve a
to
2.49
The plateau values at negative n , are roughly an order
ve - 15.35.
of magnitude
1.199 to
=
The plateau
no [even
for positive
below those
location shown in Fig. 4(c)I.
for
smaller
the radial
This is consistent with the smaller fraction
of RF power coupled to waves at negative no in the PLT waveguide grill at
relative phase
U - % / 3.
The RF power
density
plotted
11(b)] is
as
a
Srf
due to electron Landau damping
function
EY and
of
plot corresponds to the distribution
function
Fig.
no in
4(c)
in Fig.
[see Eq.
This
4(d).
at
r-9 cm.
Approximately one-fourth of the RF power density at this radial location
3
is due to waves damping at nq> (E.35 keV) and one-half is due to waves
at np< 2 damping on the suprathermal
electron tail (E,>100 keV).
Because
the initial Brambilla spectrum [Fig. l(b)] has practically no RF power at
3
nl>3, we again conclude that the wave damping at np> and maintenance of
an elevated
quasi-linear plateau
in the parallel
is due to toroidally induced increases
The RF power
refractive indices of the injected waves.
density due to wave damping at np<O is
small as would be expected
from
the launched Brambilla spectrum.
A typical ray trajectory with the associated damping and no behavior
5(a) - 5(e).
for the PLT sLmulation is shown in Figs.
m coupled
with magnetic
value of n 10 - 1.33 to n
Figure 5(e)
damping on
is the
wave
shear
result in upshifts
The variations in
of no from an initial
W 5.0 + 6.0 [see Figs. 5(a)-5(d)].
amplitude due
the distribution function
to quasi-linear
for
n j > 0.
electron
Landau
The tail damping
is
-29-
with 1.2
quasi-linear
The
1.5.
n
wave nq at
upshifts in the
velocity via
parallel phase
at
maintained
is
plateau
7
b (rad)
0
ray trajectory
of the
pass
initial
strong along the
$
35,50,
low
and
68 radians [see Fig. 5(d)].
model
The PLT
of the ray trajectories
radial pass
negative RF current
Irfm
-
-10
was
Irf
-
only a
of the plasma.
out
kA,
-10
21 kW of the injected
Approximately
kA.
in and
with
generated
allowing
simulated
also
was
example
single
A small
Irfp = 0 kA,
and
100 kW of RF power
The resulting values of the quasi-linear plateaus in this
was absorbed.
case were down by several
of magnitude from those
orders
shown in Fig.
4(c).
As in the previous section,
Te results
D.
reduction
a considerable
in
redoing the PLT model example with T1
The results of a toroidal magnetic
VB.
I.
RF
generation.
field scan for Alcator
C appear
The parameters used were identical to those given in Sec.
were
The results
obtained
by varying Bb and then adjusting the RF
power so as to maintain a constant value of RF current,
The current
current
Alcator C
Toroidal Magnetic Field Scan:
in Table
in the
-
drive efficiency
- 0.086 at 60 kG.
decreases
from
n-
Irf - Ip - 170 kA.
0.121 at
100 kG to
For the cases shown in Table I the RF power lost due
to finite electron tail confinement
ion Coulomb collisions
plus the power lost due to electron-
(Pr + Pcoll) was about
The code
22% of the injected power
results are only
in rough
agreement
at all magnetic
fields.
with the Alcator
C experiment 6 where r -0.08 at B $ -80 kG and RF current
driven discharges were difficult to maintain at B 4 <60 kG.
However, vari-
-30-
ations in
plasma
parameters
which
were
held
fixed
scan (such as the central electron temperature)
drive efficiency
as
shown
in
Sec.
VH.
in
the
toroidal
field
can also affect the current
The larger
values
of
at higher
magnetic fields in the code results can be understood by recalling that the
20 4 6
critical value of n, for wave accessibility is'9, ,
where el is given
As B
by
is increased,
electrons in
the
lower hybrid
waves.
Eq.
na_(
11/2 +
16(a)
and
e/(34)
all
quantities
the value of na is reduced
plasma
interior
to
interact
The qualitative
effect
are
evaluated
at
r-0.
(see Table I), thus allowing
with
higher
of this
phase
interaction
velocity
is easily
seen from an expression for the local current drive figure of merit readily
obtained from Eq. (6):
Jrf
Sd
c
= (-)
2
y2.1
Ve
2
1
y +1+1
1-
-
rf/(neeve), Sd
Sd have been defined by Eqs.
>>
finite electron
is
tail
assumed
(35b)
1
Y(v2 )
Sd/(voneTe),y2
11(a) and
to the lower and upper bounds
which Drf
---1n
+
2-
2
Here Jrf
(35a)
G(y 2 yY1
1
G(Y2,
(2 + Zeff),
/
11(c).
of a region
to
confinement
be
has
nonzero,
been
y1
y(v1 ),
and Jrf and
vl and v 2 refer respectively
in parallel
T1
=
Te,
neglected.
velocity space
and
the
Clearly
creases due to improved accessibility, Jrf/Sd also increases.
effect
as
v2
for
of
in-
-31-
E.
PLT
Relative Waveguide Phasing Scan:
The results of a relative waveguide phasing scan for PLT are given
in Table
II.
results were
used
The parameters
obtained
changing
by
were
given
those
relative
the
in
to the
three
- 1(d)
waveguide
that
phasings
of
hybrid waves
is
T)
Pcoll/Pin ( 0.1.
phasings
at
efficiencies
duced values
were
the
for
negligible
used
Ip - 200 kA.
(A$ - %/3,
K/
As
results
5
which
of
damping
shown
in
re-
the lower
with
II
Table
However the losses due to electron tail confinement were
(0.27-0.53).
-
A qualitative understanding
2.7
for relative
2% / 3 respectively (na-1.
of
1(b)-1(d).
power spectra occur at values of ng equal to
The maxima in the injected
and
21/3).
,
r/2 and significantly
the variation of ; with A$ can be obtained by examining Figs.
2.0,
2
with the PLT experiment
The collisional
2n /3.
at A6
significant with P-r/Pin
1.0,
and
show the three input Brambilla spectra corresponding
The results are in qualitative agreement
found highest
The
phase A
waveguide
then adjusting Prf so as to maintain a RF current of Irf
Figures 1(b)
VC.
Sec.
20
).
waveguide phasings
of -n /
Thus at lower values of
3,
t
/
2,
and
Ab, the character-
istic phase velocity of the injected RF power is higher and plasma elechaving higher parallel
trons interact with waves
sulting in improved. efficiencies.
deposition profiles
narrower and
(and
momentum
content,
re-
It is also observed that the RF power
the RF current density profiles) tend to become
exhibit maxima nearer to the plasma center as A$ is reduced.
This behavior might be expected on the basis that lower hybrid waves with
small initial values of n 1
(nearer the plasma
center)
tend to damp at a higher electron temperature
than waves
taking upshifts into account.
launched with higher np1 9 s,
even
-32-
F.
Alcator C
Electron Density Scan:
The dependence
current drive efficiency on electron density
of the
As ne was
for the Alcator C parameters of Sec. VB is shown in Table III.
varied,
'rf
the RF power was adjusted
170 kA.
~ I
so as to maintain a constant current,
from the
The values of
code are
seen to increase
only slightly over the density range, in reasonable agreement with experimental results. 6
however,
As ne increases from 2.5 x 10
3
cm-
the accessibility parameter na given by Eq.
1.196 to 1.393.
sented by Eq.
to 7.5
103cm
x
(34) increases from
Within the context of the simple analytic model repre-
(35),
we would therefore expect the current drive efficiency
to decrease substantially due to changes in v2.
The discrepancy between
the density scaling of the code and the simple analytic result arises because the latter does not account
for the effect of finite electron tail
confinement on the level of RF current generation.
was taken to be TOY3 with To
Alcator C density scan
for all densities.
Recall that for the
As ne is decreased,
5
x
10-3 sec.
the effect of the electron tail
confinement on the current moment of the distribution function increases
because tail-bulk collision processes are occuring on times scales which
are increasingly long relative to the confinement time of- tail electrons.
Evidence of
finite electron tail
confinement becoming more important
at
lower densities can be seen in Table III where P-/Pin increases from 0.13
to 0.26 as ne is lowered.
We note that although the collisional damping
of the lower hybrid waves
increases
density is
raised,
it
still
from 3 kW to 42 kW as the electron
remains
small
with
Pcoll
/Pin
(
0.07.
-33-
G.
PLT
Electron Density Scan:
Table IV shows the variation in RF power as a function of electron
density, for the PLT parameters of Sec. VC.
The relative waveguide phase
The results again demonstrate that an increase in
was chosen to be n/2.
RF power is necessary to maintain the same value of current (in this case
Irf
~ IP
-
as ne is increased,
200 kA)
with the efficiency
T
remaining
The effect of accessibility is not as significant as in
nearly constant.
the Alcator C case since na increases only from 1.142 to 1.206 as ne is
3
3.75 x 101 2 cm-
increased from
tail confinement
1.0 x 101 3 cm-
to
3
.
finite electron
The
is mitigated by increasing electron density in the code
results, but to a lesser extent than in the previous section.
Note that
P /Pin decreases from 0.44 to 0.29 as ne is raised.
The results in Table IV do not reproduce the important density cutin
off observed
cm-3.
the PLT
densities ( e
13
> 1 x 10
cm-
3
),
experiment
drive
that
do indicate
results
simulation
While
current
mentally.
5
to
; -
As discussed
a
critical
-at
T
decreases
at
in
the Introduction,
at higher
this decrease is not an
density as
0
x 1013
0.8
n
collisional damping and
due to increased
reduced accessibility of the injected RF waves,
abrupt transition
5
other
experi-
observed
physical
processes
such as parametric decay of the lower hybrid pump wave may become
impor-
2 2 23
tant at these higher densities. ,
H.
Electron Tail Confinement Scan:
The results
of
a
study
of
PLT
the
dependence
of
the
current
efficiency on the electron tail confinement time parameter (TO)
in Table V.
drive
are shown
The PLT parameters given in Sec. VC were used with Ab
=
/3,
-34-
ne
=
and
3.75 x 1012 cm-,
-
increases from 0.33 at r
sec.
1.5
going into tail losses
RF power
for
10-2 sec to 0.87
x
the efficiency
0 is reduced
As
of injected
fraction
decreases and the
y4 .
T
=
T
-
2.5
10-3
x
The RF power damped collisionally is small for the cases in Table V
For
< 0.04.
with Pcoll/Pin
the
tail
would appear that the best choice for
The resulting
for Ad -
value of
i/3 phasing
i
in
is closest
the
PLT
is 7.5
10-
3
experiment.
5
Also,
it
sec (see Table V).
reported
to the optimum efficiency
this
choice
with the electron power
amount which is consistent
an
x
here,
used
of -
RF power going into bulk electron plasma
results in 46% of the injected
heating,
model
confinement
balance
reported during lower hybrid current drive in PLT. 4 0
I.
Alcator C
Electron Temperature Scan:
Table VI
is a
central electron
set
of results which demonstrate
results
These
(Teo).
temperature
the Alcator C parameters given in Sec.
how ; varies with
were obtained
using
The current drive efficiency
VB.
improves by a factor of 1.51 as Teo is raised from I keV to 2 keV.
Col-
lisional damping is not significant for these cases with Pcoll/Pin < 0.08.
i as Teo is raised
in
The increases
qualitative fashion.
by
not affected
Do(x,k,)
0 are
-
are
velocity waves
damping at
velocity of
a
given
the
in
able
radial
current
higher efficiency.
to
the
because
Teo
As
small.
following
we note that the toroidal nq variations are
First
changes
can be understood in the
Teo
satisfy
location.
increased
is
the
thermal
however,
conditions
Consequently
carrying electrons
is
to
corrections
higher
phase
for electron
Landau
the effective
phase
increased,
leading
to
a
A second effect is the increase in the perpendicular
electron temperature Ti.
Recall
that T j
-
50 x Te has been assumed so
-35-
in-
that as Teo increases from 1 keV to 2 keV the maximum value of T,
T
in
increase
an
enhance n via
tends to
generation.
current
RF
the
a larger
VB,
in Sec.
out
pointed
As
100 keV.
50 keV to
creases from
Summary and Conclusions
VI.
In this paper we have described a simulation model for steady state
drive
current
lower hybrid
one-
relativistic,
a
incorporates
which
dimensional Fokker Planck calculation coupled with a toroidal ray tracing
(vI)
Two-dimensional
code.
in the Fokker Planck
retained
effects were
analysis in the form of an effective perpendicular electron temperature
due to
scattering
angle
pitch
reported here).
Ti
(typically
the
in
= 75 keV
results
ray tracing techniques were utilized to follow
Toroidal
the propagation and absorption of a Brambilla spectrum of injected lower
self-consistent,
The
hybrid waves.
ficient was
calculated
hybrid rays
as
flowing
by
RF
quasi-linear,
along a "tube"
with
lower
cross-section
finite
of
coef-
associated
power
the
modelling
diffusion
The simula-
glecting the focussing effects of such a tube or ray pencil.
incorporated
tion model
toroidal
index as a physical mechanism
phas.e
high parallel
electrons at
v
In Secs. VA and VB
PLT which
by which
(ve
velocities
3 ve
demonstrated
and
upshifts
<<
maintain an
the
in
refractive
parallel
at
lower hybrid waves
injected
v
and ne-
could
c),
elevated
interact
quasi-linear
with
plateau.
model examples were presented for Alcator C and
how
this
physical
"spectral gap" in parallel phase velocity.
mechanism
The
could
calculated
close
the
levels of RF
current and current drive efficiency were consistent with the experimental
6
results on PLT 5 and Alcator C .
Typically for RF waves launched with n
>O
the resulting quasi-linear plateaus on the electron distribution functions
-36-
were flat
from
extended
and
to
be
< v
< 15
/Ve
locations
radial
at
The tail damping on such distribution functions
near the plasma center.
was found
2.4
for
strong
waves
at
n
low
nfn
(1
However
2).
nn upshifts were necessary to maintain the plateaus at low phase velocity.
Without the toroidal upshifts in n 1 and allowing
for only a single pass
insignificant levels of RF current
of the rays in and out of the plasma,
were found numerically.
observed
The experimentally
efficiency as
Bb is
was
reduced
decrease
in
Alcator
C
qualitatively
reproduced
drive
current
by the model
and attributed in part to the poorer accessibility of high phase velocity
lower hybrid
waves to the plasma
the PLT experiment and model
center.
Qualitative agreement between
was demonstrated
results
VD where
in Sec.
the best current drive efficiencies were obtained for relative waveguide
2n / 3.
The
ff/2 with
n/3 and
phasings between
decrease
n decreasing
increased
; as As was
in
significantly for A$
was
due
to
a decrease
In
in the characteristic parallel phase velocity of the injected waves.
Secs.
=
VE and VF the electron density and RF power were shown to scale so
as to keep
i
The experimentally
constant.
approximately
observed
den-
sity limit for current drive in the PLT device was not reproduced by our
model.
In
Sec.
VGit was
parameter is decreased,
shown
the current
electron tail losses increase.
be consistent
with
both
that
the
as the electron tail
drive efficiency
confinement
decreases
and the
It was argued that the value of T:
0 should
experimentally
the experimentally measured electron power
observed
balance.
values
of
Finally in
6 and
Sec.
VH
the current drive efficiency was shown to improve as the central electron
temperature was increased from 1 keV to 2 keV.
-37-
In conclusion, we have presented a simulation model for the propagation and absorption
of a
realistic
spectrum
of lower
hybrid
waves
in
toroidal plasmas which accounts reasonably well for several aspects of two
Used properly, this model can hopefully
major current drive experiments.
lead to a better understanding
of the complicated interplay of physical
effects acting simultaneously in the experimental situation.
VII.
Acknowledgements
We would like to thank Professor Miklos Porkolab and Professor Thomas
M.
Antonsen
for many
insightful
comments and
suggestions and
constant encouragement during the course of this work.
edge useful discussions
with Dr.
for their
We also acknowl-
Stephen Knowlton and Dr.
Yuichi Takase
and thank Professor Porkolab and Dr. Knowlton for making the results of
their Brambilla code available to us.
This work was supported by the U.S. Department of Energy under
Contract DE-AC02-78ET-51013.
-1-
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Vol IB, ch. 13, p. 151.
confined
New York,
1981)
TABLE I
Toroidal Magnetic Field Scan:
B
(kG)
na
Pi
Pd (kW)
(kW)
Alcator C
Pcoll (kW)
P
(kW)
60
1.530
634
0.086
495
49
90
80
1.369
515
0.106
402
30
83
100
1.278
450
0.121
350
22
78
TABLE II
Relative Waveguide Phasing Scan:
A6
Pi
(kW)
Pd (kW)
PLT
Pcoll (k)T
(kW)
7/3
100
0.099
43
4
53
n/2
123
0.080
61
8
54
2 r/3
450
0.022
282
45
123
TABLE III
Electron Density Scan:
ne (cm-3 )
P1i
Alcator C
Pd (kW)
(kW)
PColl (kW)
P
(kW)
2.5 x 1013
261
0.105
190
4
67
5 x 1013
450
0.121
350
22
78
7.5 x 1013
675
0.121
542
42
91
Table IV
Electron Density Scan:
ne (CM-3)
Pi
(kW)
PLT
Pd (kW)
Pcoll ()
PT (kW)
3.75 x 1012
123
0.08
61
8
54
6.0 x 1012
195
0.081
116
14
65
8 x 1012
220
0.096
124
23
73
1 x 1013
297
0.089
162
48
87
TABLE V
Electron Tail Confinement Scan:
vo (sec)
P T(kW)
Pd (kW)
(kW)
Pi
PLT
3
190
0.052
17
166
5.0 x 10-3
130
0.076
41
84
7.5 x 10-3
100
0.099
46
50
1.5 x 10-2
82
0.121
52
27
2.5
x
10-
Table VI
Electron Temperature Scan:
Pd (kW)
Alcator C
Pcoll (kW)
P
(kW)
Teo (keV)
Pi
1.0
604
0.090
483
50
71
1.5
450
0.121
350
22
78
2.0
400
0.136
294
16
90
(kW)
(F-1)
FIGURE CAPTIONS
Fig. 1
(a)
Four
waveguide,
Alcator
waveguide phase Adb = ir/2.
(b)
Fig. 2
Sec.
A$ -
%/2, and (d)
Model
for
Alcator
results
VB.
(a)
Radial
profile
tron distribution
parallel kinetic
profile
of
RF
energy
E
C
of
at a
(d)
.
2
A) -
spectrum
RF
density
parameters
density
(kA
r
-
cm versus
density
2.48
(watts/cm3 )
Ray trajectory for Alcator C model example of Sec. VB.
in the poloidal
(c) Variation
versus toroidal angle (40. (d)
in parallel refractive index (nq) versus toroidal angle (6)0.
nil
-
1.418.
(e) Normalized wave amplitude (PN)
linear electron
Model
Fig. 4
distribution
(nil >
0),
tion at a radial location r
(d)
toroidal
results for PLT current drive parameters
file of RF current density (kA /
=
cm2 ).
(c)
Y
Initially
angle
given in Sec.
/ cm3 ).
(b)
(4).
VC.
Radial pro-
Electron distribution func-
9 cm versus parallel
RF power density (watts/cm3 ) versus E
Variation
due to damping on quasi-
versus
(a) Radial profile of RF power density (watts
versus
cross-section of the
tokamak. (b) Variation in p/a versus toroidal angle ((.
in the poloidal mode number (m)
cm3 ).
/
(c) Elec-
:
of ray trajectory
in
cm2 ).
at r=2.48 cm.
Projection
given
(watts
E
(a)
spectra.
/
radial location
RF power
relative
.
drive
power
for
PLT Brambilla
t/ 3
current
current
function
Brambilla
Six waveguide,
Ab = n/3, (c)
(b) Radial
Fig. 3
(b-d)
C
kinetic energy E.
at r-9 cm.
(F-2)
Fig. 5
:
(a)
Ray trajectory for PLT model example of Sec. VC.
Projection
of the
tokamak.
of
(b)
ray
trajectory
Variation
in
in p/a
the
poloidal
versus
cross-section
toroidal
angle
( ).
(c) Variation in the poloidal mode number (m) versus toroidal angle
(0). (d)
Variation
toroidal angle
(t).
in
parallel
Initially
refractive
nq -
1.33.
index
(e)
(nj)
versus
Normalized
wave
amplitude (PN) due to damping on quasi-linear electron distribution
(nq > 0), versus toroidal angle (4).
r-
III
I
I -
r-
0.15
P(nh ) 0.1c
0.01
(a)
0
0.4
0.3
P(n )
0.2
0.I
0
0.4
(b)
-
-
0.3
P(n )d 0.2
0.1
(-)
0
0.4
0.3
P(n )
0.2
0.1
0
-10
-8
-6
-4
-2
2
0
nt
FIG.
1
4
6
8
10
16
4
-
(1)4
-
(a)
0
E 4-2E -
-
~
0
b)
4
8
r(cm)
FIG.
2(a) - 2(b)
12
16
105
4
3
2
1.5 1.31.2 1.1
4-
3-
2-
10
43-
-2
I0
~
43
2
-3-
10
4
3-
2-
I0-4
3-5
2-
10
43-
2
-6
10
2
3 4567
2
3 4567
Ey(keV)
FIG.
2(c)
2
3 4567
nil
5 4
10
3
2
1.5 1.3 1.2 1.1
16
14-
6'-
4
2-
nhI<0~
10
10
Ey(keV)
FIG.
2 (d)
1
0.2 0.4 0.6 0.8 1.0
FIG.
3(a)
I
i
(b) -
1.0
/a
0.5-
loo--
(c)
100
m
4-
(d)
n 1-
-
2-
1.0
(e)
PN 0.5 -
0
0
6
18
12
<
(radians)
FIG'S.
3(b) - 3(e)
24
30
0.3r4,
S 0.20.1-
(a)
0
0
0.8CQE
00.6
-
<x0.4
0.2(b)
0
0
20
r(cm)
10
FIG.
4(a) - 4(b)
30
40
nil
10
I
5 4
3
2
I
I
I
I
1.51.31.2
I
I
I
1.1
I.'.I
4
3
2
I0
4
3
2
-2
10
4
3
2
3
nil
>0
10
4
3
2
-4
10
i -0
4
3
In
2
5
4
3
2
I I
34567
10
I
2
3 4567
10
2
102
Ey(keV)
FIG.
4(c)
3 4567 1
j03
10
0.32
5
K
nil
3
4
I
I
I
3[12
H.
.
1.5 1.
2
I
I
I
II
0.28
0.24
I',
E
0.20-
0
Nb
n >0
0.16
C,)
0.12
II-
0.08
0.04
*11
0
n11<0
.
I
I
-
10
102
Ey (keV)
FIG.
4 (d)
--
.
-
-
-
-
.
44
Aoooooo
0.2
FIG.
5(a)
0.4 0.6 0.8 1
(b)
1.0
p/a
0.5-
m 100
0'
6
(d)
2.
1.0
(e)
nl4
P
0.5-
0
0
.10
20
30
40
b (radians)
FIG'S 5(b) - 5(e)
50
60