PFC/JA-82-25
ON IMPURITY TRANSPORT IN TOKAMAKS
Fredrick H. Seguin and Richard Petrasso
American Science and Engineering
Cambridge, MA 02139
and
Earl S. Marmar
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
November 1982
Department of Energy Contract Nos.
This work was supported by the U.S.
DE-AC02-78ET51013,
and
MIT
contract
MIT-FC-A246206.
DE-AC02-77ET53068,
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ON IMPURITY TRANSPORT IN TOKAMAKS
Fredrick H. Seguin and Richard Petrasso
American Science and Engineering, Inc.
Cambridge, MA 02139
and
Earl S. Marmar
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
ABSTRACT
We study theoretically the effects of diffusion, convection, and sawteeth
on impurity profile evolution in tokamaks.
our' experimental
profiles
of
numerical
shapes,
observations
impurity densities
simulations
and
we
that
in the
Sawtooth effects are modeled after
internal
Alcator-C
disruptions
plasma
core.
and analytic results for confinement
predict
that
sawteeth
may
confinement times under some circumstances.
significantly
flatten
We
radial
present
times and profile
decrease
impurity
Because
impurities play a crucial role in determining plasma character-
istics in tokamaks, it is important to understand impurity transport.
investigate
and
theoretically
sawtooth
mental
oscillations
observations
disruptions
are
ions,
flattening
core
(presumably
and allowing
some general
as
known
(periodic
a guide
to
to affect
consequences
"internal
of diffusion,
disruptions"),
modeling effects
the
transport
of
Here we
convection
using
experi-
sawteeth.
of electrons
and
Internal
working-gas
radial profiles of temperatures and densities in the plasma
by momentarily
rapid
radial
providing a
transport along
radial magnetic
field lines).
field component
We
show here
that
disruptions also flatten radial profiles of individual charge states of impurities.
A numerical code is then used to simulate impurity profile evolution
in impurity injection experiments and in "steady-state" plasmas; and an analytic
model,
based
explicitly
relate
impurity
on
eigenfunctions
confinement
flux and to
of
the
transport
times and profile
sawtooth effects.
equation,
is
used
shapes to properties
of
to
the
Our earlier worki along these lines
neglected convection, but experimental results2, 3 indicate that convection may
be important -in some tokamaks,
and our theoretical results here indicate that
the effects of sawteeth on impurity confinement can be much more important if
convection is present.
Silicon
has
been
injected 4
into
the
Alcator-C
plasma
for
impurity
transport experiments; it appears first as a shell at the outer surface of the
plasma, then penetrates to the plasma core, and finally leaves the system with
an exponential decay rate.
Using a method we've described recently,
followed
absolute
silicon
the
evolution
ionization
what happens
states
of
radial
density
profiles
and of the sum of all states.1,5
of
Figure
we have
different
la shows
to the helium-like and hydrogen-like ions as a consequence of an
internal disruption during the inflow stage, when the silicon density profile
- 1 -
Ib
la
I
(i)
+
z
J(0i)
(I)
0
internal
discharge.
Observed
ion
disruptions,
for
1.
a)
1
0
density profiles
silicon
I
I
J
r/rL
injected
r/rL
(i)
just before
into
an
b)
Densities
r is the minor radius; rL is the limiter radius.
- 2 -
after
Alcator-C
of the fully-stripped ion,
experiment details.
1
and
Sum of helium-like and hydrogen-like ion densities,
after injection.
injection.
I
I
I
FIG.
%
about
(ii)
deuterium
about 6 ms
15 ms after
See Ref. [5] for
is hollow:
an abrupt
the central part of the density profile is flattened, resulting in
inward movement of particles.
the plasma core,
its
lb shows how the
After the silicon has penetrated to
distribution eventually
fully-stripped
becomes peaked on axis.
Figure
state responds to an internal disruption at
density gradients are once again reduced in the center,
though
the result of flattening this time is an outward movement of particles.
These
such a time:
results are consistent with other experimental
move
in
and
out
during
indications that impurity ions
oscillations 6
sawtooth
and
that
impurity
density
profiles are flatter in the presence of sawteeth.7
This
response
transport code.
to
sawteeth
has
been
incorporated
into
a
numerical
We assume that, apart from effects of sawteeth, transport is
governed by
aN
.at
1 a
= -r 3r
a
[r(D-
+ V)NI
;
N(rL
(1)
ar
N(r,t) is the total impurity density (summed over all ionization
rL is the limiter
radius.
Since the
diffusion coefficient
states),
and
D(r) and/or the
inward convection velocity V(r)'may be anomalous (and of unknown radial dependence)
in
many
plasmas,
we
simplify
this
discussion
by
illustrating
here
results corresponding only to the simplest choices:
r
D
Da
a
V
2 D
2)
a
where Da and Va are constants and the dimensionless
describes
the
relative
importances
of
convection
"convection parameter"
and
diffusion.
In
neo-
classical transport theory, 8 diffusion and convection coefficients vary with
- 3 -
S
I
radius but can have ratios corresponding roughly to an S of order the impurity
Some impurity profiles observed 7 '
charge Z.
have
to be
shapes which can be shown
been
some
in
fluxes
neoclassical
in
and
while certain
with S = 20,
convection
with
interpreted
Outward convection (S < 0) can
corresponding to S in the range 0 to 4.2,3,10
occur
sawteeth
in discharges without
consistent
have
experiments
injection
impurity
9
certain
predictions
assume
that
of
anomalous
transport arising from collective modesi".
To
we
injection experiments,
simulate
tained in a narrow shell just inside r = rL at time t
file
is then allowed
intervals
made
to
slightly
to evolve according to equation
(corresponding
be
flat
inside
radius
larger
td between
to the time
one
radius
rd,
with
and
to
conserving the total number of particles.
of
the region
Once
values
by
affected
of
rL,
Da'
sawteeth,
and Va
are
impurity is
0.
The density pro-
(1);
but at
the
original
specified
profile
is
at
everywhere
derivatives
1st
con-
the profile
disruptions)
join
continuous
=
an
a
and
(Note that rd is the outer boundary
the singular
and not
specified,
behavior
surface
in
the
radius.)
absence
of
sawteeth is fully determined and a "confinement" time T 0 , corresponding to the
ultimate exponential decay rate, is found.
new confinement
time
When sawteeth are incorporated, a
rd < To always results
(see Figure
In possible qualitative
with increasing S, increasing rd, and decreasing td.
confirmation
of this
result,
reduction of
a substantial
sawteeth has recently been observed on PDX (with rd
We
can
simulate
a
"steady
state"
plasma
Td/To decreases
2);
confinement
time
by
0.9 rL2 1
situation
by
continuously
injecting new impurities and waiting until a stationary profile is established
(see
Figure
Without
3).
if
aN/3r
convection,
=
0
equation
everywhere
inside
(1)
has
the
a
time-independent
source
layer
at
solution
only
surface.
Sawteeth have no effect on this flat profile of total impurity
- 4 -
the
S=-4
r=7
TO=10
5
S=o
0=3
0
11
z
30=2
'low *,wf-af
TIME
0
FIG.
2.
(ms)
me
75
Central impurity densities from sample numerical simulations
with
(solid
lines)
lines)
injection
experiments,
sawteeth.
Da = 1.48 x 103 cm2/sec, rL = 16 cm, rd = 0.7 5 rL, and td = 3 ms.
- 5 -
and
without
(broken
of
density (although the profiles of the individual ionization states, which are
not flat,
quite
will
be periodically
different
when S > 0,
the equilibrium profile
fixed
the rate
and we
by
disruptions).
contrast 3 cases:
is peaked on axis.
of inflow
"steady state" profile
flattened
The situation is
(i) Without sawteeth,
(ii) If we add sawteeth, holding
of new material,
then the
is flatter in the center,
time-average
of
while the gradient
the new
of N at
the edge remains unchanged since it determines the loss rate that must balance
the
specified
inflow.
The result
can
impurity density and radiation losses.
case
be substantial
in central
(iii) If, while sawteeth are added to
(i), we hold fixed the total number of particles in the system
respond to instantaneous
recycling),
has
in
the
same
shape
as
case
then the resulting steady-state
(ii)
but
a
consistent with the transition between cases
in
reductions
Doublet
III
(see
intrinsic nickel,
Figure
6
of
Ref.
larger
(i) and
[7],
amplitude.
(iii)
in which
with and without sawteeth,
(to corprofile
Behavior
has been observed
measured
profiles
of
can be modeled with S = 18 and
4rd = 0.4 rL).
While
the numerical
code
can
be used to predict
transport
details,
an
analytic model can be used to easily predict essential transport features and
provide physical insight.
N (r)e
-t /r
n
n
,
Equations (1) and (2) have discrete solutions
and in the absence of sawteeth any evolution can be described
through a sum
N (r,t)
AN
=
-t/Itn
(r) e
.
(3)
n
The eigenfunctions Nn (r) and eigenvalues
ues of S,
The
Tn
have been found for different val-
and some of them are shown in Figure 4 (see Ref.
"slowest" eigenvalue
T0
corresponds
to the confinement
can be approximated by the formula
- 6 -
[13] for details).
time.
Its values
--
S=-)
S=+4
z
..
----
...
.
(ii)
FIG.
Da'
r/rL
1 0
0
L'
3.
10
--
MM
00
1
Sample numerical simulations of steady-state impurity profiles;
d, and
td are as in Figure 2.
with sawteeth, as described in text.
- 7 -
(i): No sawteeth; (ii) and (iii):
77 + S22 eS - S - 1 r2
4S2
+S _
56+52
4S
Da
T
0
for numerical
which is useful for determining parameters
specific
confinement
is
desired.
The
relaxation
for
scale
No(r).
shape
the
toward
time
the
represents
time
determines
Ti
(4)
"second
of
the
an
slowest"
arbitrary
inflow
a
simulations when
time
eigenvalue
initial
in
an
T
profile
injection
experiment (without sawteeth), and also the repeaking time after flattening by
We can approximate the effects of sawteeth in the limiting case
a disruption.
td
by constraining a profile
T,
<<
will
then
= a To,
Td
stay flat
inside
the shape of N (r).
adopt
eventually
profile
to
The
rd;
outside rd, any
confinement
time is
where
rd r dr N (r
o
a
+
d) +rd
L r dr N (r)
r
r
0rN
(5)
L r dr No(r)
00
a(S,
If
rd)
td
is plotted in Figure 4; it
T
disruptions,
,
a flattened profile
and the
decreases rapidly as S and rd increase.
has time to relax
toward N (r) between
time-averaged shape of the profile leads us to the more
general expression
(a + a - W)
Td
aa
Equations
(4)
and
(6)
d
1
td
give
I
(6)
T
0,
td
-t/T1
fO dt e
.(7)
results
in
simulations.
- 8 -
good
agreement
with
numerical
-25
-10
-4
N,(r)
4
-
S=25-
0
r
8
'rL
1.0
0
-
-
.3
-1
-4
-25
FIG. 4.
0
N (r); T
0
25
0
and T1 , in units of T
0rL/Da;
- 9 -
25
r/
and a(S,
ctS
r
d/
The
quantitative
correspond
to a
results
form for
the
presented
impurity
here
flux
are
useful
(defined
because
through equations
they
(2))
which is commonly used in modeling,2,3 and because they illustrate the general
nature
of impurity profile evolution when diffusion,
effects are present in various proportions.
times,
profile
convection,
We've predicted that sawteeth can
modify
inflow
rates;
these effects can be modest under some circumstances,
quite
significant
if
the
shapes,
impurity
and sawtooth
confinement
flux
times,
contains
and radiative
a
loss
but they can be
significant
inward
convection term and/or if sawteeth affect a sufficiently large fraction of the
When analyzing experimental
plasma volume.
that
details
variations
ratio
T 1 /T
of
the
(see Ref.
0
above
[13]).
results
can
data, it
is important to remember
be altered
if
D
and
V have
radial
In particular, radial variations can change the
, which means that a comparison of inflow time to decay time in an
injection experiment is not by itself sufficient for determining whether convection
is
necessary.
is
that
behaves
present;
measurements
of
time
scales
and
radial
profiles
are
An additional complication for interpretation of real experiements
the
convection
anything
proportional
to
velocity
may
like the neoclassical
gradients
of
fluctuate
drastically
convection velocity;
background
plasma
with
time if
it
this velocity is
quantities
(such
as
temperature), and these gradients are strongly affected by sawteeth.
ACKNOWLEDGEMENTS
For their assistance
and J.
Rice,
supported by
J.
Terry,
during our experiments,
and
DOE contracts
the
rest
of
we thank N.
the Alcator
DE-AC02-77ET53068
contract IIT-FC-A-246206.
- 10 -
and
staff.
Loter of AS&E;
This work
DE-AC02-78ET51013,
was
and MIT
I
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F.H.
Seguin and R.
Petrasso,
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Controlled. Fusion Theory Conf.,
2D14
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Equipe
TFR,
Proc.
11th
Int.
Conf.
on
Plasma
Physics
and
Controlled
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3
K.
Behringer,
W.
Englehardt,
G.
Fussmann,
IAEA
Technical
Committee
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Marmar,
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J.T.
Terry,
and F.H.
Seguin,
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5
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Petrasso,
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-
11 -