PFC/RR-90-9
DOE/ET-51013-284
TSC Simulations of Alcator C-MOD
Discharges IlIl: Study of Axisymmetric Stability
J.J. Ramos
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
June 1990
This work was supported by the U. S. Department of Energy Contract No. DE-AC0278ET51013. Reproduction, translation, publication, use and disposal, in whole or in part
by or for the United States government is permitted.
TSC Simulations of Alcator C.-MOD Discharges
III: Study of Axisymmetric Stability
J. J. Ramos
PLASMA FUSION CENTER, MIT, Cambridge, MA 02139
Abstract
The axisymmetric stability of the single X-point, nominal Alcator C-MOD configuration is investigated with the Tokamak Simulation Code. The resistive wall passive growth
rate, in the absence of feedback stabilization, is obtained. The instability is suppressed
with an appropriate active feedback system.
One of the most useful applications of the Tokamak Simulation Code (1] (TSC) is
the analysis of the axisymmetric instability and its active feedback control for general
plasma cross sections and current profiles.
TSC models the time evolution of a two-
dimensional, axially symmetric plasma fluid and its electromagnetic interaction with the
external field system, through a tokamak discharge. Therefore, it provides a complete, non
linear description of the tokamak axisymmetric stability behavior.
The dynamical equations solved by TSC are those of a full two-fluid transport model
with appropriate phenomenological diffusion coefficients. Thus, this code is able to describe the plasma evolution in both the Alfvin and diffusive transport time scales. Since
these time scales are separated by many orders of magnitude, economical time integration
requires some numerical artifact. In TSC, this is accomplished by assuming an artificial ion
mass equal to a factor
a factor
ffe
fjee
times the actual mass. Hence, the Alfvin speed is reduced by
so that a practical integration time step that is still smaller than the shortest
characteristic time scale can be adopted. In order to obtain realistic predictions, a convergence study towards the physical value
ff.,
= 1 must be carried out. If no instabilities
growing on the Alfvin time scale are present, an adequate description is obtained with as
large
ffee
values (typically of the order of 10') as to make the Alfvin time comparable
to the resistive diffusion time. This is the assumption made when modeling the evolution
of a whole tokamak discharge, as done in Ref. 2 for the full 3 s duration of an Alcator
C-MOD shot. However, to test the stability against axisymmetric modes that can grow in
Alfvin-like time scales, a detailed convergence towards low
ffpe
values is necessary. Due
to computer limitations, this can be carried out only over short real time intervals.
In this work we chose to analyze the axisymmetric stability of the Alcator C-MOD
1
nominal configuration at the middle of the current flat-top, as generated by the TSC
simulation described in Ref. 2, 1.5 s into the discharge. At this point, the plasma current
equals 3 Mamp, the toroidal field at the plasma center is 9 T, the major and minor radii
are R, = 0.665m, a = 0.21m, the separatrix elongation and triangularity are x. = 1.75,
& = 0.4, and at the 95% relative to the separatrix flux surface n95 = 1.6, 95 = 0.3. The
reader is referred to Ref. 2 for further details and definitions. A TSC run is restarted
at this time with the correct currents in the active coils, but with the induced currents
in the vacuum vessel suppressed. This results in an initial transient phase of unphysical
eddy currents induced in the vacuum vessel that, since the plasma equilibrium is vertically
asymmetric, are sufficient to trigger the vertical instability. The evolution of the latter is
studied by following the plasma motion for the next 16 ms with low ffac values. This is
repeated under two different scenarios. First, the instability passive growth rate is obtained
by switching off the feedback control systems and letting the active coils carry only the
preprogrammed currents. Second, the active control of the instability is investigated by
switching the feedback systems on.
Two independent observables are used to diagnose the plasma evolution: the vertical
position of the magnetic axis and the poloidal magnetic flux difference between two observation loops located at R = 0.80 m, Z = ± 0.62 m. Figures 1 and 2 show their time
derivatives as functions of time in semi-logarithmic plots, for the passive growth simulation
without feedback. MKS units are used in all figures unless otherwise specified. The results
displayed in Figs. 1 and 2 are obtained with
ff.c
= 1500, but their characteristics are
general. Three distinct phases can be identified: an initial transient phase is followed by a
period of linear exponential growth after which the perturbation amplitude becomes too
2
large, the plasma evolution enters a non-linear regime and the equilibrium is rapidly lost.
The evolution of the plasma boundary during this 16 ms simulation with ff.c = 1500 is
shown in Fig. 3. Figure 4 displays the corresponding induced currents in the vacuum vessel. The linear phase lasts for two exponential growth times approximately, which allows
to establish a well defined linear growth rate. Such inverse linear growth rates are plotted
in Fig. 5 as functions of ff.c. Circles and triangles correspond to the values obtained with
the magnetic axis displacement and flux difference diagnostics, respectively. Two sets of
points are shown in Fig. 5. They correspond to two values of yet another artificial parameter, a numerical viscosity introduced in TSC to ensure numerical stability, that should
be extrapolated to zero. By extrapolating the data in Fig. 5 to ffac
=
1, we obtain the
physical passive growth time of about 1 ms. The characteristic L/R time constant of the
vacuum vessel is considerably larger, about 16 ms. This rather fast instability growth rate
reflects the fact that no close fitting conductor structure exists around the Alcator C-MOD
plasma, and makes the problem of its active feedback control a challenging one.
The feedback control system implemented in TSC is a two-level proportional-derivativeintegral (PDI) scheme defined in the following way. Given a preprogrammed current IP,
(function of time) in a coil or linear combination of coils, the feedback current is defined
by:
dA
Ifb = IP, +gPA+9D dt + 9 If
Adt
,
(1)
where A is the feedback signal and gp,gD,gI are the gain coefficients. The difference
between If b and the actual coil current at the time I,,, is used to determine the incremental
applied voltage which in our calculation is simply given by a proportional law:
AV=gPV(Ifb-I.c)
3
.
(2)
Five independent such systems are used in our simulation. Their characteristics and gain
coefficients are summarized in the following table.
TABLE I
I
A
IEP3
(A7P) 1
IQPU
(AP) 2
IQPL
(A03
INUL
AIplama
IEFC
AZazi
gp
gD
i
-4 x 102
0
-1.5 x 105
0
0
-1.5 x 10 5
5.2 X 102
0
0
0
1.3 x 104
-2 X 103
0
-- 4
X 105
-105
Table I: Definition of the active feedback systems. The gain coefficients are in MKS units.
The first system acts on the vertical field coil EF3 and controls the radial position of the
plasma. The second and third systems control the upper and lower elongation respectively
and act on the linear combinations:
IQPU(L) = IOH2U(L)
-
0. 8 0IEF1U(L) + 0-10EF2
(3)
that produce approximate quadrupole fields. The feedback signals for these first three
systems are the flux differences between pairs of points that are expected to lie on the
same magnetic surface and are moving in time. The fourth system controls the total
plasma current and acts on the combination:
INUL
= 1. 5 9 IOH1 + 1.49(IoH2U +
IOH2L)
(4)
+ 0.48 (IEF1u +
IEF1L)
+ 0.021EF2 + 0. 1 3 iEF3
that produces an approximate field null. Its signal is the difference between the actual
and preprogrammed plasma current. The fifth system controls the vertical position of the
4
plasma and acts on the EFC coil. This coil has its upper and lower sections connected
in anti-series to produce a radial magnetic field. Its signal is the difference between the
actual and preprogrammed vertical position of the magnetic axis. The feedback signals
used in our simulation are supposed to mimic the information gathered by the whole set of
magnetic probe measurements in the actual experiment. The voltage gain coefficient gj, is
set equal to 0.04 ohm for all systems. In addition, a maximum voltage cutoff of 20 V is set
on each coil filament of our model, regardless of the demands of the feedback algorithm,
in order to simulate realistic power supply and insulation constraints.
The axisymmetric stability simulation is now repeated with these feedback systems
in place. The resulting magnetic axis displacement and flux difference signals are shown
in Figs. 6 and 7 as functions of time for the 16 ms simulation. Results are given for four
values of the Alfven velocity slowing factor
ff.,
down to ff~c = 250. In contrast with the
passive simulation illustrated in Figs. 1 - 5, we can conclude now that the instability is
suppressed by the active feedback even after we extrapolate to low ff.c. The evolution of
the plasma boundary through this 16 ms simulation is shown in Fig. 8 for the case
ff.,
=
1500, to be compared with Fig. 3. The induced currents in the vacuum vessel are shown
in Fig. 9, to be compared with Fig. 4. The currents and voltages in the active coils are
given in Figs. 10-25, all of which show a stable behavior. We point out that the 20 V
per filament limit on the vertical control coil EFC (160 V total for its four upper and four
lower filaments) is sufficient to provide stability in our simulation. The EFC power supply
in the actual Alcator C-MOD tokamak is rated at a maximum ±500 V.
Finally we wish to call attention to the glitch observed around t = 1.512 s in the
=
ff.,
250 case. This is a spurious feature, presumably triggered by some numerical (finite
5
mesh size or truncation) inaccuracy, which is nonetheless successfully quenched by the
feedback system. Such numerical problems set a practical limit as far as the convergence
analysis with respect to ffac is concerned, because increasing the numerical accuracy and
decreasing the computational grid spacing are unaffordable with our present day supercomputer capabilities. However, the results obtained with
ff.
basis for predicting a stable extrapolated behavior at the physical
> 250 provide sufficient
ffac
= 1.
Acknowledgements
The author thanks S. Jardin for providing the TSC code and offering continuous
assistance in its running. He also appreciates the useful discussions and help provided
by the members of the Alcator group, especially S. Fairfax, R. Granetz, P. Hakkarainen,
D. Humphreys, I. Hutchinson and S. Wolfe. This work was supported by the U.S. Department of Energy under Contract No. DE-AC02-78ET51013.
6
References
[1] S.C. Jardin, N. Pomphrey and J. DeLucia, J. Computational Physics 66, 481 (1986).
[2] J.J. Ramos, Massachusetts Institute of Technology Report PFC/RR-90-2 (1990).
7
Figure Captions
[Fig. 1] Vertical displacement of the magnetic axis in the passive axisymmetric instability
simulation with ff., = 1500.
[Fig. 2] Flux difference between observation points at R = 0.80 m, Z = t0.62 m in the passive
instability simulation with ffee = 1500.
[Fig. 3] Evolution of the plasma boundary in the 16 ms passive instability simulation with
ff dc= 1500.
[Fig. 4] Vacuum vessel induced currents in the passive instability simulation with
ffhc
= 1500.
Each trace corresponds to one filament of our computational model.
[Fig. 5] Linear growth times of the passive axisymmetric instability as functions of the Alfvin
velocity slowing factor ffec.
Circles and triangles are obtained with the magnetic
axis displacement and flux difference diagnostics, respectively. The two sets of points
correspond to two values of the numerical viscosity.
[Fig. 6] Vertical displacement of the magnetic axis in the active feedback control simulations.
[Fig. 7] Flux difference between observation points at R = 0.80 m, Z = 0.62 m in the active
feedback control simulations.
[Fig. 8] Evolution of the plasma boundary in the 16 ms active feedback control simulation
with ff,, = 1500.
[Fig. 9] Vacuum vessel induced currents in the active feedback control simulations.
[Fig. 10] Current in OH1 coil with the active feedback systems on. Magnitudes plotted correspond to one filament of our model.
8
spond to one filament of our model.
[Fig. 11] Voltage in OH1 coil with the active feedback systems on. Each trace corresponds to
one filament of our model.
[Fig. 12] Current in OH2U coil with active feedback on.
[Fig. 13] Voltage in OH2U coil with active feedback on.
[Fig. 14] Current in OH2L coil with active feedback on.
[Fig. 15] Voltage in OH2L coil with active feedback on.
[Fig. 16] Current in EF1U coil with active feedback on.
[Fig. 17] Voltage in EF1U coil with active feedback on.
[Fig. 18] Current in EFIL coil with active feedback on.
[Fig. 19] Voltage in EF1L coil with active feedback on.
[Fig. 20] Current in EF2 coil with active feedback on.
[Fig. 21] Voltage in EF2 coil with active feedback on.
[Fig. 22] Current in EF3 coil with active feedback on.
[Fig. 23] Voltage in EF3 coil with active feedback on.
[Fig. 24] Current in EFC coil with active feedback on.
[Fig. 25] Voltage in EFC coil with active feedback on.
9
102
IdAZ I
'p
'p
'p
-i
/
10
TR
L
'p
NL
00p*
1
000
10~
I
I
1.500
1.504
1.508
Figure 1
10
1.512
1.516
10
dAqi
dt
1TR
L
NL
10-
10
1.500
1.504
1.508
Figure 2
11
1.512
1.516
.7
.6
.4D
.3Z9
.2
.0
-.
-.
-3.
1l
2~
o
0
C
Figre
12
C
n
'
10
__
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0
C:)
-10
LnUl)
Ul)
U-)
Figure 4
13
)U
L
Y<
2 xv
0
A
0
SW0
.003
0
A
.002
0
0
A
A
.001
I
I
0
A
ff ac
I
I
125250
500
1000
Figure 5
14
1500
dAZ
dt
10-1
ffac= 250
1000
1C-2
1.500
1500
I
I
I
1.504
1.508
1.512
Figure 6
15
It
1.516
d Aq
dt
10~
fac=250
o-2
103
1000
t
1.500
I
I
I
1.504
1.508
1.512
Figure 7
16
I
1.516
I
7
T
I
.11 L.L-L L 1.
.6
I LLLII
I
I
c
,I
I
D
EM
P99M
CKDOM
LM
.4
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DQXDOM
Domm
DZKDM
DONN
.3
.2
.0
-. 4
D
-. 5
-. 6
-. 7
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CN
-~
*
I~~
C"
CD0
Figure 8
17
C.
U,
~O
2.5
1 0
-
-----
-
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---~----
---
.
ffac =
1 .0
1500
.5
c:
.01
e.>
r
2
. I
I
TIME(SEC)
-
fac = 1000
C
0
Q=
ifIME (SEC)
f ac
o
00
500
0
oc
0
0~
II ME
0
(SEC)
2
ffac
a-
o
a
cc
C
TIME(SEC)
Figure 9
18
=250
0
-- 20
-40
ffac
= 1500
S -60
Q
o
o
-J
e
a
4,
4,
4,
fl
40
a
0
- 20
-40
3
____
____
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0
4,
4,
a
4,
UTUIME( SEC)
40
I--i
____
44
ifl
____
is
*
4~
4,
ffac = 1000
____
-60
I
o
o
4S
I
-.
a
~
a
4~
h
SIE(
0
-~
C
4,
a
1
0
--
SC)
a
_____
20
_____
_____
ffac
-40
7t717171.
S -60
-.-
o
o
4,
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a
C
a
-
a
4,
--
*0
0
.0
~
-
tIIl
-
C
0
r4
.0
4,
4,
0
= 500
(SEC)
4,
0
-20
ffac = 20
-40
-60
CD,
Figure 10
19
_
0 00
.
I
-. 005
- .010
i- i
-. 015
o
020
-.
I
U
-
-0
I
E
C*4
*
1 .1 i
=
S
ffac
TM(
E
= 1500
C)
.000 1-
-.005
-J-.010
- 0,0
015-
--
-
______
I I I I
-
-
-
ffac =
1000
ac
= 500
-
.
o- .020
* -
-_-
o
o
r~
.
to
~
o
r.
-
to
-
-
-
a,
a,
0
a,
0
a,
a~
a,
-
-
-
-
-
-
-
-
4
I
I
o
o
a,
a,
.000
-.
I
a~
I
005
-. 010
-+
4
4-4-4-4
-. 015
o -. 020
ur(~rr~
ri-S,-sI
T
o
a
0
0
0
-
-
-
-
a,
a,
Sn
tC~
U~
a,
a,
a,
a,
-
-
-
.
-
-
-
-
4
.000 !-. 005
7-7
I
fac = 250
-. 010
-. 015
oL.
TIME(SEC)
S-.020
go.-
-
Figure 11
20
0
-10
ffac =
1500
-20
CL
0T
IME(SEC
0
0
0
0
-
-
-
-00
f ac = 1000
-20
I_
a
I
.
1&I
-
-
I ME(
SEC)
, )0
0
: -10
ffac
= 500
-20
.0
(N
0
.0
.0
0
-I--
-b-
_-~
-
o
o
0
.0
C
C
0
0
.0
.0
,ME ( SEC)
-
(N
.0
.0
~1
0
-10
-t
a
ffac = 250
-20
1e
-30
-
-
-
a
~~
~
-
-
U
-
~
-
-
I
*4
C lo,-
Figure 12
21
-S-
-~4
TIN(SEC)
(
.000
I-
-
I
-
-. 005
ffac = 1500
- 010
I
-. 015
-
cc
.
I(SEC)
0
4
-
.000
f ----- -I
-. 005
0- .010
- .015
020
-
.000
0
0
.
0
IMr(SEC)
-
10
-
4
-
I
-. 005
-
1
_
0
ffac = 1000
/I(
I
.ini
~
ffac
010
=
500
* -. 015
-
I IVE(SEC)
-. 020
-. 020
-o
ri
IC
a
a
N
f
.000 1 7,
-. 005
ffac = 250
.010
-. 015
CL.
nuTI
S -.020
us~W
us
U
Us
us
in
-
Figure 13
22
4
(sEC)
0
3
-20
ff ac = 1500
-30
0t
-50
T IU~ I ~fi~
......
I
0
0
5
. . -.-
.
..
-
0
-
I
-Z
-
71W7L-
0
-10
S -20
ffac = 1000
-30
-40
CD
I
-50
-
____
o
~
0
~~
*
-
- 0
eI
C
-
*
E
*
IOEC
0
5
ffac = 500
-20
-30
3
-40
-50
ed '0
C
C
d
*
T
iME(SEC)
0
-10
ffac = 250
-20
-30
a-.
3
-40
-50
TIME(SEC)
o
o
ai
~
0
fl
C
0
C
0
C
0
~
0
.4,
.~
.4,
Figure 14
23
C
.4,
C
-
C
.000
-. 005
-
ff ac = 1500
010
- .015
cc
o
Im
-. 020
((SEC)
.000
ff ac = 1000
o
-
'N
-
t
e
C
.-
rim
E(SCC)
.000
-. 005
ffac = 500
- 010
- .0
--
5
020-
-
TIM
-
(SEC)
.000 1
0
-.
003
-.
010
ffac =
- .02j
.o
ot
in
t
o
o
t
t
Figure 15
24
ot
s
e
250
~w
~-
10
ffac = 1500
4
2
0
C
'.4
o
4
0
'0
'0
C
0
40
0
0
'0
'0
IME(SEC)
0
49
'0
-
'0
.0
4
40
-
-
.0
'0
ffac = 1000
6
-'
4
aD
2
IME (SEC)
0
0
0
C.
0
0
Z;?0
0
.
-
-
-
.
0
0
-
'
10
at
8
f fac = 500
6
4
o
0
I
L..........
.4.
'
________
0
0
40
C
0
0
'0
0
.0
'0
~
'N
~
O
-
12
4
4
2
-
0
'0
-w
-
(.4
.0
.0
-
ME (SEC)
0
'0
-
10
f ac = 250
6
a-
4
2
A
-
o
o
a
.0
..
4-
40
.d~
.0
'0
a
a
e
Figure 16
25
a
~
4
7 IME(SEC)
T
.005
.000
- 005
ffac
= 1500
>-. 010
-. 015
TIME(SEC)
o- .020
e~.
=
=
a
=
C
0
0
-.
0
fl
0
u~
0
10
-
C'.
U)
C
tO
0
t~
C
-
-4
005
-
ffac = 1000
005
0
-. 010
1
-
o
-. 020
lMt(
___
___
______
___
'-
EC)
40
-
.01
00
-9-
ffac =
500
01
-.
-02
-
02'
cc
0
IME(SEC)
C
C
C
C
C
AVh,
.01
fc
.00
-. 01
-.
02
I
CL.1
TIN(( s EC)
-
-
-
-
-
Figure 17
26
-
-
= 250
20
15
10
a-
ff ac
o
44
-.
40
C
0
44
*
o
o
o
0
0
-
-
-
= 1500
5
0
IME(
0
-v
- ~
-w-
-
SEC)
-
-~
20
ffac = 1000
x
C.
15
TI W
SEC)
0
40
040
44
*
4
20
I
x
1
e
10
5
ffac
=500
ac
=250
0.
IME(SEC)
1 5
20
--
----
---
-
-
-
-----
-
10
0
-
-
0
o
44
a
4,,
0
440
0
4 ,
j
C
0
40
0
40
Figure 18
27
T IMC(SEC)
.005
.000
'
-. 005
ffac
= 1500
ff ac
= 1000
-. 010
U->
-. 015
o -.
020
-
IME(SEC)
m--
.005
000
005
-
-. 010
-. 015
0
020
-.
'i-l
_E__'_______)
.01
00
ffac = 500
01
_________
________
a
-
r~.
________
_________
a-
a
U~
(0
_________
a
________
r
-
(
U~
(
IME(SEC)
01
o
a
U,
U
0
U
~
00
ffac = 250
.01
0.
S- .02
02.
a
a
a3
IMC(SEC)
a
('4
.
w
U,
U,
UI
-
-
4
Figure 19
28
12
=~w
~3
=1
=~= =~
10
S
8
ffac
= 1500
4
CD
2
A
IM(SEC)
--
'2
U
0
10
C
-
-~
-4
-~
~-
-~
10
S
8
ffac = 1000
6
4
(c2
-L
-
0
12
-
-~
-- TIlMO(SIC)
9-
-.---
-~
-U-
-~
10
S
8
ffac = 500
6
CL
4
2
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Figure 20
29
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30
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Figure 22
31
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Figure 23
32
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33
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Figure 25
34
25