PFC/RR-9-9 DOE/ET-5113-284 TSC Simulations of Alcator C-MOD Discharges ll: Study of Axisymmetric Stability J.J. Ramos Plasma Fusion Center Massachusetts nstitute of Technology Cambridge, MA 2139 June 199 This work was supported by the U. S. Department of Energy Contract No. DE-AC2-78ET5113. 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 : Study of Axisymmetric Stability J. J. Ramos PLASMA FUSON CENTER, MT, Cambridge, MA 2139 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 twodimensional, 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. n TSC, this is accomplished by assuming an artificial ion mass equal to a factor fjee times the actual mass. Hence, the Alfvin speed is reduced by a factor ffe so that a practical integration time step that is still smaller than the shortest characteristic time scale can be adopted. n order to obtain realistic predictions, a convergence study towards the physical value ff., = 1 must be carried out. f 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 1') 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. n 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, =.665m, a =.21m, the separatrix elongation and triangularity are x. = 1.75, & =.4, and at the 95% relative to the separatrix flux surface n95 = 1.6, 95 =.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 =.8 m, Z = ±.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 = 15, 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 = 15 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 (PD) scheme defined in the following way. Given a preprogrammed current P, (function of time) in a coil or linear combination of coils, the feedback current is defined by: da fb = P, +gpa+9d dt + 9 f Adt, (1) where A is the feedback signal and gp,gd,g are the gain coefficients. The difference between f b and the actual coil current at the time,,, is used to determine the incremental applied voltage which in our calculation is simply given by a proportional law: AV=gPV(fb-.c). (2) 3
Five independent such systems are used in our simulation. Their characteristics and gain coefficients are summarized in the following table. TABLE A gp gd i EP3 (A7P) 1 -- 4 X 15-4 x 12 QPU (AP) 2-1.5 x 15 QPL (A3-1.5 x 1 5 NUL Aplama 5.2 X 12 1.3 x 14 EFC AZazi -15-2 X 13 Table : 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: QPU(L) = OH2U(L) -. 8 EF1U(L) + -1EF2 (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: NUL = 1. 5 9 OH1 + 1.49(oH2U + OH2L) +.48 (EF1u + EF1L) +.21EF2 +. 1 3 ief3 (4) that produces an approximate field null. ts 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. ts 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.4 ohm for all systems. n addition, a maximum voltage cutoff of 2 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 = 25. n 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., = 15, 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. 1-25, all of which show a stable behavior. We point out that the 2 V per filament limit on the vertical control coil EFC (16 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 ±5 V. Finally we wish to call attention to the glitch observed around t = 1.512 s in the ff., = 25 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. > 25 provide sufficient basis for predicting a stable extrapolated behavior at the physical 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,. Hutchinson and S. Wolfe. This work was supported by the U.S. Department of Energy under Contract No. DE-AC2-78ET5113. 6
References [1] S.C. Jardin, N. Pomphrey and J. DeLucia, J. Computational Physics 66, 481 (1986). [2] J.J. Ramos, Massachusetts nstitute of Technology Report PFC/RR-9-2 (199). 7
Figure Captions [Fig. 1] Vertical displacement of the magnetic axis in the passive axisymmetric instability simulation with ff., = 15. [Fig. 2] Flux difference between observation points at R =.8 m, Z = t.62 m in the passive instability simulation with ffee = 15. [Fig. 3] Evolution of the plasma boundary in the 16 ms passive instability simulation with ff dc= 15. [Fig. 4] Vacuum vessel induced currents in the passive instability simulation with ffhc = 15. 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 =.8 m, Z =.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,, = 15. [Fig. 9] Vacuum vessel induced currents in the active feedback control simulations. [Fig. 1] 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 EFL coil with active feedback on. [Fig. 19] Voltage in EF1L coil with active feedback on. [Fig. 2] 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
12 daz -i / 'p 'p 'p 'p 1 TR L NL p* 1 1~ 1.5 1.54 1.58 1.512 1.516 Figure 1 1
1 daqi d t 1- TR L NL 1-1 1.5 1.54 1.58 1.512 1.516 Figure 2 11
.7.6.4D.3Z9.2. -. 1l -. 2~ -3. o C C n ' Figre 12
1 C:) -1 LnUl) Ul) U-) )U L Figure 4 13
.3 Y< A 2 x v.2 A A A A SW.1 12525 5 1 15 ff a c Figure 5 14
daz dt 1-1 ffac= 25 1C-2 1 15 1.5 1.54 1.58 1.512 1.516 t Figure 6 15
1~ d A q d t fac=25 o-2 13 1 1.5 1.54 1.58 t 1.512 1.516 Figure 7 16
7 T.6.4.11 L.L-L L 1. LLL c, LM EM P99M CKDOM DQXDOM Domm DZKDM DONN D.3.2. -. 4 -. 5 D -. 6 -. 7 CN -~ * ~~ C" CD C. U, ~O Figure 8 17
2.5 1 1..5 c:.1 e.> - ----- - ---- - --- ~---- -. r. TME(SEC) ffac = 15 2 - fac = 1 C Q= ifme (SEC) f ac 5 o o c ~ ME (SEC) 2 ffac =25 a- o cc C a TME(SEC) Figure 9 18
-- 2-4 ffac = 15 Q S -6 o -J fl 4 4 44 * is o e a a a - - - - 4, 4, 4, 4, 4, 4, ifl 4~ 4, UTU ME( SEC) - 2-4 --i ffac = 1 3-6 h o -. -~ o a a a a 4S ~ 4~ C 4, SE( SC) -- 2-4 S -6 1 7t717171. -.- - a -- ~ - - tl (SEC) '-.~ - * C r4 o a a - - - - 4, C 4,.. 4, 4, 4, ffac = 5-2 -4 ffac = 2 CD, -6 Figure 1 19
. _ -. 5 -.1 -. 15 i - i 1.1 i E U o -. 2 TM( - C) - S = C*4 * E ffac = 15. 1- -.5 -J-.1 -, -- 15-. o-.2 - - * - -_- o r~. to ~ o r. - to o o o - - - - a, a, a, a, a, a, a~ a~ a, - - - - - - - - 4 - - - ffac = 1. -. 5 -. 1 -+ 4 4-4-4-4 ac = 5 -. 15 o -. 2 T ur(~rr~ ri-s,-s o a - - - - a, a, Sn tc~ U~ a, a, a, a, - - -. - - - - 4.!- -. 5 -. 1 7-7 fac = 25 -. 15 ol. S-.2 TME(SEC) go.- - Figure 11 2
CL -1-2 ffac = 15 T ME(SEC - - - - f ac = 1-2 _ 1&. ME( SEC) a - -, ) : -1-2 - - - - - - - - U -~- -Sffac = 5 - _-~ -b- --- o (N - C C (N - o - - - -....... ~1.,ME ( SEC) -1 a -t ffac = 25 1- -2 e -3 a ~ ~ ~ ~ C lo,- *4 4 ( TN(SEC) Figure 12 21
. -. 5 - - - - 1 ffac = 15 - -. 15 cc. (SEC) - 4. -. 5 -.1 -.15 - f -- --- ffac = 1-2 _ 1. - 1 Mr(SEC) - 4. - -. 5 ~ / (.ini - 1 ffac = 5 * -. 15 - -. 2 -. 2 o - ri C a a N f VE(SEC). 1 7, -. 5.1 ffac = 25 CL. -. 15 S -.2 nut (sec) us~w us U Us us in - 4 Figure 13 22
3-2 -3 ff ac = 15 t -5 T U~ ~fi~..... -.-... - 5 - -Z - -1 71W7L- S -2-3 ffac = 1 CD -4-5 - * - * - o ~ ~~ e C - E * OEC 5-2 -3 ffac = 5 3-4 -5 ed ' C C d * T ime(sec) -1-2 -3 ffac = 25 a-. 3-4 -5 o ~ C C C.~ C C o - - - - ai fl ~.4,.4,.4, C TME(SEC) Figure 14 23
. -. 5-1 - ff ac = 15 cc -.15 o -. 2 m ((SEC). ff ac = 1 - - o 'N - t C e.- rim E(SCC). -. 5-1 ffac = 5 -. 5 -- 2- - - TM (SEC). 1 -. 3 -. 1 ffac = 25 -.2j.o o t in t o o t t ot s e Figure 15 24
1 ~w ~- 4 ffac = 15 2 C '.4 4 4 C 49 4 4 o - - - - ' ' ' '. ' '. ' ME(SEC) a- -' 6 4 D 2 ffac = 1 C. - - - - ME (SEC) Z;?.. ' 1 at 8 6 f fac = 5 4 o 2 4 4 L....4. ~ 'N - 4 C (.4 ~ - - - - O '. ' '.. ' ' ME (SEC) 12 1 - - - -w - - 6 f ac = 25 a- 4 2 A - 7 o.. 4-4 o a a a e..d~. ' ME(SEC) a ~ 4 T Figure 16 25
.5. - 5 > -. 1 ffac = 15 -. 15 o-.2 TME(SEC) = e~. -. C = C C'. - t~ = - - - - a u~ fl 1 to U) C - -4 5-5 -. 1 ffac = 1-1 o -. 2 '- - 4 lmt( EC).1-9- ffac = 5 -. 1-2 2' - C cc C C C C ME(SEC).1 AVh,. fc = 25 -. 1 CL.1 -. 2 TN(( sec) - - - - - - - Figure 17 26
2 15 a- 1 o 44 -. 4 C 44 * 5 ME( SEC) ff ac = 15 o o o - - - - - ~ -v - -w- -~ 2 ffac = 1 x C. 15 T W SEC) 4 4 44 * 4 2 x 1 5 e 1 ffac =5. ME(SEC) 1 5 2 1 -- --- ---- - - - ----- - ac =25 - - T 44 - - C o a - 4,, 44 j 4, 4 4 MC(SEC) Figure 18 27
.5 '. -. 5 -. 1 U-> -. 15 ffac = 15 o -. 2 - m-- ME(SEC).5-5 -.1 ff ac = 1 -. 15 -. 2 'i-l _E ' ).1 1 ffac = 5 a r~. a - a a r - ( ME(SEC) - 1 o a U, U U~ ( U ~ U~ ( ffac = 25.1. S-.2 2 ȧ a a3 a ('4. w U, U, U - - 4 MC(SEC) Figure 19 28
12 1 ~3 =~w =~= =~ =1 S 8 4 ffac = 15 CD 2 A -- C 1 M(SEC) '2 1 U -~ - -~ ~- -~ -4 S 8 6 4 ffac = 1 (c2 - -L -- TlMO(SC) 12 1 - -~ 9- -~ -.--- -~ -U- S 8 6 ffac = 5 CL 4 at 2 A~ o -. - ~...... ME(SEC),2 -. -. 1. ffac = 25 8e CD 2 L o (N.4" (N t C - - -.(i.fl... ~... itiwc(sec) Figure 2 29
. -. 5 o -. 1 ffac = 15 o. -. 15 TME(SEC) - a - - -. 1------r -. 5 1 f ac =1 c C -. 5 o r- ( -. = C~. P ME( SEC). -.5 1 f r w/ia l- ffac = 5 -. 15 + + o -. 2 -- ME( SEC) -.3 - -. 5 -.1 = 25 -. 15. o -. 2 C= a C. - 4 q 4 Figure 21 3
m -1 3-2 -3 ff ac = 15-4 S -5 - - -- 1 o 'N '4- ' ' * T M((S(C) 3-2 -3 ff ac = 1. 3-4 -5 _ ~= -'= -- - _ 'ME(SEC) o z - m -1-2 -3 ffac = 5. S -4 <3-5 _- - - - a, a, a a, a,, a, a, a - -~ -o T t ME (SEC) S -1-2 -3 ~- -4 ----------- ffac = 25 (3-5 C ('4 '4- ' *" *~ - o - - - - ~ U, U, 4~ ' ' ime(sec) Figure 22 31
.1. -. 1 ------ ffac = 15 o -. 2 o - to ~ C tn ~ to C C C - - - - 4~. n n 1 to U, to - - - - - - - - 4 MC(S[C) 5 ]1 1 -- 7. 1 - ffac = 1 -. 15 M - - -.,= - 2 At C C-------- M (str A 1 1 1 ffac = 5 _2 M SEC).5.1 -. 1 - fac = 25 1 -. 15 -. 2 ~ ME(SEC) Figure 23 32
fac = 15 CD, ME(SEC) --- N - - - - - - - - - - - T ~ ffac = 1 a. NE(SEC) o ------ - o+ - d --- - -* 1 T ffac = 5 1. x.5 o a a - - - --- 4E (SEC) -5 CD, 4) 4) 4).5 4) ) 4)All4 fac =25-1. - -MF 1-T Ftr 1 ~F a ox CW 1!N Figure 24 33
2. -. 1 -~ - fac = 15..2 QD d t = C~~4 4. 1 C C tt tt g~ - - - - 4 ME(SEC) 2 1T.1 - - ~- -4- ~ ffac = 1 -. 2 o N. * C = C (N - i o o o C C - - - - ft C U- Cl n tt n tt t - - - - - - - - -.4 T dm(sec) 2 1 - +Q~ -4- ffc = 5 o -. 2 H U - f (WE (SEC).2 - o m -.1. ffac = 25 an -. 1 344 o-.21 4 Figure 25 34