of HIGH betan*h SIMULATIONS TOKAMAK ADVANCED IN DIII-D WITH DISCHARGES 3D NONLINEAR CODE NFTC. THE N.N.Popova A.M.Popov, State University Moscow La Haye, A.D.Turnbull V.S.Chan,R.J. Atomics General M.Murakami Ridge National Laboratory Oak APS meeting, Quebec Canada,23-27 October 2000
ABSTRACT Nonlinear self-consistent MHD stability simulations of neoclassical tearing modes (NTM) in high betan Advanced Tokamak (AT) are presented. Radially localized electron cyclotron current drive (ECCD) current profile control is considered based on DIII- D discharge # 99411 in which fi N = 3:9 and H89P-factor=2.9 are reached. Simulations were performed with the full 3D nonlinear code NFTC. Neoclassical terms are included in the basic equations for the magnetic field and pressure. An effective fully implicit numerical scheme allows the transport profile to evolve self-consistently with the nonlinear MHD instabilities and an externally applied source such as ECCD. NTM activity with m/n=2/1 is found in simulations to correspond to experiment. It is shown that magnetic islands can be quickly suppressed by localized ECCD at q=2 before the mode grows substantially. The time response and nonlinear evolution of the 2/1 island width for ECCD and required modulation phasing, the CD location with respect to the q=2 surface, and the width of the spatial distribution are determined. The possibility of q-profile modification by ECCD well before MHD activity to keep q min > 2, so that the discharge evolves stably is also discussed. 2
discharge AT in DIII-D by three-dimensional MHD code NFTC. ffl Simulations of the nonlinear MHD stability of high fi N # 99411 at the moment t=1800ms (q min = 1:67) -Discharge The problem of critical seed island necessary to triger NTMs. - Comparison with experiment. - of NTM instability by radially localized ECCD around -Suppression with q min > 1:5 and NCS (Negative Discharges Shear) are the leading scenario for AT (Advanced Toka- Central operation in DIII-D. mak) NTM modes still remain a significant obstacle to approaching - ideal fi limit. the High fi N (fi N ο 3) is oserved in long pulse NCS discharges.no - GOAL q=2.(magnitude of ECCD, width of driven current, location) ffl Motivation. sawteeth or fishbons to provide the seed island.
MODEL ffl The nonlinear three dimensional evolution of a tokamak plasma is described by the full (nonreduced, compressible) MHD system of equations in general toroidal geometry. ffl We seek the solutions fv; B; Pg by decomposition: V(ρ; ; '; t) = V 0 (ρ; )+ ^V(ρ; ; t) z } n=0 B(ρ; ; '; t) = B eq (ρ; )+ ^B(ρ; ; t) z } n=0 P (ρ; ; '; t) = P eq (ρ)+ ^P (ρ; ; t) z } n=0 + V(ρ; ; '; t) z } n6=0 + B(ρ; ; '; t) z } n6=0 + P (ρ; ; '; t) z } n6=0 ffl Our problem is to find fv; B; Pg for a given fv 0 ; B eq ;P eq g. ffl The solutions satisfy the fixed boundary conditions: B n = 0; V n = 0; P = 0: Neoclassical terms in the model are included : ffl Helical and axi-symmetric bootstrap current terms are added into the magnetic field equation. ffl Polarization current effect is included in the magnetic field equation as a helical current perturbation. ffl Radially localized toroidal current density from ECCD is included in the magnetic field equation. ffl The equation for pressure is modified by incorporated the perpendicular transport across a seed magnetic island and along magnetic field lines. 4
BASIC EQUATIONS. ρ @V @t = ρ[(v 0 r)v +(V r)v 0 ] fi 0 rp + +[[r B eq ] B] +[[r B] B eq ]+ν 4 V ρ[( ^V r)v +(V r) ^V] +[[r ^B] B] @B @t @P @t +[[r B] ^B] ρ[(v r)v] +[[r B] B]; (1) = [r [V B eq ]] + [r [V 0 B]] [r ( [r B])] + +[r [ ^V B]] + [r [V ^B]] + +[r [V B]] + [r E P ]; (2) = r (PeqV) r (P V 0 ) ( 1)[Peq(r V) +P (r V 0 )] r ( ^P V) r (P ^V) ( 1)[ ^P (r V) +P (r ^V)] r (P V) ( 1)[P (r V)] +r? (χ? r? P )+r k (χ k r k P ); (3) ρ @ ^V @t = ρ[(v 0 r)v 0 +(V 0 r) ^V +(^V r)v 0 ] fi 0 r ^P + +[[r B eq ] ^B] +[[r ^B] B eq ]+ν 4 ^V (4) ρ( ^V r) ^V +[[r ^B] ^B] + @ ^B @t @ ^P @t +[[r B] B] n=0 ρ[(v r)v] n=0 ; = [r [ ^V B eq ]] + [r [V 0 ^B]] [r ( [r ^B])] + +[r [ ^V ^B]] + [r [V B] n=0 ]+[r ^E P ]; (5) = r (Peq ^V) r ( ^P V 0 ) ( 1)[Peq(r ^V) + ^P (r V 0 )] r ( ^P ^V) ( 1) ^P (r ^V) (r (P V)) n=0 ( 1)[P (r V)] n=0 +r? (χ? r? ^P )+rk (χ k r k ^P ) + Q; (6) P = f (ρ; T ): (7) 5
MODEL. The sources of the current density The total parallel current density is the sum of Ohmic current and the non-inductive current: ~j = j eq + ^j + ~ j = j Ω + j BS + j pol + j cd ; Axisymmetric and helical components of current density perturbation: ^j + ~ j = ( j eq + ^j Ω + ^j BS + ^j cd ) + j ~ Ω + ~ j BS + ~ j pol + ~ j cd z } z } n=0 n6=0 Source term for the equations is included as [r E P ], where E P = ^E P = ( j eq + ^j BS + ^j cd ) for n = 0 E P = ( ~ j BS + ~ j pol + ~ j cd ) for n 6= 0 form: The bootstrap current j BS is included in the simplest model ^j BS = Ag 1:46 p "[ @ ^P=@ρ B pol ]; ~j BS = Ag 1:46 p "[ @P n6=0=@ρ B pol ]; where Ag = O(1) is the geometric parameter of the model. The polarization current ~ j pol is included in the model form: ~j pol = W 2 th W 2 ~j BS where W th is the parameter of the model. 6
ffl Representation of current density from ECCD Current density j cd from ECCD is represented as a radially localized toroidal current Jcd = j cd e';j cd = j(ψ? ) on the perturbed helical magnetic flux ψ?. A normalized helical flux function : ~ψ? = (ρ ρ s ) 2 ± W mn 2 cos(m n' + ffß) 8 represents the local behaviour near the resonant surface ρ s. The island width W is defined as W =4 s q 2 s Y 1 (ρ s ) mρ s q 0 ; where Y 1 = p gb 1 and g is a metric element. Define j cd ( ~ ψ? )=I cd 1 2ß 3=2 W cd e ~ ψ? =W 2 cd The following simplified form of j cd ( ~ ψ? ) is used in NFTC code. 1 j cd = I cd e (ρ ρ 2 0 ) =Wcd[1 2 ± 1 2ß 3=2 W cd ( Wmn 8 W cd ) 2 cos(m n' + ffß)] Parameters: - I cd (tkw ) is the total ECCD current.this can have any arbitrary dependence on time. A square wave is used in the simulations presented here. - W cd is the width of radial distribution, - ρ = ρ 0 ρ s denotes the offset of the driven current layer from the rational surface. -Φ(t) = ffß is the phase fitted to the magnetic island rotation. 7
MODEL ffl Modification of the transport equations @P @t @ ^P @t = + r? (χ? r? P )+r k (χ k r k P ); = + r? (χ? r? ^P )+rk (χ k r k ^P )+[rk (χ k r k P )] n=0 + Q; The unit vector along magnetic field line is defined as : b tot B tot jb tot j ο = B eq + ^B + B jb eq j b eq + ^b + b: The differential operator along magnetic field line is: r k P tot = (b tot r)p tot = (b eq r+ ^b r+ b r)(p eq + ^P + P ) The total heat conductivity operator is: fi r 2 P k totfi n6=0 r 2 P k tot = b eq r(b eq rp + b rp eq ) + + b eq r(b rp )+b r(b eq rp )+b r(b rp eq )+ + b r(b rp ) + + [^b r(b eq rp + b rp eq )+b r(b eq r^p + ^b rp eq )+ + b eq r(^b rp )+b eq r(b r^p )+ + ^b r(b rp )+b r(^b r^p + ^b rp + b r^p )+ + ^b r(^b rp + b r^p )]; fi fi n=0 = b eq r(b rp)+b r(b eq rp)+b r(b rp eq ); r 2 k ^P fi fi fi fi = (b eq + ^b) r((b eq + ^b) r^p ): n=0 Underlined terms in the cylindrical approximation: + r 2 k P m 2 4 Y 1 1 p g @ @ρ ( 1 8 2 fi fi < fi cos(m n') = 1 4 n6=0 : F 2 P + FP 0 jb eq j 2 eq 39 1 Y @P = p jb eq j 2 g @ρ ) 5 ; mn 3 1 p Y 1 5 + g cos(m n'); F mn = B 3 eq (m q n): 8
10 3 10 4 10 6 1.25 0.00 1.0 ADVANCED TOKAMAK DISCHARGE IN DIII D (AT DURATION ENDED BY 2/1 TEARING MODE) SHOT 99411 BT(set) = 1.60 T 0.0 4.0 0.0 2.5 2.0 1.5 5.0 IP(A) P INJ (KW) D α β N 4 l i q 0 mse q min mse n=2 rms (T) n=1 rms (T) 0.0 0 250 500 750 1000 Time (ms) DIII D NATIONAL FUSION FACILITY SAN DIEGO 1250 1500 1750 2000 282-00/jy
Helical magnetic perturbations in # 99411 discharge. ffl First splash at t=1300ms after q min ο 2:0 and fi N ο 3:0 ffl Second low level splash when q 0 decreases to the value 2.0. ffl The third, basic splash at t=1800ms. Perturbation increases by 20 times.fin = 3:78; qmin = 1:67. Mirnov signal, rms Btheta. n=1 B,T x10-3 0 1 2 3 4 5 6 1.0 1.2 1.4 1.6 1.8 2.0 2.2 a Time, msec Mirnov signal, rms Btheta. n=2 x10 3 x10-3 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 B, T 1.0 1.2 1.4 1.6 1.8 2.0 2.2 b Time x10 3 10
Equilibrium radial profiles in # 99411 discharge. ffl At t=1800ms only one rational surface q=2 located on ρ s = 0:74a H and q = q 0 q min = 0:13, Shear S=1.28. ffl At t=1915ms two rational surfaces located on ρ 1 s ρ 2 s = 0:68.S(1) = 0:16; S (2) = 0:43. q-profile = 0:42 and q 2.0 2.5 3.0 3.5 1 - t=1915 0 t=1800 1 0 0.0 0.2 0.4 0.6 0.8 1.0 a ρ Pressure Peq x10-2 0 1 2 3 4 5 6 7 Peq 0 1 0.0 0.2 0.4 0.6 0.8 1.0 b ρ 11
Nonlinear calculations of NTM for eq.1800ms. ffl Find W th for m/n=2/1 as amplitude of j pol with Eq.1.8s". ffl Two parameters of the model: Ampl and W th with W d = 0. - First, increase Ampl eigefunction (withw th = 0until NTM is exited. Ampl, W th. -Second, with fixed W cr change W th until W(t), W sat saturation will be found. ffl Then found W th a ffl W cr a = 0:045 and W sat a = 0:2 = 0:0235 and this is used for all equilibriua. Islands evolution, Eq.1800 msec, qmin=1.67 (dw,w)-diagram 0.0 0.5 1.0 1.5 2.0 Wm/n m/n=2/1 3/1 4/1 0.0 0.5 1.0 1.5 2.0 x10 3 a Time x10-4 -3-2 -1 0 1 2 3 dw/dt 0.0 0.5 1.0 1.5 2.0 b Wm/n Jbs Evolution t=1800 Jbs -6-5 -4-3 -2-1 0 1 0.0 0.5 1.0 1.5 2.0 x10 3 c Time 12
Time evolution of nonlinear solution. ffl Helical perturbations of magnetic field ρ B ρ (ρ; t) and pressure P (ρ; t) for m/n=2/1. ffl Axisimmetric component of quasilinear corrections for poloidal field and pressure. ffl Profiles alter from green to red Time evolution of Y1s(2/1) profile( Eq.1800) Pol.magn.field Y2^(m=0) qmin=1.67 r.brho~ x10-3 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 a Helical pressure Pc(2/1) ρ Y2^ -1.0-0.5 0.0 0.5 1.0 x10-2 5 6 7 8 9 b ρ Pressure. Quasilinear corr.p^(m=0) P~ -3-2 -1 0 1 x10-3 P^ -3-2 -1 0 1 x10-3 0.5 0.6 0.7 0.8 0.9 1.0 c rho 5 6 7 8 9 d rho 13
Critical island width evaluation. ffl Magnetic islands m/n=2/1 evolution with different seed islands for eq.1.8s. ffl Bootstrap current density for m/n=2/1 in the fixed radial point. W m/n=2/1 islands evolution, Eq.1800 msec 0.5 1.0 1.5 2.0 2.5 Wm/n 4 3 2 1 2 - W_0=0.045 1 - W_0=0.030 3 - W_0=0.052 4 - W_0=0.060 0.0 0.5 1.0 1.5 2.0 x10 3 Time Bootstrap current Jbs 2/1 evolution Jbs m/n=2/1-1.4-1.2-1.0-0.8-0.6-0.4-0.2 0.0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 x10 3 Time 14
Time evolution of basic harmonics in nonlinear solution. ffl m/n=2/1 NTM instability excites toroidal modes n=2 with same helicity. ffl Harmonic m/m=4/2 increases to the level W 4=2 ο 0:09aH and m/n=6/3 has W 6=3 ο 0:07aH. ffl The double NTM 5/3 has the large island width W 1 5=3 = 0:1a H and W 2 5=3 = 0:07a H located in the middle of plasma area with low shear on q=5/3. Low toroidal coupling with 2/1. Islands evolution, Eq.1800 msec, qmin=1.67 Wm/n 0.0 0.2 0.4 0.6 0.8 1.0 1.2 3/1 4/1 7/3 5/2 6/2 6/3 m/n=2/1 4/2 5/3 5/3 0.0 0.2 0.4 0.6 0.8 1.0 1.2 x10 3 Time 15
Magnetic islands due to NTM instability. Magnetic islands t=200 t 0A Eq."1800msec" ρ 0.4 0.6 0.8 1.0 4/1 6/2 5/2 7/3 2/1 5/3 5/3 0 1 2 3 4 5 6 a θ q-profile t=200 t 0A (Eq.1800) q 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 d ρ 16
Magnetic islands due to NTM instability. Magnetic islands t=600 t 0A Eq."1800msec" ρ 0.4 0.6 0.8 1.0 4/1 6/2 5/2 7/3 2/1 5/3 5/3 0 1 2 3 4 5 6 b θ q-profile t=600 t 0A (Eq.1800) q 1.8 2.0 2.2 2.4 2.6 2.8 3.0 0.2 0.4 0.6 0.8 1.0 e ρ 17
Magnetic islands due to NTM instability. Magnetic islands t=1200 t 0A Eq."1800msec" ρ 0.4 0.6 0.8 1.0 4/1 6/2 5/2 7/3 2/1 5/3 5/3 0 1 2 3 4 5 6 c θ q-profile t=1200 t 0A (Eq.1800) q 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 f ρ 18
Comparison of NTM calculations with experiment. ffl Amplitude of magnetic perturbation n=1 and magnetic island width W 2=1 as a function of time. ffl The m/n=2/1 harmonic dominates in magnetic perturbation. ffl The level and radial profile of quasilinear correction of equilibrium poloidal magnetic field Eq.1.8s corresponds to MSE diagnostic. Comparison with experiment.signal on Mirnov coils Bp. Bp/Btor 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x10-3 Bp/Btor experiment W_2/1 X 0.1 Bp/btor NFTC 0 20 40 60 80 100 120 140 Time, msec ; simul.timex30 Comparison with MSE diagnostic.t=1.80-1.84s. (Bpol(t)-Bpol(1.8s))/Btor x10-2 -0.5 0.0 0.5 1.0 NFTC experiment 0 2 4 6 8 rho 19
Conventional double tearing instability when q min crosses 2.0. ffl Succsesive changes of eigenfunctions for decreasing q min. ffl When q 0 = 2:25 ( q min = 1:993) and q 0 = 2:2 (q min = 1:95 ) conventional double tearing mode 2/1 is weakly unstable. Corresponds to t=1300ms in discharge.nonlinearly stabilizes. ffl When q 0 = 1:87(q min = 1:67) conventional tearing mode is stable.nonlinear NTM unstable. Corresponds to t=1800ms in discharge. qmin=1.99 Radial magnetic field Y1(r,m,n) m/m=2/1 qmin=1.95 Y1 sin x10-7 0 2 4 6 8 Y1 sin -2.5-2.0-1.5-1.0-0.5 0.0 0.5 x10-6 0.0 0.2 0.4 0.6 0.8 1.0 ρ qmin=1.67 0.0 0.2 0.4 0.6 0.8 1.0 ρ λ vs. qmin 4 Y1 sin -8-6 -4-2 0 x10-4 0.0 0.2 0.4 0.6 0.8 1.0 ρ, m/n=2/1-8 -6-4 -2 0 λ 1.7 1.8 1.9 2.0 q min 20
fi N limit due to NTM instability. ffl Comparison of theoretical limit fi cr N = C ps with experimental data for # 99411 discharge. ffl Equilibrium profiles and papameters are fitted to the moment t=1.8s in experiment.linear approximation of q min (t) = 2:075 0:225t in the period of instability. ffl When t=1.3 parameter q 0 = 2:0 and q min = 1:78 the first MHD burst happens. When t=1.8 the main NTM instability is excited. Time evolution of Beta_N (exp.) and NTM Beta_N limit. Beta_N 1.5 2.0 2.5 3.0 3.5 4.0 Beta_N limit Beta_N q0=2.0 qmin q0=1.87 1.0 1.2 1.4 1.6 1.8 2.0 Time, s Critical Beta_N vs. qmin Beta_N, m/n=2/1 0 1 2 3 4 sqrt(s) Beta_N=C sqrt(s) 1.7 1.8 1.9 2.0 qmin 21
Decreasing of central pressure due to NTM instability. ffl Time evolution of quasilinear correction of equilibrium pressure profile shows the decreasing of central pressure due to NTM instability. ffl Heat conductivity is artificially increased by two orders in the area of magnetic islands overlapping.(4/2,6/3,5/3,..). Pressure collapse. Quasilinear correction P^(m=0) P^ -1.5-1.0-0.5 0.0 x10-3 0.0 0.2 0.4 0.6 0.8 1.0 rho 22
Magnitude of ECCD current needed for NTM suppression. ffl a) W(t) with different magnitude of Icd/Ip. W cd = 0:06a H. Island width begin to oscilate. Critical seed island W cr = 2=1 0:045a H. If I cd =I p > 0:055 then W sat = W max = 0:02a H. ffl b) W sat 2=1 and W max 2=1 vs. Icd/Ip. Islands evolution, Eq.1800 msec, qmin=1.67 0.0 0.5 1.0 1.5 2.0 Wm/n m/n=2/1 3/1 4/1 I cd =0.0 Icd/Ip=0.011 Icd/Ip=0.023 Icd/Ip=0.034 I cd /I p =0.0057 0.0 0.5 1.0 1.5 2.0 a Time x10 3 Island Suppression vs.icd. 0.5 1.0 1.5 2.0 Wmax, m/n=2/1 Wmax Wsat, t=2000ta 0 1 2 3 4 5 b Icd/Ip x10-2 23
The effect of driven current layer width Wcd. ffl The effectiveness of magnetic islands suppression increases with decreasing of driven current layer Wcd, when the magnitude of Icd/Ip is constant (here 0.023). ffl The less Wcd the less period of oscillations.w max ß W cd. The best conditions: if W cd > 0:5W cr seed then no oscillations during period of suppression. Islands evolution, Eq.1800 msec, qmin=1.67 Wm/n 0.0 0.5 1.0 1.5 2.0 Icd=0.0 I cd /I p =0.023 Wcd=0.1 m/n=2/1 Wcd=0.08 Wcd=0.06 3/1 Wcd=0.02 4/1 0.0 0.5 1.0 1.5 2.0 a Time x10 3 Wmax, m/n=2/1 0.2 0.4 0.6 0.8 1.0 Island Suppression vs.wcd. Wmax Wsat 0.2 0.4 0.6 0.8 1.0 b Wcd 24
The effectiveness of magnetic islands suppression. ffl Area of magnetic islands suppression on the planes: (I cd =I p ; W cd =a H ) and (I cd =I p ; W=W cd ). ffl Each point on curves corresponds to the same maximal magnetic islands width equal W 2=1 = 0:06a H obtained during calculations. Icd/Ip vs. Wcd/a 0 1 2 3 4 5 x10-2 Icd/Ip Wmax > 0.06a Wmax < 0.06a 0.0 0.2 0.4 0.6 0.8 1.0 Wcd/a Icd/Ip vs. W/Wcd x10-2 Icd/Ip 0 2 4 6 8 Area of suppression 0.4 0.6 0.8 1.0 1.2 1.4 1.6 W/Wcd 25
The effect of max(jcd) Location ffl Maximum of stabilization effect occurs when ρ = 0.When ρ W cd =a H, then the stabilization effect is lost.wcd = 0:06a H for these cases. Islands evolution, Eq.1800 msec, qmin=1.67 Wm/n 0.0 0.5 1.0 1.5 2.0 I cd /I p =0.034 ρ=0.0 ρ=+0.05 ρ=-0.05 ρ=0.0, I cd =0.0 0.0 0.5 1.0 1.5 2.0 x10 3 Time Wmax vs. ρ = ρ0 - ρs Wmax m/n=2/1 0.5 1.0 1.5 2.0 t=1000 t=500-1.0-0.5 0.0 0.5 1.0 ( ρ 0 -ρ s ) 26
ECCD suppression vs. islands rotation. ffl Magnetic islands width during ECCD suppression with different islands rotation. ffl Comparison of ECCD efficiency for only axi-symmetric component and axi-symmetric and helical components of j cd. Islands evolution, Eq.1800 msec, qmin=1.67 x10-2 Wm/n 0 2 4 6 8 I cd /I p = 0.034 ω = 0.02 Wcd/a H = 0.03 ω = 0.0 0.0 0.5 1.0 1.5 2.0 Time x10 3 Wsat vs. Icd/Ip Wsat, m/n=2/1 x10-2 1 2 3 4 5 6 Axi-symmetric mode Axi-symmetric + helical mode 2 3 4 5 6 Icd/Ip x10-2 27
CONCLUSIONS ffl Stability calculations by NFTCofhighfi N?H AT indiii-d(# 99411 discharge ) were performed. Comparison of nonlinear NTM calculations with experimental data shows - correspondence with the level of the helical magnetic perturbations with Mirnov signal - correspondence of quasilinear correction of equilibrium poloidal field with MSE diagnostic. ffl Explanation of critical seed island were done as aftereffect of conventional tearing instability when q min crosses the value q min = 2. ffl A radially localized current from ECCD is included for NFTC code as a source term. Driven current j cd depends on perturbed magnetic surfaces and has helical and axisymmetric components in its representation. Self-consistent calculations using the NFTC code show the effectiveness of ECCD suppression of neoclassically destabilized magnetic islands. ffl ECCD current which is needed for the island suppression increases parabolically with the driven current layer width W cd The less W cd is, the less total current is needed for suppression. Magnetic island can be suppressed to the level of order W cd. For W cd < 0:05a H and I cd = 0:05I p the seed magnetic islands are suppressed and NTM stabilizes at t=1800ms in discharge # 99411. ffl Localization of max(j cd ) relating rational surface of mode is very important fro stabilization. Maximum shift is of order ρ max ο W cd =a H, which is equal to 0:05a H for # 99411 discharge. 28