EFFECT OF ION CYCLOTRON HEATING ON FAST ION TRANSPORT AND PLASMA ROTATION IN TOKAMAKS

Transcription

1 EFFECT OF ION CYCLOTRON HEATING ON FAST ION TRANSPORT AND PLASMA ROTATION IN TOKAMAKS by V.S. Chan, S.C. Chiu, and Y.A. Omelchenko Presented at the American Physical Society Division of Plasma Physics Meeting Quebec City, Canada The authors acknowledge many useful discussions with F.W. Perkins and the use of the ORBIT code from R.B. White 57 00/VSC/wj October 3 7, 000

2 MOTIVATION Recently, Alcator C Mod has discovered that plasmas with fast wave minority heating can develop an appreciable co-current toroidal rotation in the vicinity of the magnetic axis, even though the heating method provides negligible angular momentum [Rice, J.E., et al., Nucl. Fusion 38 (1998) 75; Rice, J.E., et al., Nucl. Fusion 39 (1999) 1175] A mechanism proposed by Perkins and White (invited talk DPP00) and demonstrated using Monte-Carlo orbit simulations showed a number of relevant features No angular momentum input from the energetic minority ions Using a non-slip boundary condition at the plasma edge, co-rotation is computed with low-field side (LFS) ion cyclotron resonance heating The angular rotation scales with stored energy 57 00/VSC/wj

3 PRESENT STUDY Issues An explanation of a net co-current torque with LFS resonance and a net counter-current torque with HFS resonance in Perkin s model The model predicts counter-current toroidal rotation when the resonance is on the high field side (HFS); recent C Mod experiment (000) did not see this Approach: An rf diffusion operator is added to the Monte-Carlo orbit code (ORBIT originally developed by R.B. White) Stochastic kick in perpendicular energy consistent with quasi-linear diffusion Wave zone modeled after full wave code; electric field E + consistent with code Finite N (symmetric or asymmetric spectrum) and Doppler resonance included (Omelchenko DPP99, see detailed description of code later in this poster) 57 00/VSC/wj

4 KEY RESULTS A more realistic model of rf heating can produce a toroidal drift in the velocity distribution for minority species with symmetric wavenumber spectrum i.e. can induce angular momentum torque (consequence of finite orbit). However, it is small for the cases studied Two cases were studied Case 1: RF created initial energetic ions, magnetic and frictional torque allowed to balance 3 Co-current rotation with LFS resonance heating Due to favorable frictional torque of energetic, co-passing minority ions 3 Counter-current rotation with HFS resonance heating Due to favorable frictional torque of barely-untrapped, counterpassing minority ions Case : Using a model for a steady-state minority ion source, similar features to Case 1 are reproduced 57 00/VSC/wj

5 A MORE REALISTIC RF HEATING MODEL CAN PRODUCE AN ASYMMETRIC VELOCITY DISTRIBUTION RF heating increases the minority ion orbit from thin to fat bananaution Constant of motion: Ψ * = Ψ I v /Ω Collisions break orbit: v (co-passing) > v (counter-passing) Z/a 1.0 Resonance v co v ctr The spatial distribution of fast ions peaks near the resonance sdn (x10 6 ) ? 00/VSC/wj Density Source Poloidal Flux PDF X/a Resultant distribution is asymmetric in v with net momentum in the co-direction t = v ζ

6 CASE 1: RF CREATED INITIAL ENERGETIC IONS, MAGNETIC AND FRICTIONAL TORQUE ALLOWED TO BALANCE Collisions scatter energetic particles inward and outward creating radial currents Magnetic Torque Poloidal Flux Collisional Torque Poloidal Flux J bulk xb torque on the bulk due to Frictional torque on bulk exerted by return current in response to minority minority ions due to combination of ions moving in (out): VxB and non-uniform radial distribution J minority out (in) J bulk xb counter- (co-) of co- and counter-passing ions 5? 00/VSC/wj

7 LOW FIELD SIDE RESONANCE PRODUCES CO-CURRENT ROTATION 0.4 Total Torque 0.3 Rotation Poloidal Flux Poloidal Flux Sum of JxB and frictional torque Balancing torque against angular gives a net positive torque near momentum diffusion (τ E = τ M ) the magnetic axis and using non-slip boundary condition, co-rotation profile is obtained 5? 00/VSC/wj

8 POSITIVE FRICTIONAL TORQUE NEAR THE AXIS FOR LFS IS DUE TO MORE CO-PASSING IONS VxB force on minority ions inside resonance is counter-direction frictional force on bulk is negative, cannot explain result LFS resonance (with N = 0) heats deeply trapped particles (not easily detrapped by coilisions). From finite drift orbit analysis (Chiu DPP99, see details below), particles scattered inwards can either remain trapped or become co-passing co-current frictional torque is favored Z/a 1.0 LFS Resonance Deeply trapped particle X/a 5? 00/VSC/wj

9 An orbit illustrating this process is shown here v ζ Trajectory 4 r/a Trajectory 4 Z/a 1.0 Trajectory Time Time X/a Toroidal velocity becomes positive with time co-passing Radius moves inward on average Cross section Fat banana orbit becomes co-passing orbit 5? 00/VSC/wj

10 HIGH FIELD SIDE RESONANCE PRODUCES COUNTER-CURRENT ROTATION 0.5 Magnetic Torque Collisional Torque Poloidal Flux Poloidal Flux J bulk xb torque is similar to Frictional torque clearly negative LFS both due to collisional near the axis scattering of resonance ions inwards and outwards 5? 00/VSC/wj

11 Total Torque 0.1 Rotation Poloidal Flux Poloidal Flux Sum of JxB and frictional torque gives negative torque near the center (frictional torque larger) Rotation is in the counter direction 5? 00/VSC/wj

12 NEGATIVE FRICTIONAL TORQUE NEAR THE AXIS FOR HFS IS DUE TO MORE COUNTER-PASSING IONS The VxB frictional torque exerted on the bulk is largely cancelled by J bulk xb (in zero banana width approximation, they exactly cancel) HFS resonance heats barely trapped particles (easily detrapped). Because of finite orbit width, counterpassing particles moved inside and co-passing particles moved outside negative frictional torque inside Z/a 1.0 Trajectory 1 8 HFS Resonance Barely trapped particle X/a 5? 00/VSC/wj

13 An orbit illustrating this process is shown here 1.0 Trajectory Trajectory Trajectory 1 4 v ζ r/a 0. Z/a Time Toroidal velocity becomes negative with time Time Radius decreases with time on average X/a Cross section Barely trapped particle becomes counter-passing 5? 00/VSC/wj

14 ANALYSIS OF ORBITS NEAR THE AXIS α p α pc 1 α pc 1.0 (a) α p n E n E ε ε α p 3 (b) Co-particle start at ε 0 = 0.4, θ = 0; (a) enlarged at lower n E, (b) at higher n E 57 00/VSC/wj α po 4 (c) 0.6 α p n E ε Counter-particle start at ε 0 = 0.4, θ = 0 ψ ε = Ia R 1 E η E = Ω0 R 1 µb 0 η = Ω0 R 1 η p = η E η α p = η p /η E

15 MONTE-CARLO ORBITS Ion trajectories are calculated by solving the Hamiltonian guiding center (drift) equations [9]: d dt where Pθ = I ρ + α { Pς, Pθ} = ς, θh, ( ) + ( ) Ψ, Pς = g ρ + α d dt { } =, (1) ςθ, P ς, P θ H 1 Ψ p, H = ρ B + µ B + Φ () The axisymmetric equilibrium field is expressed through its contravariant and covarient forms: B = ς ΨP + q ( Ψp) Ψp θ, B = g( Ψp) ς + I( Ψp) θ + δ Ψp (3) where the poloidal angle, θ is chosen so that d3 x = JdΨ p dθς d, J = gq + I B ( ) (4) 57 00/VSC/wj

16 MONTE-CARLO RF AND COLLISION OPERATORS Each time an ion passes through the ion cyclotron resonance layer [ω rf k v = Ω(B)] its perpendicular velocity component undergoes a random change, v. This increment can be readily obtained from the quasi-linear equation governing the rf-induced particle diffusion in velocity space [Chiu (1999)]: (k = 0, E_ = E = 0): 1 e µ = µ rf ± µ µ rf, µ i rf = + τ mb E rf, (5) i { τ rf = τ uc = ( π 1 Ω ) τc = πai( ς) Ω 13, ς = Ω Ω 3 if τ uc τ c if τ uc τ c (6) Pitch angle scattering and slowing-down collisions are modeled with simple operators [Boozer, Kuo-Petravic (1981)]: λ ν tλ 1 λ ν t, v = ν tv (7) = ± ( ) 57 00/VSC/wj

17 NON-SLIP BOUNDARY PLASMA ROTATION In a straight cylinder the steady-state equation for the parallel momentum density is: 1 r r r χ r nmv i i ς = F ς( r ), Anomalous momentum diffusion: χ ~ a/6 τ M : τ M = τ E Non-slip boundary condition: v ς (r=a) = 0, Zero angular momentum input: a Fς( r ) rdr = /VSC/wj

18 CASE : STEADY-STATE SIMULATION-THERMALIZED FAST IONS ARE RE-INJECTED INTO THE INITIAL ANNULUS AS MAXWELLIAN PARTICLES Initial and final fast ion velocity distributions are approximately the same indicating a good approximation to steady state PDF PDF 8 t = t = V s 5? 00/VSC/wj

19 BOTH FAST ION SPATIAL DENSITY DISTRIBUTION AND ICRH ABSORBED POWER DENSITY ALSO REACH STEADY STATE Spatial fast ion density peaks around the resonance in a driven system 3.0 Density Absorbed power density similar to calculation from full-wave code Power Density.5 8 dn Poloidal Flux pdn (x10 8 ) Poloidal Flux 5? 00/VSC/wj

20 fdn The magnetic torque shows the ± features from in/out diffusion of fast ions 5? 00/VSC/wj THE STEADY STATE TORQUES SHOW THE SAME FEATURES AS IN THE CASE WITH NO RE-INJECTION The frictional torque shows the ± features as explained in Case 1 (HFS) Collisional Torque Poloidal Flux jdn tdn Magnetic Torque Poloidal Flux The net torque is negative resulting in counter-rotation for HFS resonance Total Torque Poloidal Flux

21 CONCLUSIONS Using a realistic quasi-linear rf operator to self-consistently simulate fast ion heating in a Monte Carlo drift orbit code, we found: Co-current rotation with LFS resonance heating 3 Due to favorable scattering of potato co-passing orbits toward the magnetic axis producing co-frictional torque on bulk Counter-current rotation with HFS resonance heating 3 Due to favorable detrapping of boundary layer particles into counterpassing particles inside the resonance location leading to counterfrictional torque on bulk These features remain in a steady-state rf-driven system The co-rotation observed in Alcator C Mod for HFS resonance remains to be explained. Future studies: higher power, finite k, edge condition 57 00/VSC/wj

22 ORDER OF MAGNITUDE DIII D C Mod R M 0.67M a 0.65 M 0. M B T 8 T q s.1.1 R M 0.16 M ε0 0.1 (0 cm) 0.15 (10 cm) η Ec ρ 0.79 cm 0.53 cm Ec Ec (kev) ~15 ~ /VSC/wj

23 1. ANALYSIS OF ORBITS NEAR THE AXIS A Orbits are well described by constants of motion: E u = + µ B, µ= u, Pζ B Iu = ψ, Ω where B B, 0Ω0 ( Ω) = ( ) 1+εcosθ, I = ± I a = RBT Basic length scale: (δ l = elongation) R R1 = δ l 0 qs 57 00/VSC/wj

24 Parameters of orbits: (i) Flux surface parameter (ε = inverse aspect ratio): ε = ψ IR a 1 (ii) Invariant energy parameters: E µ η E = Ω 0 R 1, η = B 0 0 R Ω 1, ηp = ηe η. We can also define pitch angle αp=ηp/η E. (iii) Angular momentum parameters: η ζ ζ = P IR a /VSC/wj

25 Then orbit is well described by ε ηζ ηp ηeε cos θ ( ) = + Alternatively, this can be written as ηζ = ε σ ηp + ηeε cos θ η E, ηp, η ζ and σ completely determines the orbits and thus forms the phase space of orbits σ = + 1 is for a co-current moving particle, and σ = 1 is for a counter-current moving particle 57 00/VSC/wj

26 It can be shown that for a given set of η s, a passing co-orbit always encloses a passing counter-orbit The orbits of co-and-counter orbits are split, in contrast to thin banana theory, in which they are degenerate in the lowest approximation. This split results in asymmetry in power deposition and subsequent radial excursions 57 00/VSC/wj

27 B Analysis of orbits: On the equatorial plane, the orbit intercepts are given by ε ηζ ηp ηeε ( ) = + ε > 0 (<0) represent intercepts at θ = 0 (= π). Orbit topology is easily deduced graphically An alternative label for η ζ is intercept at θ = 0, i.e, ε0 ηζ = ε 0 σ ηp + ηeε0, then [( ε ε ) 0 ( ε+ ε0 ) + σ ηp + ηeε0( ε + ε0) ] ηe = /VSC/wj

28 (a) Passing ε (b) Barely Trapped ε ε (c) Trapped 57 00/VSC/wj

29 (a) σ = + 1 (co-current) where 3 δε Critical energy η 0 EC =. There exists two transitions of orbit topologies for 7 η E < η EC as a particle pitch angle is increased above the transition values αpc and αpc1. 4εα 3 αpc1 = ε 0 Tci 0 1 i = 1, 9η, E 1 1 π αtci = + α θr 4 1cos i 1 E + ( ) 3 θr = α α 3 tan E( 4 E) 10α α E. E 57 00/VSC/wj

30 For αp < αpc, particles are trapped. For αpc1 > αp > αpc, a co-passing orbit encloses a counter-passing orbit. For αp > αpc1 only one co-passing orbit exits There is a line in the αp-η E plane: αp3 η ε E = 0 ε0 η E. For values of η E greater than those on this line, co-passing particles are confined on the low-field side of the magnetic axis (b) σ = + 1 (counter-current) There is only one transition of orbit topologies marking the trapped-passing boundary of co-and-counter particles at αp = αpc4 which is given by 4εα 3 αpc4 = ε 0 Tc η E 57 00/VSC/wj

31 where αtc4 θt4 α E αE 1 3 α α α α E + 10 E α α 4 13 = + + E ( E ) + E E for αe 4 4 ( ) > = α θt4 α 4 E + 1 cos ( ). for E > α α 3 tan E( 4 E) = α E. + 10αE There is an η E4 at which αpc4 = 1. For η E >η E4, all counter-particles starting from (θ=0, ε=ε0) are trapped 57 00/VSC/wj