Modelling of Frequency Sweeping with the HAGIS code S.D.Pinches 1 H.L.Berk 2, S.E.Sharapov 3, M.Gryaznavich 3 1 Max-Planck-Institut für Plasmaphysik, EURATOM Assoziation, Garching, Germany 2 Institute for Fusion Studies, University of Texas at Austin, Austin, Texas, USA 3 EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, UK Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 1
Structure of Talk What is frequency sweeping? Experimental evidence Theoretical understanding Numerical modelling Description of the HAGIS code Simulations of frequency sweeping Summary Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 2
Experimental Observations Frequency [khz] Frequency sweeping in MAST #5568 140 120 100 80 64 68 66 Frequency sweep δω/ω 0 ~ 20% Time [ms] 70 72 Chirping modes exhibits simultaneous upwards and downwards frequency sweeping More experimental details in talk by M. Gryaznavich this afternoon Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 3
JET Observations Shear optimised D-T pulse TAE modes during current ramp phase Frequency sweep δω/ω 0 ~ 5% Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 4
Frequency Sweeping Universality in nonlinear response of resonant particles to low amplitude wave [Berk, Breizman, Pekker (1997)] Particle distribution satisfies a 1-dimensional equation (two phase-space coordinates) Constants of motion for wave E(r,t) = C(t) E(r,θ,nφ - ω 0 t) Magnetic moment, µ (if ω 0 «ω c and L ω > ρ i ) Energy in rotating frame, H = H - (ω 0 /n) P ζ (if 1/C dc/dt «ω 0 ) Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 5
Wave-Particle Interaction Define: As changes due to interaction at fixed Equations of particle motion for fixed - sinξ Hence, Pendulum equation Trapping frequency, F is a phase space dependent form factor Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 6
Nonlinear Trapping in TAE Trapping frequency is related to TAE amplitude Frequency sweep is related to trapping frequency [Berk et al., (1997)] Amplitude related to frequency sweep Analytic estimates give correct order of magnitude. Numerical simulation required for more accurate estimate. Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 7
Aim Use experimentally observed rate of frequency sweeping to determine wave amplitude In general, numerical modelling is needed to establish the form factor that relates δω and δb Validate HAGIS for model case Employ HAGIS to establish δb in general case General geometry (including tight-aspect ratio) Mode structure: global mode analysis Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 8
The HAGIS Code [Pinches, Thesis (1996)] Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 9
Code Overview Straight field-line equilibrium Boozer coordinates Hamiltonian description of particle motion [White & Chance 1984] Fast ion distribution function δf method Evolution of waves Wave eigenfunctions computed by CASTOR Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 10
Equilibrium Representation Coordinates chosen to produce straight field lines General toroidal geometry Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 11
Particle Description Exact particle Lagrangian, is gyro-averaged and written in the form, with leading to 4 equations Particle trajectory Guiding centre trajectory [White & Chance 1984] Magnetic field line Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 12
Fast Particle Orbits ICRH ions in JET deep shear reversal On axis heating : Λ = µb 0 /E = 1 E = 500 kev Produces predominately potato orbits z [m] J. Hedin, Thesis 1999 R [m] Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 13
Distribution Function Represented by a finite number of markers Markers represent deviation from initial distribution function - so-called δf method Dramatically reduces numerical noise where Parker & Lee, 1993 Denton & Kotschenreuther 1995 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 14
Wave Equations Linear eigenstructure assumed invariant Introduce slowly varying amplitude and phase: Gives wave equations as: Additional mode damping rate, γd where Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 15
HAGIS Code Performance Run time [s] 6000 5000 4000 3000 2000 1000 1/n scaling Linux Cluster Wall-clock time to calculate TAE linear growthrate in ITER-like case 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Number of processors, # HAGIS code parallelises very well relatively low level of inter-processor communication traffic Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 16
Self-Consistent Frequency Sweeping Equilibrium: n=3 TAE a/r 0 = 0.3 q0 = 1.1 f Radially peaked fast ion profile E 0 = 3.5 MeV s q p f Slowing down distribution s s Energy [MeV] Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 17
Linear Growthrate γ d /ω 0 = 0, β f = 3 10-4 Mode saturates at δb/b~10-3 γ d /ω 0 =2.7% n p = 52,500 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 18
with additional damping γ d /ω 0 = 2%, β f = 3 10-4 Mode saturates at much lower level, δb/b~10-4 n p = 210,000 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 19
Frequency Sweeping Fourier spectrum of evolving mode ω 0 Frequency sweep δω/ω 0 ~ 10% Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 20
Linear Growthrate γ d /ω 0 = 0, β f = 7.5 10-5 Mode saturates at δb/b~4 10-5 γ d /ω 0 =0.45% n p = 52,500 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 21
with additional damping γ d /ω 0 = 0.4%, β f = 7.5 10-5 Mode saturates at much lower level, δb/b~3 10-6 n p = 210,000 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 22
Frequency Sweeping Fourier spectrum of evolving mode ω 0 Frequency sweep δω/ω 0 ~ 2% Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 23
Fast Ion Redistribution Resonant energy changes as mode sweeps in frequency Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 24
MAST #5568 Obtain factor relating ω b and δb E b = 40 kev a/r 0 = 0.7 B 0 = 0.5 T R 0 = 0.77 m Global n=1 TAE Monotonic q-profile Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 25
Particle Trapping in MAST Particles trapped in TAE wave All particles have same H = E - ω/n P ζ = 20 kev TAE amplitude: δb/b = 10-3 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 26
Scaling of Nonlinear Bounce Frequency 4.00E+04 3.50E+04 3.00E+04 2.50E+04 HAGIS 1 Linear (HAGIS 1) MAST #5568 Monotonic q profile H = 20 kev ω b 2.00E+04 1.50E+04 1.00E+04 5.00E+03 ω b = 1.156 10 6 (δb/b) 0.00E+00 0.00E+ 00 5.00E- 03 1.00E- 02 1.50E- 02 2.00E- 02 2.50E- 02 3.00E- 02 3.50E- 02 (δb/b) Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 27
Scaling of Nonlinear Bounce Frequency 3.00E+04 2.50E+04 HAGIS 1 HAGIS 2 Scaling MAST #5568 Reversed shear 2.00E+04 H = 20 kev ω b 1.50E+04 1.00E+04 5.00E+03 ω b = 7.2 10 5 (δb/b) for δb/b < 2 10-4 0.00E+00 0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02 2.50E-02 3.00E-02 3.50E-02 (δb/b) Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 28
Mode Amplitudes For monotonic q-profiles we now know: where For a single resonance, where Therefore, Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 29
TAE Amplitude in MAST 140 Frequency [khz] 120 100 df = 18 khz dt = 0.8 ms 80 64 68 66 70 72 Time [ms] Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 30
Conclusions Frequency sweeping has been modelled using the HAGIS code Benchmarked against analytic theory The amplitude of a frequency sweeping mode in MAST has been calculated to be δb/b = 4 10-4 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 31