Modelling of Frequency Sweeping with the HAGIS code

 Everett Rose
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1 Modelling of Frequency Sweeping with the HAGIS code S.D.Pinches 1 H.L.Berk 2, S.E.Sharapov 3, M.Gryaznavich 3 1 MaxPlanckInstitut 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
2 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
3 Experimental Observations Frequency [khz] Frequency sweeping in MAST # Frequency sweep δω/ω 0 ~ 20% Time [ms] 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
4 JET Observations Shear optimised DT 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
5 Frequency Sweeping Universality in nonlinear response of resonant particles to low amplitude wave [Berk, Breizman, Pekker (1997)] Particle distribution satisfies a 1dimensional equation (two phasespace 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
6 WaveParticle 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
7 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
8 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 tightaspect ratio) Mode structure: global mode analysis Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 8
9 The HAGIS Code [Pinches, Thesis (1996)] Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 9
10 Code Overview Straight fieldline 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
11 Equilibrium Representation Coordinates chosen to produce straight field lines General toroidal geometry Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 11
12 Particle Description Exact particle Lagrangian, is gyroaveraged 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
13 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
14 Distribution Function Represented by a finite number of markers Markers represent deviation from initial distribution function  socalled 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
15 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
16 HAGIS Code Performance Run time [s] /n scaling Linux Cluster Wallclock time to calculate TAE linear growthrate in ITERlike case Number of processors, # HAGIS code parallelises very well relatively low level of interprocessor communication traffic Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 16
17 SelfConsistent 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
18 Linear Growthrate γ d /ω 0 = 0, β f = Mode saturates at δb/b~103 γ d /ω 0 =2.7% n p = 52,500 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 18
19 with additional damping γ d /ω 0 = 2%, β f = Mode saturates at much lower level, δb/b~104 n p = 210,000 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 19
20 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
21 Linear Growthrate γ d /ω 0 = 0, β f = Mode saturates at δb/b~ γ d /ω 0 =0.45% n p = 52,500 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 21
22 with additional damping γ d /ω 0 = 0.4%, β f = Mode saturates at much lower level, δb/b~ n p = 210,000 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 22
23 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
24 Fast Ion Redistribution Resonant energy changes as mode sweeps in frequency Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 24
25 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 qprofile Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 25
26 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 = 103 Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 26
27 Scaling of Nonlinear Bounce Frequency 4.00E E E E+04 HAGIS 1 Linear (HAGIS 1) MAST #5568 Monotonic q profile H = 20 kev ω b 2.00E E E E+03 ω b = (δb/b) 0.00E E E E E E E E E 02 (δb/b) Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 27
28 Scaling of Nonlinear Bounce Frequency 3.00E E+04 HAGIS 1 HAGIS 2 Scaling MAST #5568 Reversed shear 2.00E+04 H = 20 kev ω b 1.50E E E+03 ω b = (δb/b) for δb/b < E E E E E E E E E02 (δb/b) Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 28
29 Mode Amplitudes For monotonic qprofiles we now know: where For a single resonance, where Therefore, Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 29
30 TAE Amplitude in MAST 140 Frequency [khz] df = 18 khz dt = 0.8 ms Time [ms] Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 30
31 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 = Simon Pinches, 8th IAEA Technical Meeting on Energetic Particles, San Diego 31
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