Energetic particle modes: from bump on tail to tokamak plasmas M. K. Lilley 1 B. N. Breizman 2, S. E. Sharapov 3, S. D. Pinches 3 1 Physics Department, Imperial College London, London, SW7 2AZ, UK 2 IFS, University of Texas at Austin, Austin, Texas, 78712, USA 3 EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK Page 1 Imperial College London
Outline Experimental observations Marginal stability and the role of collisions Bump-on-tail model including drag Generalisation to toroidal systems Page 2 Imperial College London
Experimental observations Page 3 Imperial College London
Question: Why disparity between NBI and ICRH? ICRH drive (JET) NBI drive (MAST) Increasing ICRH power Heeter et.al PRL 85, 3177 (2000) Pinches et.al PPCF, 46, S47 (2004) Page 4
Marginal stability and the role of collisions Page 5 Imperial College London
Marginal stability System evolves through a threshold, L d L d P H Page 6 Imperial College London r
Collisionality - previous analysis Marginal stability allows collisions to compete with mode growth ~ L Krook and diffusion were studied in bump on tail d eff df dt coll F F 0 df dt F F coll k v v 3 2 2 0 2 2 2 eff max, Page 7 Imperial College London Berk, Breizman et.al PRL, 76, 1256 (1996) Breizman, Berk et.al PoP, 4 1559 (1997)
Bump on tail - Basic ingredients F 0 Particle injection and effective collisions, eff, create an inverted distribution of energetic particles F 0 (v) Discrete spectrum of unstable electrostatic modes Instability drive, L ~ df 0 /dv, due to wave-particle resonance (ωkv=0) Background dissipation rate, d, determines the critical gradient for the instability Critical slope L= d kv-ω B kx- ωt Page 8 v=ω/k v Separatrix
Collisionality affects mode saturation ˆ L d? Heeter et.al PRL, 85, 3177 (2000) Page 9 Imperial College London
Low collisionality Frequency chirping δω~ t Page 10
kv-ω Frequency chirping Holes and Clumps Slowly accelerating trapped particles release energy that balances dissipation δω~ t Page 11 kx-ωt
Low collisionality Frequency chirping δω~ t? Collisionality not low enough to explain MAST Page 12
Yet another collisional effect to analyse NBI-produced fast ions slow down due to electron drag Dynamical friction (drag) should be included df dt 2 F F coll k v v Could this explain the bursting behaviour? 0 Page 13 Imperial College London
Bump-on-tail model including drag Page 14 Imperial College London
Bump on tail - formalism Linear cold background with sinusoidal field E V e E cv t m Kinetic fast particle population F F e k ˆ i F df u E t e c.c. t 2m u dt 1 ˆ i E te c.c. 2 kx t ukv F F0 f0 f exp.. n 1 n in c c coll Current from cold background obtained perturbatively using smallness of wave growth and dissipation Page 15 Eˆ e f ˆ 2 1du d E t k 0 d c /2
Near threshold ordering Perturbative approach applied time scales shorter than non-linear bounce period of the wave ω -1 B 2 ekeˆ / m B Can be maintained indefinitely if collision frequency is much larger than bounce frequency Marginal stability allows truncation of expansion at cubic order in wave amplitude Page 16 Eˆ t Imperial College London ˆ 2 ˆ ˆ 3 d ~ E 1 c E E L
Mode evolution equation Normalised wave equation. First term gives exponential growth, second term is cubic nonlinearity /2 2z da 1 3 2 2 2 ˆ 2 /3 ˆ2 ˆ A dz z A z dxe z z x zx i zzx d 2 ˆ - Diffusion coefficient 0 0 A z x A 2z x * ˆ ˆ - Krook coefficient - Drag coefficient Drag gives oscillatory behaviour, in contrast to the Krook and diffusive cases. Page 17 Imperial College London
Sign of cubic nonlinearity First term leads to exponential growth, we must have a negative second term to have saturation. /2 2z da 1 3 2 2 2 ˆ 2 /3 ˆ2 ˆ A dz z A z dxe z z x zx i zzx d 2 0 0 A z x A 2z x * For Krook and diffusion Sign can only flip for low collisionality For drag The oscillatory nature allows the sign to flip often Page 18 Imperial College London
Diffusion + drag For diffusion drag steady state solutions do exist For an appreciable amount of drag these solutions become unstable (pitch fork splitting etc.) Explosive solutions again when drag dominates Page 19 Imperial College London Lilley et.al PRL, 102, 195003 (2009)
Collisionality affects mode saturation This was only a perturbative analysis (cubic order in E) Fully non-linear treatment requires numerical techniques Techniques should take advantage of separation of times scales i.e. Use BOT code: Fourier space code that runs in a couple of minutes on a laptop What happens in the explosive regime? Page 20
Pure drag Page 21
Pure drag No steady state Holes grow Clumps die Page 22 Lilley et.al PoP, 17, 092305 (2010)
Pure drag Growing holes Drag collision operator has a slowing down force and a sink 2 df dt F F coll k v v Slowing down + sink returns distribution to equilibrium E field however can hold the hole in place working against slowing down force Sink still acts to lower F deeper hole over time Deeper hole larger density perturbation larger E 0 Page 23
Now add some diffusion Page 24
Drag + diffusion Steady state hole Page 25 Lilley et.al PoP, 17, 092305 (2010)
Add a bit more diffusion Page 26
Drag + diffusion Undulating frequency Page 27 Lilley et.al PoP, 17, 092305 (2010)
Keep adding diffusion Page 28
Drag + diffusion Hooked frequency chirp BOT Hooked frequency chirp seen in BOT Also seen in MAST (NBI) and JET (ICRH) Page 29 MAST (NBI) JET (ICRH)
Drag + diffusion Hooked frequency chirp Page 30 Hooks for the holes, clumps die sooner Lilley et.al PoP, 17, 092305 (2010)
Drag Diffusion competition Page 31 F ~ F g 0 Power balance Poisson Equation g t 4 2 g d B L B t 3 g t 2 B L B g Diffusion fills, chirping and drag deepen 2 B 2
0-D Equations Page 32 2 1 1 y x y xy y y a x x=y=1 is steady state Unstable for a<1 Stable for a>1 Lilley et.al PoP, 17, 092305 (2010)
Generalisation to toroidal systems Page 33 Imperial College London
Toroidal systems A first glance (low freq.) Phase space resonance is more sophisticated u kv 0 Location of resonance varies n P, E p P, E 0 P φ Page 34 E Imperial College London Resonance Motion across resonance Motion due to wave E p n const. Breizman et.al PoP, 4, 1559 (1997) Chirikov Phys. Rep. 52 263 (1979)
Toroidal systems Reduction to 1-D model (low freq.) Particle motion along resonance does not lead to large gradients in F, so neglect them Need projection of motion and collisions across resonance P φ Page 35 E Imperial College London Resonance Motion across resonance Motion due to wave E p n const. Breizman et.al PoP, 4, 1559 (1997) Chirikov Phys. Rep. 52 263 (1979)
Toroidal systems Reduction to 1-D model (low freq.) Transform to coordinates that straighten resonance Motion across the resonance 1-D for given E and μ Must integrate over all E and μ to get the result Ω 0 Resonance Motion across resonance Motion due to wave n P, E p P, E 0 Page 36 E Imperial College London Breizman et.al PoP, 4, 1559 (1997) Chirikov Phys. Rep. 52 263 (1979)
Experimental estimate for MAST and ITER 3/4 1/6 T ~ e n e B TAE 5/6 TAE 0 Drag vs diffusion depends on plasma parameters TAE TAE 0.2-1.6 MAST - beams - Drag can dominate explosive Page 37 TAE TAE 1.4 Imperial College London ITER alphas drag and diffusion comparable Lilley et.al PRL, 102, 195003 (2009)
Drag in toroidal systems HAGIS HAGIS shows symmetric chirp without collisions HAGIS shows hooking tendency with drag and Krook collisions Page 38
Conclusions and future work Drag is destabilising and acts as a seed for energetic particle modes Drag gives asymmetry as a lowest order effect, giving steady state holes and hooks Possible explanation of NBI vs. ICRH TAE observations Experiments underway (ITPA) to get drag/difusion database HAGIS now includes drag Page 39 Imperial College London
BOT Code Email bumpontail@gmail.com from your work email Please give your name, institution and your position It is free for you to use, modify and also distribute, but I encourage others to contact me for the code so that I can send updates as they become available Page 40