Tokamak Physics Jan Mlynář 6. Neoclassical particle and heat transport

1. Tokamak Physics • Jan Mlynář • 6. Neoclassical particle and heat transport Random walk model, diffusion coefficient, particle confinement time, heat transport, high and low collisionality regimes, thermal diffusion, relaxation times 1: Úvod, opakování

2. Random walk model average step between collisions average time between collisions (1 dim case) [m2/s] Fick’s Ist law + transport eq. Fick’s IInd law 6: Neoclassical particle and heat transport

3. Particle confinement time Fick’s IInd law Cylindrical geometry: Bessel functions J0 ,J1 ,J2 Coulomb collisions: This estimate is wrong by 5 orders of magnitude !! 6: Neoclassical particle and heat transport

4. Particle confinement time 6: Neoclassical particle and heat transport

5. Heat transport convective loss conductive loss work doneby pressure viscousheating heat generation conductive loss: heat flux no convection, no heat sources: • cis thermal diffusion coefficient [ m2s-1 ] cylindrical geometry 6: Neoclassical particle and heat transport

6. Ion and electron temperatures thermal equilibrium: the slowest relaxation process Typical tokamaks: wrong by 3 orders of magnitude, in fact 6: Neoclassical particle and heat transport

7. Neoclassical transport mean free path hydrodynamic length (~ banana, field line) Larmor radius ~ correlation length collisional regime collisionless regime also notice: O.K. drift approximation classical diffusion coefficient: 6: Neoclassical particle and heat transport

8. High collisionality regime Particles do not close full poloidal rotation i.e. cold and dense plasmas (e.g. the plasma egde) (freq. of poloidal rotation) Pfirsch –Schlüter diffusion: Ohm’s law: Due to the Pfirsch-Schlüter current “correction” factor of ~ 10 O.K. 6: Neoclassical particle and heat transport

9. Low collisionality regime Galeev-Sagdeev (banana) transport Banana orbits: Banana width: Banana period: Effective collision frequency: Condition: physics behind the effective collision frequency i.e. most particles close full banana orbit before collision Galeev – Sagdeev diffusion: ratio of trapped particles increase by factor ~5 compared to high collisionality 6: Neoclassical particle and heat transport

10. Neoclassical diffusion coefficient summary: high collisionality low collisionality In between np and nb : plateau In the plateau, diffusion coeff. D is independent of nei 6: Neoclassical particle and heat transport

11. Neoclassical thermal diffusion high collisionality : Pfirsch-Schlüter low collisionality : Galeev-Sagdeev main loss channel: thermonuclear core plasma: i.e. it is in the low collisionality regime 6: Neoclassical particle and heat transport

12. Thermal diffusion in experiments however in special regions (transport barriers) i.e. it indeed sets the theoretical limit for tokamak confinement !! but in theory it should be lower!! and are anomalous. Notice: Functional dependencies are wrong, too. e.g. Instead of the externally heated plasmas follow rather (see also the next talk) 6: Neoclassical particle and heat transport

13. Summary: Relaxation times Relaxation times (~ Maxwellisation, thermalisation) Te ,Ti equilibration notice that : also notice : ( OK sound reasonable ) 6: Neoclassical particle and heat transport

14. Neoclassical thermal diffusion 6: Neoclassical particle and heat transport