1. Fyzika tokamaků1: Úvod, opakování1 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

2. Tokamak Physics2 Random walk model 6: Neoclassical particle and heat transport average step between collisions average time between collisions (1 dim case) [m 2 /s] Fick’s I st law Fick’s II nd law+ transport eq.

3. Tokamak Physics3 Particle confinement time 6: Neoclassical particle and heat transport Bessel functions J 0, J 1, J 2 Fick’s II nd law Cylindrical geometry: Coulomb collisions: This estimate is wrong by 5 orders of magnitude !!

4. Tokamak Physics4 Particle confinement time 6: Neoclassical particle and heat transport

5. Tokamak Physics5 Heat transport 6: Neoclassical particle and heat transport convective loss conductive loss work done by pressure viscous heating heat generation conductive loss: heat flux no convection, no heat sources:    is thermal diffusion coefficient [ m 2 s -1 ] cylindrical geometry

6. Tokamak Physics6 Ion and electron temperatures 6: Neoclassical particle and heat transport thermal equilibrium: the slowest relaxation process Typical tokamaks: wrong by 3 orders of magnitude, in fact

7. Tokamak Physics7 Neoclassical transport 6: Neoclassical particle and heat transport mean free path hydrodynamic length (~ banana, field line) Larmor radius collisional regime collisionless regime also notice: classical diffusion coefficient: O.K. drift approximation ~ correlation length

8. Tokamak Physics8 High collisionality regime 6: Neoclassical particle and heat transport O.K. 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

9. Tokamak Physics9 Low collisionality regime 6: Neoclassical particle and heat transport physics behind the effective collision frequency Galeev-Sagdeev (banana) transport Banana orbits: Banana width: Banana period: Effective collision frequency: Condition: 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

10. Tokamak Physics10 Neoclassical diffusion coefficient 6: Neoclassical particle and heat transport summary: high collisionality low collisionality In between p and b : plateau In the plateau, diffusion coeff. D is independent of ei

11. Tokamak Physics11 Neoclassical thermal diffusion 6: Neoclassical particle and heat transport i.e. it is in the low collisionality regime high collisionality : Pfirsch-Schlüter low collisionality : Galeev-Sagdeev main loss channel: thermonuclear core plasma:

12. Tokamak Physics12 Thermal diffusion in experiments 6: Neoclassical particle and heat transport 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)

13. Tokamak Physics13 Summary: Relaxation times 6: Neoclassical particle and heat transport Relaxation times (~ Maxwellisation, thermalisation) T e,T i equilibration notice that : also notice :( OK sound reasonable )

14. Tokamak Physics14 Neoclassical thermal diffusion 6: Neoclassical particle and heat transport