Fyzika tokamaků 1: Úvod, opakování 1

Tokamak PhysicsJan 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

Tokamak Physics 2

Random walk model

6: Neoclassical particle and heat transport

1 1 1j j j Nj j

x x x x x x 0x

2 221

1 1

N N

j j jj j

x x x x

2

lN

x

tN

22 l

t

D

x

2

1 20.5( )x S n nt

2 1dnn n xdx

D n Γ

0n nD nt t

Γ

average step between collisions

average time between collisions

(1 dim case) [m2/s]

Fick’s Ist law

Fick’s IInd law+ transport eq.

2 0x

Tokamak Physics 3

Particle confinement time

6: Neoclassical particle and heat transport

Bessel functions J0 , J1 , J2

0( , ) ( , ) expp p

n n tn t n tt

r r

0

p

nD n

Fick’s IInd law

Cylindrical geometry: 1 1 0p

nr nr r r D

0 02.4 exp

p

r tn n Ja

2

22.4paD

Coulomb collisions:2Le

nei

rD

3 5 2 -120

22 10 10 m sn

nD

B T

This estimate is wrong by 5 orders of magnitude !!

Tokamak Physics 4

Particle confinement time

6: Neoclassical particle and heat transport

Tokamak Physics 5

Heat transport

6: Neoclassical particle and heat transport

-33 3 [ Wm ]2 2

iij

j

VnT nT Q p

t x

V q V

convective loss

conductiveloss

work doneby pressure

viscousheating

heat generation

conductive loss: -2 [ Wm ]n T qheat flux

no convection, no heat sources:23

T Tt

is thermal diffusion coefficient [ m2s-1 ]

cylindrical geometry0 0

2.4 expH

r tT T Ja

H E

Tokamak Physics 6

Ion and electron temperatures

6: Neoclassical particle and heat transport

i e thermal equilibrium:

( )1 3 02

e i ee e

eq

T n T Trn S

r r n

( )1 3 02

i i ei

eq

T n T Trn

r r n

i eT T ieq ie ei

e

mm

the slowest relaxation process2Li

iii

r

2Le

eee

r

i e i

20 202 2

3 2 -1

0.1 0.048

1.8 10 m s

cl cli e

cli

n nB T B T

Typical tokamaks: wrong by 3 orders of magnitude, in fact i nD

Tokamak Physics 7

Neoclassical transport

6: Neoclassical particle and heat transport

m.f.p.vT

mean free path

hydrodynamic length (~ banana, field line)

Larmor radius

HL,L Lr

collisional regime

collisionless regime

also notice:

classical diffusion coefficient:

m.f.p.D L HL

m.f.p.D L HL

1L

HL

1L

D

O.K. drift approximation

22

0

eeei L ei

nTD r

B

D ~ correlation length

Tokamak Physics 8

High collisionality regime

6: Neoclassical particle and heat transport

O.K.

m.f.p.v v2

2Te Te

ei pei

q Rq R

2 vn Rds

2/ip nen B

BE v B j

2 2v e eiE m pn n

B e B

22v 1 1

2

e i

e

ei

nT T nn qB T r

D

2

L eiq r D

Particles do not close full poloidal rotationi.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

Tokamak Physics 9

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 collisionGaleev – Sagdeev diffusion:

ratio of trapped particles

increase by factor ~5 compared to high collisionality

v 1v

rR

Lebqr

v vbqR qR

eieff

31 eei b

b e

TqR m

32 22

. .G S b eff L eiD q r

Tokamak Physics 10

Neoclassical diffusion coefficient

6: Neoclassical particle and heat transport

32

ei p b

vTeei pqR

summary: high collisionality

low collisionality

In between p and b : plateau

In the plateau, diffusion coeff. D is independent of ei

2 2 ( const.)p L p pD q r

Tokamak Physics 11

Neoclassical thermal diffusion

6: Neoclassical particle and heat transport

iTi ie

e

mD

m

3 32 2

3 32 2

2 22 2 2

2 22 2 2

0.89

0.68

clLe Lee e e

ee ee

clLi Lii i i

ii ii

r rq q q

r rq q q

i* 0.01eff

b

i.e. it is in the low collisionality regime

high collisionality :Pfirsch-Schlüter

low collisionality :Galeev-Sagdeev

main loss channel:

thermonuclear core plasma:

Tokamak Physics 12

Thermal diffusion in experiments

6: Neoclassical particle and heat transport

exp teorin experiments, 10 higher than the values on the previous slide i i

however in special regions (transport barriers) exp teori i

i.e. it indeed sets the theoretical limit for tokamak confinement !!

in experiments, , 3x lower than e i iD

but in theory it should be lower!!42i

e

mm

e nDand are anomalous.

Notice: Functional dependencies are wrong, too.

e.g. Instead of the externally heated

plasmas follow rather

22p

E

T Bn

2

1.8pE

BnT

(see also the next talk)

Tokamak Physics 13

Summary: Relaxation times

6: Neoclassical particle and heat transport

Relaxation times (~ Maxwellisation, thermalisation)

: : : : :

1 : : 1 : : :

E E E Eee ii ei ie ie ei

m m mm i i iim m mm e e ee

Te ,Ti equilibratione i nD

notice that : 2 2/ // /

i Li Le i e i ii n

n ii ei e ei e

r r m m m mD

D m mm m

p E also notice : ( OK sound reasonable )

Tokamak Physics 14

Neoclassical thermal diffusion

6: Neoclassical particle and heat transport