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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 !!
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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
