Tokamaks 读 读读 书 4.17-4.19 读 读 2013.10.24
Feb 22, 2016
Tokamaks 读书报告4.17-4.19
叶 磊2013.10.24
4.17 Fluctuationsanomalous transport : turbulent diffusion caused by fluctuations.
1. electrostatic : E X B drift velocity +
Particle flux :
heat flux:
2. Electromagnetic : +rBparaV
n or T
B
E E
rV
rV
edge Langmuir probes Langmuir probes Langmuir probes Mirnov coils
interior MS,FIR,HIBP,BES,MR
HIBP Corss polarization scattering
Observationsn T B
4/ ~ 10B B
/ ~ /ee T n n
K spectrum: 0.3sk ~1k qR
Mixing-length estimate: / ~ 1/ nn n k L
Characteristic frequency: 100kHZ
Boltzmann relation
Correlations with transport 2/n n and increase with r
electrostatic fluctuations provide the dominant transport mechanism.
4.18 Turbulence-induced transport
• given turbulence spectra to calculate transport. [random walk model, quasi-linear theory]
• attempt to calculate nonlinear saturated state.[scale invariance , strong turbulence theory, weak turbulence theory, nonlinear scattering]
two kinds of theoretical approach
Why random walk?
Expected value & Variance• the expected value E(X) of a random variable X is a weighted average
of all possible values.
e.g. Let random variable X represent the outcome of a roll.
The expectation of X is:
• Variance measures how far a random variable deviate from its excpected value.
e.g.
2( ) ( )V X E X E X
1/2 20(v )(v) exp
2 2m umf n
KT KT
0E f u KTV fm
Drunkun sailor ---- 1D random walk and diffusion
:
::
time step t
walk distancex
0 2 33 2
After N steps t= Nτ, what is the probability distribution of the man?1
N
ii
x
1
0N
ii
E E x
2
2
1 , 1 1 , 1
2N
i ji
NN N
i i jj
i j
iii i j
V E E x E x x E x xx E
2N, 1
( ) ( )N
i ji ji j
E x E x
22 2( / / /)E t t N ND
D: diffusion coeffcient
turbulent diffusion coeffcient• turbulence spectra: ikx
kke
2v kk
ik BB
22
22 / v k
kk
ik BD
B
• correlation time is determined by :
and in a strong turbulence by
k1/ k 1/ para Tk V 1/ d 1/ eff
1/ k2
v kk k
kkB
21 kk
effk
kDB
kk
DB
Quasi-linear theory
Saturation levels and transport fluxes
scale invariance
If the turbulence is on a microscopic scale, and the fluctuations have a scale then , ,l l a ~e l
T a
mixing-length estimate
/ ~ /ee T n n Boltzmann relation
quasi linear diffusion coeffcient
mixing-length estimate
The linear growth balanced by a stabilization from turbulent diffusion
Radial electric field and transport
Velocity shear may increase the effective
k
k
Orbit squeezing effect
It is shown that ion thermal conductivity is reduced by a factor of
reflecting an increase of the fraction of trapped particles by a factor of
, and the reduction of the orbit size in cby a factor of
Shaing, Hazeltine 1997
Thank you!