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!