Neoclassical Transport

PhD Network, Garching, 8.10.2009 R. Dux

Neoclassical Transport

R. Dux

• Classical Transport

• Pfirsch-Schlüter and Banana-Plateau Transport

• Ware Pinch

• Bootstrap Current


Why is neoclassical transport important?

Usually, neoclassical (collisional) transport is small compared to the turbulent transport.

Neoclassical transport is important:

• when turbulent transport becomes small - transport barriers (internal, edge barrier in H-modes) - central part of the plasma, where gradients are small

• to understand the bootstrap current and the plasma conductivity

• transport in a stellarator (we do not cover this)

• transport of fast particles (there seems to be also some turbulent contribution)


The gradients of density, temperature and electric potential in the plasma disturb the Maxwellian velocity distribution of the particles, which would prevail in thermodynamic equilibrium.

The disturbance shall be small.

Coulomb collisions cause friction forces between the different species and drive fluxes of particles and energy in the direction of the gradients.

Coulomb collisions drive the velocity distribution towards the local thermodynamic equilibrium and the fluxes try to diminish the gradients.

We seek for linear relations between the fluxes and the thermodynamic forces (gradients).

We concentrate on the particle flux:

Transport of particles, energy …due to collisions

un


Moments of the velocity distribution

We arrive at moments of the velocity distribution by integrating the distribution function times vk over velocity space

vdfn aa3

vdvfn

u aa

a31

aBaaaaa

a TknvduvuvfTrm

p 3

3

Ipvduvuvfm aaaaaa

� 3

0. moment: particle density

1. moment: fluid velocity

2. moment: pressure and viscosity

The formulation of the neoclassical theory is based on fluid equations, whichdescribe the time evolution of moments of the velocity distribution.

3. moment: (random) heat flux vduvuvfm

q aaaa

a32

2


Fluid equations = moments of the kinetic equation

0

aaa unt

n

Integrating the kinetic equation times vk over velocity space yields the equations of motion for the moments of the velocity distribution (MHD equations)

03vdCab

babababaab FvdCvmvdCvmF

33

ab

abaaaaaaaa

aaa

aa FpBuEneuut

unm

dt

udnm

0. moment: particle balance (conservation)

1. moment: momentum balance

In every equation of moment n appears the moment n+1and an exchange term due to collisions (here: momentum exchange, friction force)


The friction force due to collisions

The force on particle a with velocity va due to collisions with particles b with velocity vb averaged over all impact parameters

• formally equal to the attractive gravitational force (in velocity space)

• This result for point like velocity distributions can be extended to an arbitrary velocity distribution fb of particles b using a potential function h.

ab

abbaabb

ab

ababb

ba

ba

ab

abbaab

m

eeAn

v

vAn

vv

vv

m

ee

dt

pd20

22

3320

22

4

ln

4

ln

bbbb

vabab vdvf

vvvhvhA

dt

pd 31)()(

vdvhvfAF vaabab3)(

The average force density on all the particles a with velocity distribution fa is obtained by integrating the force per particle over the velocity distribution

fa

fb

vx

vy


The friction force for nearly Maxwellian distributions

202

2

32/3

21)(exp)(

TvuTT v

uvvf

v

uv

v

nvf

T

baababaaab FuunmF

the average force density on the species a due to collisions with b is.

For Maxwellian velocity distributions with small mean velocity u<<vT

a

b

B

ababbaab m

n

Tk

mee2/3

22

20

ln

)4(3

24

The collision frequency is for Ta=Tb:

mTkvv

v

v

nvf BT

TT

/2exp)(2

2

32/30

For undisturbed Maxwellian velocity distributions with thermal velocity vT

The friction forces are zero.


Closure of the fluid equations

• In every equation of moment n appears the moment n+1.

• At one point one has to close the fluid equations by expressing the higher order moments in the lower ones

• In neoclassical theory one considers the first four moments: density, velocity, heat flux, ???-flux

• To estimate classical particle transport we use a simple approximation and just care about density and velocity (first two moments)

ab

ababaaaaaaa

aa uunmpBuEnedt

udnm

momentum balance

ababaaab

a

uunmF

0


The ordering

)()(

)( 23

O

abababaa

O

aaaa

O

aaa uunmpBuEnedt

udnm

momentum balance

1Be

m

a

aa

ca

a

We assume :

• the strong magnetic field limit (magnetized plasma)

• to be close to thermal equilibrium

• temporal equilibrium

1,,

Tnca

Ta

Tn

a

L

v

L

Lowest order of : no friction

aaaa pBuEne

)0(0

Next order of : include friction with lowest order fluid velocities

ab

ababaaaaa uunmBune )0()0()1(0

)1()0(aaa uuu

Expand fluid velocity


The lowest order perpendicular fluid velocities

Cross product with B-field yields perpendicular velocities:

aaaa pBuEne

)0(0 ab

ababaaaaa uunmBune )0()0()1(0

22)0(

, Bne

Bp

B

BEu

aa

aa

ababab

a

aa BuBu

Be

mu

)0()0(2

)1(,

aa

aa ne

pEBu

)0( ab

ababa

aa uu

e

mBu )0()0()1(

2BuBBu

ab aa

a

bb

bab

a

aa ne

p

ne

p

Be

mu

2)1(,

ExB drift + diamagnetic drift

Velocity in direction of pressure gradients(ExB drops out, friction only due to diamagnetic drift)

zero order first order


The lowest order perpendicular current density

22)0(

, Bne

Bp

B

BEu

aa

aa

The zero order perpendicular current is consistent with the MHD equilibrium condition.

22

alityQuasineutr 0

2)0(

,)0(

B

Bp

B

Bpne

B

BEunej

a

a

aaaa

aaa

Bjp

)0(

BFB

uneab

abaaa

2

)1(,

1

ab

abab

ababaaaaa FuunmBune )0()0()1(

01

onconservati momentum 0

2)1(,

)1(

BF

Bunej

a abab

aaaa

The first order perpendicular current is ambipolar.

zero order

first order


Particle picture for the ambipolarity of the radial flux

2qB

Bprrgc

A collision between a and b changes the directionof the momentum vector and the position of the gyro centre changes by.

The displacement is ambipolar.

The position of the gyro centre and the gyrating particle are related by.

gcb

a

b

a

b

a

agca r

e

e

Be

Bp

Be

Bpr

2onconservati momentum

2

0 gcbb

gcaa rere

0 gcb

gca rr

No net transport for collisions within one species (Faa=0 in the fluid equation).

The same argument does not hold for the energy transport (exchange of fast and slow particle within one species).


The classical radial particle flux (structure)

ab aa

a

bb

bab

a

aa ne

p

ne

p

Be

mu

2)1(,

ab a

b

b

b

b

aaaab

a

Ba

ab aa

aa

bb

bbab

a

Baaa

e

e

T

T

n

n

e

enn

Be

Tkm

ne

nTTn

ne

nTTn

Be

kmn

122

2

ab

aba

abab

ca

Ta

abab

aa

aB

abab

a

BaaCL

v

mBe

mTk

Be

TkmD

22

2

2

2

22222 aaa

CLD 2

2

The radial particle flux density is thus (for equal temperatures):

aaCLa

aCLa nvnD

It has a diffusive part and a convective part like in Fick’s 1st law:

The classical diffusion coefficient is identical to the diffusion coefficient of a ‘randomwalk’ with Larmor radius as characteristic radial step length and the collision frequency as stepping frequency.

TTTTknp baaBaa


A cartoon of classical flux

2, Bne

Bpu

aa

aadia

Diamagnetic velocity depends onthe charge and causes frictionbetween different species, thatdrive radial fluxes.

Be

mTk

mBe

v

a

aaB

aa

Taa

2


The classical diffusion coefficient

ab

aba

BaaCL Be

TkmD

22 a

b

B

ababbaab m

n

Tk

mee2/3

22

20

ln

)4(3

24

ab

bbabab

B

aCL nem

TkBD 2

2/1220

ln1

)4(3

24

• The classical diffusion coefficient is (nearly) independent of the charge of the species.• In a pure hydrogen plasma DCL is the same for electrons and ions.• For impurities, collisions with electrons maeme can be neglected compared to collisions with ions.• The diffusion coefficient decreases with 1/B2 due to the quadratic dependence on the Larmor radius

Our expression for the drift is still not the final result, since the frictionforce from the shifted Maxwellian is too crude...


The perturbed Maxwellian (more than just a shift)

We calculate the perturbation by an expansion of the Maxwellian in the x-direction:

The B-field shows in the y-direction. All gyro centre on a Larmor radius around the point of origin contribute to the velocity distribution. There shall be a gradient of n and T in the x-direction.

mTkvv

v

v

nvf BT

TT

/2exp)(2

2

32/30

22

2

02

2

01

2'

2

5''

2

3'

T

z

a

B

Tdia

c

z

T v

v

Be

Tk

v

vuf

v

T

T

v

v

T

T

n

nff

czgc vx

cL vr B

x

z

Tn

201

2

T

diaz

v

uvff

100

00

0

00 ff

v

x

ffx

x

fff

c

z

xgc

x

The perturbation of the Maxwellian has an extra term besides the diamagnetic velocity, which we have neglected so far. It leads to the diamagnetic heat fluxand an extra term in the friction force, the thermal force.

old


The thermal force

There is a diamagnetic heat flux connected with the temperature gradient

2)0(

, 2

5

Be

BTkpq

a

aBaa

2

flux)heat (diamagn. force-thermo velocity)(diamagn. force slipping

, 2

3

B

B

me

m

me

mTk

ne

p

ne

pnmF

aa

ab

bb

abB

aa

a

bb

babaaab

This leads to new terms in the friction force which are proportional tothe temperature gradient and are called the thermo-force.

Ion-ion collisions, equal directions of p and T :For ma>mb the thermo-force is in the opposite direction than the p-term

2, 2

3

B

BTk

n

p

e

nmF B

eieei

Also for a simple hydrogen plasma the two forces are opposite.


The thermal force

The reason for thermal force isthe inverse velocity dependence of the friction force:

Collisions with higher velocity difference are less effective than collisions with lower velocity difference.

This lowers the friction force due to the differences in the diamagneticvelocity.


The classical radial particle flux (final result)

abaab

CLab a

ab

a

b

b

ab

b

b

b

aabCL

aba

abCLa

aCL D

m

m

e

e

m

m

T

T

n

n

e

eDnDn

21

2

31

2

3 2

T

T

n

nZnn

H

HZZ

ZHZZCL 2

1

2

2

iCL

eieeieeCL T

Tnn

T

T

n

nnn

22

22

1

2

22

For a heavy impurity in hydrogen plasma (collisions with electrons can be neglected):

inward outward (temperature screening)

For a pure hydrogen plasma:

ieieie 22 ion and electron flux into the same direction and of equal size!

T

T

n

nZ

n

n

H

H

Z

Z

2

1In equilibrium the impurity profile is much more peaked than thehydrogen profile (radial flux=0)


End of classical transport


Look ahead to neoclassical transport

Similar:

•Classical and neo-classical particle fluxes have the same structure: - diffusive term + drift term - larger drift for high-Z elements (going inward with the density gradient) - temperature screening

• The neo-classical diffusion coefficients are just enhancing the classical value by a geometrical factor.

Different:

• A coupling of parallel and perpendicular velocity occurs due to the curved geometry.

• The neo-classical transport is due to friction parallel to the field (not perpendicular).

• additional effects due to trapped particles: bootstrap current and Ware pinch


The Tokamak geometry

ttt eZReRBB

),(

Helical field lines trace out magnetic surfaces.

the poloidal flux is 2 and ||=RBp

• The safety factor q gives the number of toroidal turns of a field line during one poloidal turn. • The length of a field line from inboard to outboard is: qR

• The transport across a flux surface is much slower than parallel to B.• We assume constant density and temperature on the flux surface.


Flux surface average

dSG

VdS

GVG

'

11

Density and temperature are (nearly) constanton a magnetic flux surface due to the much faster parallel transport and the transport problemis one dimensional.

V VV

dSG

VdSrG

VGdV

VG

11

pp

dlB

G

VG

'

2pp dlRddSRB

t

n

We calculate the flux surface average of a quantity G

Tokamak:


Flux surface average of the transport equation

dS

G

VG

'

1

''

1V

Vt

n

t

n

'VdSSddVV

'

''

''

V

VV

V

VV

The average of the divergence of the flux is calculatedusing Gauss theorem:

The one dimensional equation is then:

We have to determine the surface averages:

q

which are linear in the thermodynamic forces

a

aa T

np


Two contributions to the radial flux

Take the toroidal component of the momentum equation, multiply with R and forma flux surface average. This leads to an expression for the radial flux due to toroidal friction forces:

ab

abp

abt

aabtab

aa B

FRB

B

FRB

eRF

e

transportclassical

,

transportalneoclassic

||,,

11

We can calculatethis term by just forming the flux surface average from the old result

We need to know the differences ofthe parallel flowvelocities to get the friction forces.

ab

aba

a

Bapa

n

e

Tkm

B

BR

22

22

Diffusion CL

The classical diffusion flux with correct flux surface average:

eB

Be

B

Be ptt

||


Divergence of the lowest order drift

22222

)0(, B

B

Bne

Bp

B

B

Bne

Bp

B

BEu a

aa

a

aa

aa

a

aaa

p

ne

1

B

Bu

BB

B

BB

BB

BB

u

a

a

aa

aa

)0(,

3

0

22

0

2)0(

,

2

2

1

0

00

j

BBB

pressure and particle density andelectric potential are in lowest order constant on flux surface

One contribution two the parallel flows arrises from the divergenceof the diamagnetic and ExB drift:


The lowest order drifts are not divergence free

)0(,

)0(, aaaa

a ununt

n

B

Buu aa

)0(,

)0(, 2

we find, that ions pile up on the top and electrons on the bottom of the flux surface (reverses with reversed B-field).

In the particle picture this is found from the torus drifts (curvature, grad-B drift).

This leads to a charge separation.

From the continuity equation:

and the divergence of the diamagn. drift

B

Bun

t

naa

a

)0(

,2


Coupling of parallel and perpendicular dynamics

The separation of charge leads to electric fields along the field linesand a current is driven which preventsfurther charge separation.

Parallel electron and ion flows build upto cancel the up/down asymmetry.The parallel and perpendicular dynamics are coupled.

The remaining charge separation leadsin next order to a small ExB motion andcauses radial transport.


Coupling of parallel and perpendicular heat flows

A similar effect appears for the diamagnetic heat flow, which causestemperature perturbations inside theflux surface which is counteracted byparallel heat flows leading in higherorder to a radial energy flux.


The Pfirsch-Schlüter flow

diamagnetic velocity

Pfirsch-Schlüter velocity

eB

RBp

neu pa

aaa

1)0(,

||2

)0(||,

11e

B

B

BRB

p

neu t

a

aaa

form of total velocity (divergence free)

BKu ta

eR)0(

0)0(||, Bu a

• not completely determined • another velocity will be added later this is also divergence free since div(B)=0• it is caused by trapped particles (Banana-Plateau transport)

2ˆ BBu


The Pfirsch-Schlüter transport

Pfirsch-Schlüter velocity

||2

)0(||,

11e

B

B

BRB

p

neu t

a

aaa

We use the shifted Maxwellian friction force and calculate the radial flux

ab

a

aa

b

bbab

a

aatPSa

p

ne

p

nee

nm

BBRB

1111

22

2

ab

abt

aa B

FRB

e||,1

22222

2 11

BBBBR

RBg

p

tPS

...8

29

16

3912

231

11

1 4222

2

2

22

qqgPS

The result has the same structure as in the classical case. The fluxes are enhanced by a geometrical factor.

For concentric circular flux surfaces with inverse aspect ratio =r/R

The Pfirsch-Schlüter flux is a factor 2q2 larger than the classical flux.


The Pfirsch-Schlüter flux pattern

The Pfirsch-Schlüter velocity

||2

)0(||,

11e

B

B

BRB

p

neu t

a

aaa

changes its direction at the top/bottomof the flux surface.

Also the radial fluxes change direction.

The flux surface average is theintegral over opposite radial fluxesat the inboard/outboard side.

the flux vectorscan also showinward/outward at the outboard/inboardside


Strong Collisional Coupling

Temperature screening in the Pfirsch-Schlüter regime similar to classical case:

• consequence of the parallel heat flux, which develops due to the non-divergence free diamagnetic heat flux.

• temperature screening is reduced for strong collisional coupling of temperatures of different fluids (that happens typ. for T < 100eV) - energy exchange time comparable to transit time on flux surface - up/down asymmetry of temperatures reduced due to collisions - weaker parallel heat flows - reduced or even reversed radial drift with temperature gradient


Lets have a coffee

Intermission


Regime with low collision frequencies

For the CL and PS transport, we were just using the fact, that the mean free path is large against the Larmor radius (a<<ca)

The mean free path increases with T2 and can rise to a few kilometerin the centre. Thus, we arrive at a situation, where the mean free pathis long against the length of a complete particle orbit on the flux surfaceonce around the torus.

The trapped particle orbits become very important in that regime, sincethey introduce a disturbance in the parallel velocity distribution for a givenradial pressure gradient. This extra parallel velocity ‘shift’ will lead to a new contribution in the parallel friction forces and to another contributionto the radial transport, the so called Banana-Plateau term.


Particle Trapping

22|| 2

1

2

1 mvmvE

B

mv

2

2

leads to particle trapping. At the low field side, v||has a maximum.

LFSLFSLFS B

Bvvv 122

||2||

21

111||

LFS

HFS

LFSB

B

v

vv|| becomes zero on the orbit for all particles with

For a magnetic field of the form

cos1/cos10

0

0

B

Rr

BB

Conservation of particle energy and magn. moment


Fraction of trapped particles

In all these estimates the inverse aspect ratio=r/R is considered to be a small quantity.

The fraction of trapped particles is obtainedby calculating the part of the spherical velocitydistribution, which is inside the trapping cone.

46.046.114

31

max/1

0

2

B

tB

dBf

tf 46.046.1

ft only depends on the aspect ratio.


Trapped particle orbits

The bounce movement together with thevertical torus drifts leads to orbits with a banana shape in the poloidal cross section.

The trapped particles show larger excursionsfrom the magnetic surface, since the verticaldrifts act very long at the banana-tips.

On the outer branch of the banana the current carried by the particle is always inthe direction of the plasma current (co).


Trapped particle orbits

Conservation of canonical toroidal momentum

consteRvmp aa

yields for low aspect ratio an estimate for the radial width of the banana on the low-field side

papa

Tb

T

bpaaaa

mBe

vw

vvv

wRBeRvmeRvm

||

022

The width scales with the poloidalgyro radius (= Larmor radius evaluatedwith the poloidal field).

bw


The banana current

apa

TbtTt mBe

vwnnvv ||,

dr

dp

eB

dr

dn

meB

vv

dr

dnwvun

p

p

TT

tbtt

12/3

||,||

The banana current is in the co-direction for negative radial pressure gradient: dp/dr < 0.

Collisions try to cancel the anisotropy in the velocity distribution.

Consider a radial density gradient.• On the low-field side, there are more co-movingtrapped particles than counter moving particles,leading to a co-current density • effect is similar to the diamagnetic current


Time scales

The velocity vector is turned by pitch angle scattering.The collision frequency is the characteristic value for an angle turn of1:

b

aba

To scatter a particle out of the trappedregion it needs on average only an angle. Due to the diffusive nature of theangle change by collisions the effectivecollision frequency is:

aa

effa

2,

The distance from LFS to HFS along the field line is: LqR

A passing particle with thermal velocity vTa needs a transit time:Ta

T v

qR

A trapped particle has lower parallel velocity and needs the longer bounce time: Ta

TB

v

qR


Collisionality

The collisionality is the ratio of the effectivecollision frequency to the bounce frequency

2/3

,

02/300,*

amfpTa

a

Ta

a

b

effaa

qR

v

qR

v

qR

• The summation includes a. • higher collisionality for high-Z. • strong T-dependence

322

2

2/3024

22

22/3

020

*

mlnkeV

m109.4

ln1

)4(3

4

bbbabab

a

a

bbbabab

aB

aa

nZmm

Z

T

qR

nemmTk

eqR

Schlüter-Pfirsch:high

plateau :medium1

banana:low 1

2/3*

2/3*

*

a

a

a

PSedge @ Imp

plateaucenter @ Imp

plateauedge @ He,

bananacenter @ He,


Random walk estimate

If the collisionality is in the banana regime, wecan estimate the diffusion coefficient. The diffusion is due to the trapped particles.

qwr

nn

pb

eff

t

2rn

nD eff

tBP

CLBP Dqq

D2/3

22

In the banana regime the transportof trapped particles dominates by a large factor. This banana-plateau contribution becomes small at highcollisionalities.

This estimate works only if the step length (banana width) is small againstthe gradient length.


Exchange of momentum: trapped passing

The loss of trapped particles into the passingdomain creates a force density onto the passing particles

dr

dp

eBmunmF a

p

aatataeffpt

12/3,||

The passing particles loose momentum to the trapped particles in a fraction nt/n of all collisions

aaaatp unmF ||

aa

paa

aa

paaaatppt

udr

dp

Ben

udr

dp

BennmFFF

||

coeff.viscosity

||

1

1

In the fluid equations, this is the contribution of the viscous forces to the parallel momentum balance. The contribution increases with the collision frequency in the banana regime and decreases with 1/ in the PS regime.

Simple model for banana regime:


The parallel momentum balance

The PS flow drops out and one gets a system of equations for the û (here it is written for the shifted Maxwellian approach):

||||ˆ aa BFB

ab

ababaaa

aataa uunm

p

neRBu ˆˆ

1ˆ

||2||2

)0(||, ˆ

11e

B

Bue

B

B

BRB

p

neu at

a

aaa

This integration constant û of the parallel fluid velocity is calculated from the flux surface averaged parallel momentum balance.

Thu û are functions of the viscosity coefficients, collision frequencies and the pressure gradients. Once a solution has been obtained, one can calculatethe banana-plateau contribution to the radial transport.

ab

abt

aa B

FRB

e||,1

ab

ababaat

aa uunm

B

RB

eˆˆ

12


Radial banana plateau flux (Hydrogen)

We calculate the û for a Hydrogen plasma using the simple viscosity estimate inthe case of low collisionality.

ieieit

iieii

eieiet

eeiee

uup

en

RBu

uup

en

RBu

ˆˆˆ

ˆˆˆ

i

eeiie

i

eeeiieeei m

m

m

m

ietei

pp

en

RBuu

1ˆˆ

0ˆ

ˆˆˆ

itiii

eieiet

eei

p

en

RBu

uup

en

RBu

p

e

m

B

RBuunm

B

RB

eeiet

eieiet

e 22

2

2BP 1ˆˆ

1

2/3

2

2

2

2222

2

qq

BBRB

RBg

p

tBP

This flux has the same form as the classical hydrogen flux enhanced by a rather large geometrical factor: This is just the same estimate,

we got with our random walkarguments.

For large collisionalities the viscosity decreases with collision frequency and the banana-plateau flux becomes small.


The bootstrap current

The difference of the û for a Hydrogen plasma from the simple viscosity estimateyields a parallel current.

r

p

Br

p

RBB

BRB

B

Bjj

pp

t

1

1

ˆ22||

2|| ˆˆ

B

Buuenj ei

ietei

pp

en

RBuu

1ˆˆ

dr

dp

Buenj

ptt

12/3||,banana||,

This is the a very rough estimate for the bootstrap current density. It is a factorof 1/ larger, than the banana current which is initiating the bootstrap currentof the passing particles.

r

Tnk

r

Tnk

r

nTTk

Bj i

Be

BieBp

bs 42.069.044.2

A better expression correct to order

Finally, a dependence on the collisionality has to enter .


Effects on the conductivity

Trapped particles do not carry any current.

Only the force on the passing particlesgenerates a current.

The corrections disappear for high collisionalityof the electrons.

2

*11

CSP

1nnp

These two effects lead to a neo-classical correction on the Spitzer conductivity due to the trappedparticles.

Momentum is lost by collisions with ionsor by collisions with trapped particles.

|||||| uu

dt

dueeei


Ware Pinch

Conservation of canonical toroidal momentum

constemRvp a

At the banana tips, the toroidal velocityis zero. All turning points of the bananaare on a surface with

The movement of this surface of const.flux yields a radial movement of the banana orbit.

const

pt RBvREvtdt

d

field electric induced

p

ttware B

Enf)( *

2B

BEv ptE

The Ware pinch is much larger than the classical pinch:

• co: acceleration• counter: de-acceleration• no equal stay above/below midplane• radial drifts do not cancel


The total radial flux due to collisions

The total radial flux induced by collisions is a sum of three contributions: classical(CL) , Pfirsch-Schlüter(PS) and banana-plateau (BP) flux.

BPPS,CL,

DriftDiffusion

11

x aba

abx

b

bb

aaabx n

T

TH

n

ne

enD

temperature screening

For low collisionalities the BP-term dominates at high collisionalities the PS-term.For each term the drifts increases with the charge ratio times the diffusioncoefficient.

There are numerical codes available to calculate the different contributions(NCLASS by W. Houlberg, NEOART by A. Peeters).


Transport coefficients due to collisions (example 1)

62.0

T5.2

240

52

keV5005.0

m101 320

,

trap

t

De

eSiHe

f

B

.ε

.q

T

nn

nn

change of collisionality bychange of T at a fixed position


Transport coefficients due to collisions (example 2)

T7.5

3

keV32)0(

m101

m45.8

95

320

t

e

B

q

T

n

R

T5.2

3.3

keV5.2)0(

m101

m65.1

95

320

t

e

B

q

T

n

R

ITER-FDR (old)

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Standard neoclassical theory does not work, ...

• near the axisthe banana width is assumed to be smallagainst the radial distance to the axis

• for very strong gradients, with gradientlength smaller than the banana width

• for high-Z impurities in strongly rotatingplasmas, with toroidal Mach numbers >> 1 - leads to asymmetries of the density on the flux surface• ...


The End


(Neo-)classical transport can be explained by the combination of particle orbits and

Coulomb collisions

The particle density in phase space is given by a velocity distribution

The kinetic equation is the Fokker-Planck equation

The kinetic equation

vdxdtvxfNd a336 ,,

bbaab

a

a

aaa ffCv

fBvE

m

e

x

fv

t

f,

the time derivative alongthe particle orbit

prescribed macroscopicfields

Collision operator

The electric and magnetic fields are static and only fluctuations with a length scale smaller than the Debye length are considered. These fluctuations are considered within the collision operator.

Documents

Neoclassical Transport