Transcript Neoclassical Transport

Neoclassical Transport
R. Dux
• Classical Transport
• Pfirsch-Schlüter and Banana-Plateau Transport
• Ware Pinch
• Bootstrap Current
PhD Network, Garching, 8.10.2009
R. Dux
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)
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R. Dux
Transport of particles, energy …due to collisions
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:
PhD Network, Garching, 8.10.2009


  nu
R. Dux
Moments of the velocity distribution
The formulation of the neoclassical theory is based on fluid equations, which
describe the time evolution of moments of the velocity distribution.
We arrive at moments of the velocity distribution by integrating the distribution
function times vk over velocity space
0. moment: particle density
na   f a d 3v
1. moment: fluid velocity

1
ua 
na
2. moment: pressure and viscosity
3. moment: (random) heat flux
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

f a v d 3v
 
 
ma
Tr  f a v  ua   v  ua d 3v  na k BTa
3

 
  3
 a  ma  f a v  ua   v  ua d v  pa I
pa 

m
qa  a
2

  2  
f a v  ua  v  u a d 3v
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Fluid equations = moments of the kinetic equation
Integrating the kinetic equation times vk over velocity space yields the equations of
motion for the moments of the velocity distribution (MHD equations)
0. moment: particle balance (conservation)

na
 na ua   0
t
C
ab
d 3v  0
1. moment: momentum balance


  


 
dua
 ua
ma na
 ma na 
 ua   ua   ea na E  ua  B  pa     a   Fab
b a
dt

t








3
3
Fab   ma v Cab d v    mb v Cba d v   Fba
In every equation of moment n appears the moment n+1
and an exchange term due to collisions (here: momentum exchange, friction force)
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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

d pab
ea2eb2 ln  ab

dt
402 mab
 
va  vb
  3 nb   Aab
va  vb

vab
 3 nb
vab
ea2eb2 ln  ab
Aab 
402 mab
• 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.


dpab
  Aab v h(v )
dt


1
h(v )     f b vb d 3vb
v  vb
vy
fa
vx
fb
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



Fab   Aab  f a v  v h(v )d 3v
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The friction force for nearly Maxwellian distributions
For undisturbed Maxwellian velocity distributions with thermal velocity vT
 v2 
f 0 (v)  3 / 2 3 exp  2 
 vT
 vT 
n
vT  2k BT / m
The friction forces are zero.
For Maxwellian velocity distributions with small mean velocity u<<vT
 2
 





n
v u
2v  u 

f (v)  3 / 2 3 exp 
 
 f 0 (v)1  2 
2
 vT
vT  u vT
vT 


the average force density on the species a due to collisions with b is.


 
Fab  ma na ab ub  ua   Fba
The collision frequency is for Ta =Tb:
 ab
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2 2
4 2 ea eb mab ln  ab nb

3(40 ) 2
ma
k BT 3 / 2
R. Dux
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)
  a  0

 
Fab  ma na ab ub  ua 
momentum balance

  
 
dua
ma na
 ea na E  ua  B  pa   ma na ab ub  ua 
b a
dt

PhD Network, Garching, 8.10.2009

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The ordering

  
 
dua
ma na
 e a na E  ua  B  pa   ma na ab ub  ua 
 b
a
dt





O ( )

momentum balance
We assume :

O ( 2 )
O ( 3 )
 a  a ma

 1
• the strong magnetic field limit (magnetized plasma)  

e
B
ca
a
a
vTa
• to be close to thermal equilibrium  

 1
Ln,T ca Ln,T
• temporal equilibrium



ua  ua( 0)  ua(1)
Expand fluid velocity
Lowest order of : no friction


  ( 0) 
0  ea na E  ua  B  pa
Next order of : include friction with lowest order fluid velocities
 (1) 


0  ea naua  B   ma na ab ub( 0)  ua(0)
b a
PhD Network, Garching, 8.10.2009


R. Dux
The lowest order perpendicular fluid velocities
first order
zero order


  ( 0) 
0  ea na E  ua  B  pa
 p a
 (0) 
ua  B   E 
ea na
 (1) 


0  ea naua  B   ma na ab ub( 0)  ua(0)

b a

 (1) 


m
ua  B   a  ab ub( 0 )  ua( 0 ) 
ea b a
  

Cross product with B-field yields perpendicular velocities: u  B  B  u B 2

u( 0,a)

 
E  B pa  B


2
B
ea na B 2
ExB drift + diamagnetic drift



 (1)
 (0)   (0) 
ma
u,a 
 u  B  ua  B
2  ab b
ea B b  a

 p p 

m
u(1,)a  a 2  ab  b  a 
ea B b a  eb nb ea na 
Velocity in direction of pressure gradients
(ExB drops out,
friction only due to diamagnetic drift)
PhD Network, Garching, 8.10.2009
R. Dux
The lowest order perpendicular current density
zero order

 

E  B pa  B
u( 0,a) 

2
B
ea na B 2

 

 (0)

p a  B
EB
p  B
j   ea na u( 0,a) 
e
n



a a a a B 2
2
B2
B
a

 (0) 
 p  j  B
 0 Quasineutrality
The zero order perpendicular current is consistent with the MHD equilibrium condition.
first order

 (1) 
 ( 0)  ( 0)
ea naua  B    ma na ab ub  ua    Fab

ba


 (1)
1
ea na u ,a  2  Fab  B
B ba
 (1)

1
j   ea na u(1,)a  2
B
a


Fab

a ba



b a

B 0
 0 momentum conservation
The first order perpendicular current is ambipolar.
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R. Dux
Particle picture for the ambipolarity of the radial flux
The position of the gyro centre and the gyrating particle are related by.
 

 p B
rgc  r 
qB 2
A collision between a and b changes the direction
of the momentum vector and the position of the gyro
centre changes by.
 
 

p  B
pb  B
eb  gc
ragc  a 2




rb

2
ea B momentum conservation
ea B
ea
The displacement is ambipolar.


ea ragc  eb rbgc  0
No net transport for collisions within one species
(Faa=0 in the fluid equation).


ragc  rbgc  0
The same argument does not hold for the energy transport
(exchange of fast and slow particle within one species).
PhD Network, Garching, 8.10.2009
R. Dux
The classical radial particle flux (structure)
 p p 

m
u(1,)a  a 2  ab  b  a 
ea B b a  eb nb ea na 
pa  na k BTa
Ta  Tb  T
The radial particle flux density is thus (for equal temperatures):

 nbT  Tnb na T  Tna 
ma k B

a  na
 

2  ab 
ea B b  a 
eb nb
ea na


ma k BT
ea  nb T  eb  
 2 2  ab  na  na 

1   

ea B b  a 
eb  nb
T  ea  
It has a diffusive part and a convective part like in Fick’s 1st law:

a
a
a  DCL
na  vCL
na
The classical diffusion coefficient is identical to the diffusion coefficient of a ‘random
walk’ with Larmor radius as characteristic radial step length and the collision
frequency as stepping frequency.
a
CL
D
2
ma k BT
k BT ma
vTa
 a2
 2 2  ab  2 2 2  ab 
 
 ab
2  ab
ea B b  a
ea B ma b  a
2ca b  a
2 ba
PhD Network, Garching, 8.10.2009
a
CL
D

 a2
2
a
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A cartoon of classical flux
2k BTa ma
vTa

ea B ma
ea B


p  B
udia,a   a 2
ea na B
a 
Diamagnetic velocity depends on
the charge and causes friction
between different species, that
drive radial fluxes.
PhD Network, Garching, 8.10.2009
R. Dux
The classical diffusion coefficient
a
CL
D
mk T
 a2 B2  ab
ea B b a
a
DCL

 ab
2 2
4 2 ea eb mab ln  ab nb

3(40 ) 2
ma
k BT 3 / 2
4 2
1
2
m
ln

e
nb

ab
ab
b
1
/
2
2
2
b
a
3(40 ) B k BT 
• 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 maeme 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 friction
force from the shifted Maxwellian is too crude...
PhD Network, Garching, 8.10.2009
R. Dux
The perturbed Maxwellian (more than just a shift)
 v2 
f 0 (v)  3 / 2 3 exp  2 
 vT
 vT 
n
vT  2k BT / m
rL  v c
z
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
xgc  vz c
n and T in the x-direction.
B
x
n T
We calculate the perturbation by an expansion of the Maxwellian in the x-direction:
f  f0 
f 0
x
xgc  f 0 
x 0
f 0
x
vz
x 0
c
 f 0  f1

 n' 3 T ' v 2 T '  v z
 v 2 5  k BT '  2v z
 2
f1  f 0  
 2 
 f 0  udia   2  

 n 2 T vT T  c
 vT 2  ea B  vT

old
f1  f 0
2v z u dia
vT2
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 flux
and an extra term in the friction force, the thermal force.
PhD Network, Garching, 8.10.2009
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The thermal force
There is a diamagnetic heat flux connected with the
 temperature gradient

5 pa k B Ta  B
q( 0,a)  
2
ea B 2
This leads to new terms in the friction force which are proportional to
the temperature gradient and are called the thermo-force.

F,ab

 





p

p
m
m
3
B
b
a
ab
ab


 ma na ab

 k BT 

 2

eb nb ea na
2
e
m
e
m
B
b b
a a

















slipping
force
(diamagn.
velocity)
thermo
force
(diamagn.
heat
flux)


Ion-ion collisions, equal directions of p and T :
For ma>mb  the thermo-force is in the opposite direction than the p-term
Also for a simple hydrogen plasma the two forces are opposite.

me n ei
F,ei  
e
PhD Network, Garching, 8.10.2009

 p 3
 B
 k BT   2

n
2

 B
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The thermal force
The reason for thermal force is
the 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 diamagnetic
velocity.
PhD Network, Garching, 8.10.2009
R. Dux
The classical radial particle flux (final result)

a
eb
T  3mab
ab
ab ea  nb
CL  na  DCL  na  DCL


1


ba
ba
eb  nb
T  2mb
ea
 3mab  

 1
 2ma
 
ab
CL
D

 a2
2
 ab
For a heavy impurity in hydrogen plasma (collisions with electrons can be neglected):
 Z  Z2  ZH
CL 
2

 n H 1 T  




n

n
Z


Z
Z 
2 T 
 nH

inward
In equilibrium the impurity profile
is much more peaked than the
hydrogen profile (radial flux=0)
outward (temperature screening)
 n
nZ
1 T 

 Z  H 
nZ
2 T 
 nH
For a pure hydrogen plasma:
2
e
e2  ei 
n T   i
 n 1 T   e  ei 
CL 

 
 n  n
 2n 
  CL
2 
n
2
T
2
2
T




 e2  ei   i2  ie ion and electron flux into the same direction and of equal size!
PhD Network, Garching, 8.10.2009
R. Dux
End of classical transport
PhD Network, Garching, 8.10.2009
R. Dux
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
PhD Network, Garching, 8.10.2009
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The Tokamak geometry
Helical field lines trace
out magnetic surfaces.



B  RBt  et   ( R, Z )  et
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.
PhD Network, Garching, 8.10.2009
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Flux surface average
Density and temperature are (nearly) constant
on a magnetic flux surface due to the much faster
parallel transport and the transport problem
is one dimensional. n

t
  
We calculate the flux surface average of a quantity G
G 
1
1

G
GdV

G

r
dS

dS




V V
V V
V V 
 V
G  
 



1

G
1
G
dS  
dS

V ' 
Tokamak:
  RB p
dS  Rddl p
PhD Network, Garching, 8.10.2009
G 
2
V'
G
 Bp dl p
R. Dux
Flux surface average of the transport equation

n
  
t
G 
1
G
dS

V ' 
The average of the divergence of the flux is calculated
using Gauss theorem:

 
 

V   dV   dS     dS  V ' 
V '  


 
V '  


  

 V ' 



V
  

 V ' 
V ' 



The one dimensional equation is then:


n
1 

V ' 
t
V ' 
PhD Network, Garching, 8.10.2009

We have to determine the surface averages:



q
which are linear in the thermodynamic forces
pa

na
Ta

R. Dux
Two contributions to the radial flux
Take the toroidal component of the momentum equation, multiply with R and form
a flux surface average. This leads to an expression for the radial flux due to
toroidal friction forces:

1
a  
ea

ba
 Bt  B p 
et  e|| 
e
B
B
RFab,t



Fab,||
Fab, 
1

   RBt
 RB p
ea b  a 
B
B
  
neoclassical transport classicaltransport 
We need to know
the differences of
the parallel flow
velocities to get
the friction forces.
We can calculate
this term by just
forming the
flux surface average
from the old result
The classical diffusion flux with correct flux surface average:

a 
CL Diffusion
PhD Network, Garching, 8.10.2009
R 2 B p2 ma k BT na

 ab
2
2
B
ea  b a
R. Dux
Divergence of the lowest order drift
One contribution two the parallel flows arrises from the divergence
of the diamagnetic and ExB drift:

u( 0,a)

 



E  B pa  B
   B pa   B
  B










a
B2
ea na B 2
 B 2
 ea na B 2
B2
a    

1 pa

 ea na 
pressure and particle density and
electric potential are in lowest order
constant on flux surface

 a  

 ( 0 )   a  
 1 
  u,a 
   B  a     2     B 
    B
2
2
B
B 



B



0

B
 2a   3   B
B

B
 2u( 0,a) 
B
PhD Network, Garching, 8.10.2009

0







    B  B           B  0



0
0 j
R. Dux
The lowest order drifts are not divergence free
From the continuity equation:


na
   na u( 0,a)  na   u( 0,a)
t


and the divergence of the diamagn. drift


B
  u( 0,a)  2u( 0,a) 
B
 B
na
 2na u( 0,a)
t
B
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.
PhD Network, Garching, 8.10.2009
R. Dux
Coupling of parallel and perpendicular dynamics
The separation of charge leads to
electric fields along the field lines
and a current is driven which prevents
further charge separation.
Parallel electron and ion flows build up
to cancel the up/down asymmetry.
The parallel and perpendicular
dynamics are coupled.
The remaining charge separation leads
in next order to a small ExB motion and
causes radial transport.
PhD Network, Garching, 8.10.2009
R. Dux
Coupling of parallel and perpendicular heat flows
A similar effect appears for the
diamagnetic heat flow, which causes
temperature perturbations inside the
flux surface which is counteracted by
parallel heat flows leading in higher
order to a radial energy flux.
PhD Network, Garching, 8.10.2009
R. Dux
The Pfirsch-Schlüter flow
diamagnetic velocity
 ( 0)
1 pa RB p 
u,a  
e
ea na  B
Pfirsch-Schlüter velocity
1
 (0)
1 pa
B 
u||,a 
RBt   2  e||
ea na 
 B B 
 ( 0) 
u||,a B  0
form of total velocity (divergence free)


 ( 0)
ua   Ret  KB
• not completely determined

2
• another velocity uˆ B B will be added later
this is also divergence free since div(B)=0
• it is caused by trapped particles (Banana-Plateau transport)
PhD Network, Garching, 8.10.2009
R. Dux
The Pfirsch-Schlüter transport
Pfirsch-Schlüter velocity
1
 (0)
1 pa
B 
u||,a 
RBt   2  e||
ea na 
 B B 

F
1
a  RBt  ab,||
ea
B
ba
We use the shifted Maxwellian friction force and calculate the radial flux

a 
 RBt 
2
PS
 1
1  ma na
 2  2 
B  ea
 B
 1 pb
1 pa 


 eb nb  ea na  
 ab 
ba
The result has the same structure as in the classical case. The fluxes are
enhanced by a geometrical factor.
g PS 
RBt 2
R 2 B p2 B 2
 1
1 
 2  2 
B 
 B
For concentric circular flux surfaces with inverse aspect ratio =r/R
g PS
1   2 
1   2 
39 2 29 4

2
q
1


2
q
1





...


 2  1  3 2 2 
8
 16

2
The Pfirsch-Schlüter flux is a factor 2q2 larger than the classical flux.
PhD Network, Garching, 8.10.2009
R. Dux
The Pfirsch-Schlüter flux pattern
The Pfirsch-Schlüter velocity
1
 (0)
1 pa
B 
u||,a 
RBt   2  e||
ea na 
 B B 
changes its direction at the top/bottom
of the flux surface.

Also the radial fluxes change direction.
The flux surface average is the
integral over opposite radial fluxes
at the inboard/outboard side.
the flux vectors
can also show
inward/outward
at the
outboard/inboard
side
PhD Network, Garching, 8.10.2009
R. Dux
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
PhD Network, Garching, 8.10.2009
R. Dux
Lets have a coffee
Intermission
PhD Network, Garching, 8.10.2009
R. Dux
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 kilometer
in the centre. Thus, we arrive at a situation, where the mean free path
is long against the length of a complete particle orbit on the flux surface
once around the torus.
The trapped particle orbits become very important in that regime, since
they introduce a disturbance in the parallel velocity distribution for a given
radial pressure gradient. This extra parallel velocity ‘shift’ will lead to a
new contribution in the parallel friction forces and to another contribution
to the radial transport, the so called Banana-Plateau term.
PhD Network, Garching, 8.10.2009
R. Dux
Particle Trapping
Conservation of particle energy and magn. moment
1
1
E  mv||2  mv2
2
2
mv2

2B
leads to particle trapping.
At the low field side, v|| has a maximum.

B
v||2  v||2LFS  v2 LFS 1 
 BLFS



For a magnetic field of the form
B
B0
B0

1  r cos  / R0 1   cos 
v|| becomes zero on the orbit for all particles with
v||
v
 1
LFS
BHFS
1 
 1
 2
BLFS
1 
PhD Network, Garching, 8.10.2009
R. Dux
Fraction of trapped particles
The fraction of trapped particles is obtained
by calculating the part of the spherical velocity
distribution, which is inside the trapping cone.
ft  1 
3 B2
4
1/ Bmax

0
d
  1.46  0.46 
1  B
In all these estimates the inverse aspect ratio
=r/R is considered to be a small quantity.
 1.46  0.46 
ft
ft only depends on the aspect ratio.
PhD Network, Garching, 8.10.2009
R. Dux
Trapped particle orbits
The bounce movement together with the
vertical torus drifts leads to orbits with a
banana shape in the poloidal cross section.
The trapped particles show larger excursions
from the magnetic surface, since the vertical
drifts act very long at the banana-tips.
On the outer branch of the banana the
current carried by the particle is always in
the direction of the plasma current (co).
PhD Network, Garching, 8.10.2009
R. Dux
Trapped particle orbits
Conservation of canonical toroidal momentum
p  ma Rv  ea  const
yields for low aspect ratio an estimate
for the radial width of the banana on
the low-field side
2ma Rv  ea   2ma Rv  ea RB p wb  0
v  v||   vT
wb  
vT
  p
ea B p ma
wb
The width scales with the poloidal
gyro radius (= Larmor radius evaluated
with the poloidal field).
PhD Network, Garching, 8.10.2009
R. Dux
The banana current
v||,t   vT
nt n  
wb  
vT
ea B p ma
Consider a radial density gradient.
• On the low-field side, there are more co-moving
trapped particles than counter moving particles,
leading to a co-current density
• effect is similar to the diamagnetic current
nt u||  v||,t wb
  3/ 2
dnt
vT
 dn
  vT 
dr
eB p m dr
1 dp
eB p dr
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.
PhD Network, Garching, 8.10.2009
R. Dux
Time scales
The velocity vector is turned by pitch angle scattering.
The collision frequency is the characteristic value for
an angle turn of1:
 a   ab
b
To scatter a particle out of the trapped
region it needs on average only an angle
. Due to the diffusive nature of the
angle change by collisions the effective
collision frequency is:
 a ,eff 
a
a

 2 
The distance from LFS to HFS along the field line is: LqR
A passing particle with thermal velocity vTa needs a transit time:
A trapped particle has lower parallel velocity
and needs the longer bounce time:
PhD Network, Garching, 8.10.2009
B 
T 
qR
vTa
T
qR


 vTa
R. Dux
Collisionality
The collisionality is the ratio of the effective
collision frequency to the bounce frequency
 a ,eff  a qR0
 qR
qR0 3 / 2
 

 a 0  3 / 2 

b
  vTa
vTa
mfp,a
*
a
 a*  1
low : banana
1   a*   3 / 2 medium : plateau
 a*   3 / 2
high : Pfirsch - Schlüter
ea2
4 
*
3 / 2
a 
R0 q
2
3(40 )
k BT 2
 4.9 10
 24
1
ma
R0 mq 3 / 2 Z a2
T keV2 ma


mab ln  abeb2 nb
b
 
mab ln  ab Z b2 nb m 3
e, H @ center
e, H @ edge
Imp @ center
Imp @ edge
banana
plateau
plateau
PS
b
• The summation includes a.
• higher collisionality for high-Z.
• strong T-dependence
PhD Network, Garching, 8.10.2009
R. Dux
Random walk estimate
If the collisionality is in the banana regime, we
can estimate the diffusion coefficient. The
diffusion is due to the trapped particles.
DBP 
nt
 eff r 2
n
nt n  
 eff 


r  wb    p 
q


 q 
q2
DBP   
   3 / 2 DCL
   
2
In the banana regime the transport
of trapped particles dominates by a
large factor. This banana-plateau
contribution becomes small at high
collisionalities.
This estimate works only if the step length (banana width) is small against
the gradient length.
PhD Network, Garching, 8.10.2009
R. Dux
Exchange of momentum: trapped passing
Simple model for banana regime:
The loss of trapped particles into the passing
domain creates a force density onto the passing particles
Ft  p   eff ma nt u||a ,t  ma
 a 3 / 2 1 dpa


eB p dr
The passing particles loose momentum to the
trapped particles in a fraction nt/n of all collisions
Fpt   a ma nau||a
F  Ft  p  Fp t
 1 dpa


  a ma na
 u||a 
 en B dr

 a p

 1 dpa


 a
 u||a 
  en B dr

a
p


viscositycoeff.
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.
PhD Network, Garching, 8.10.2009
R. Dux
The parallel momentum balance
1
 (0)
1 pa
B 
B 
u||,a 
RBt   2  e||  uˆa 2 e||
ea na 
B
 B B 

F
1
a  RBt  ab,||
ea
B
ba
This integration constant û of the parallel fluid velocity is calculated from the flux
surface averaged parallel momentum balance.
B  ˆ a ||  BFa||
The PS flow drops out and one gets a system of equations for the û
(here it is written for the shifted Maxwellian approach):

a uˆa  RBt

1 pa 
   ma na ab uˆb  uˆa 
ea na   b a
Thu û are functions of the viscosity coefficients, collision frequencies and
the pressure gradients. Once a solution has been obtained, one can calculate
the banana-plateau contribution to the radial transport.

1 RBt
a  
ea B 2
PhD Network, Garching, 8.10.2009
 m n  uˆ
ba
a a ab
b
 uˆa 
R. Dux
Radial banana plateau flux (Hydrogen)
We calculate the û for a Hydrogen plasma using the simple viscosity estimate in
the case of low collisionality.

  ee  ei uˆe 


  ii  ie uˆi 

uˆi  uˆe 

e
RBt pe 
  ei uˆi  uˆe 

en  
RBt pi 
  ie uˆe  uˆi 

en  
RBt  pe  pi 

1   en


 ei uˆe 


 ii uˆi 

 ei   ee
RBt pe 
  ei uˆi  uˆe 

en  
RBt pi 
0
en  
me
m
 ii   ee
 ie   ei e
mi
mi
1 RBt
 RBt  me ei p
ˆ
ˆ



m
n

u

u

e
ei
i
e
e B2
e 2 
1  B2
2
BP
This flux has the same form as the classical hydrogen flux enhanced by a
rather large geometrical factor:
This is just the same estimate,
g BP 
 RBt 2
B 2 R 2 B p2 B 2
 q2 q2

 3/ 2
2


we got with our random walk
arguments.
For large collisionalities the viscosity decreases with collision frequency and
the banana-plateau flux becomes small.
PhD Network, Garching, 8.10.2009
R. Dux
The bootstrap current
The difference of the û for a Hydrogen plasma from the simple viscosity estimate
yields a parallel current.
j||  enuˆi  uˆe 
B
B2
uˆi  uˆe 
RBt  pe  pi 

1   en

ˆjB
 RBt B 1 p
 p
j||  2 

1   B 2 RB p r B p r
B
This is the a very rough estimate for the bootstrap current density. It is a factor
of 1/ larger, than the banana current which is initiating the bootstrap current
of the passing particles.
j||,banana  ent u||,t   3 / 2
1 dp
B p dr
A better expression correct to order 
jbs 

Te
Ti 
n


2
.
44
k
T

T

0
.
69
nk

0
.
42
nk


B e
i
B
B
Bp 
r
r
r 
Finally, a dependence on the collisionality has to enter .
PhD Network, Garching, 8.10.2009
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Effects on the conductivity
Trapped particles do not carry any current.
Only the force on the passing particles
generates a current.

np  n 1 

Momentum is lost by collisions with ions
or by collisions with trapped particles.
du||
dt
  ei u||   eeu||
These two effects lead to a neo-classical correction
on the Spitzer conductivity due to the trapped
particles.
2

 

   SP 1 
* 
 1  C 
The corrections disappear for high collisionality
of the electrons.
PhD Network, Garching, 8.10.2009
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Ware Pinch
Conservation of canonical toroidal momentum
p  mRv  ea  const
At the banana tips, the toroidal velocity
is zero. All turning points of the banana
are on a surface with
  const
The movement of this surface of const.
flux yields a radial movement of the
banana orbit.

d  

 v   
RE t
 v RB p

dt
t
induced electric field
   ware ( * ) f t n
Et
Bp
The Ware pinch is much larger than the
classical pinch:
EB
vE 
PhD Network, Garching, 8.10.2009
t
• co: acceleration
• counter: de-acceleration
• no equal stay above/below midplane
• radial drifts do not cancel
p
B2
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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.








n
e

n
1
1

T
b
    Dxab   a  a 
 H xab
na 

  eb  nb 
T   
x  CL,PS,BP b  a



 Diffusion  
Drift


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 diffusion
coefficient.
There are numerical codes available to calculate the different contributions
(NCLASS by W. Houlberg, NEOART by A. Peeters).
PhD Network, Garching, 8.10.2009
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Transport coefficients due to collisions (example 1)
change of
collisionality by
change of T at
a fixed position
nHe, Si  ne
ne  nD  1 10 20 m 3
T  0.05  50 keV
q  2.5
ε  0.24
Bt  2.5T
f trap  0.62
PhD Network, Garching, 8.10.2009
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Transport coefficients due to collisions (example 2)
ITER-FDR (old)
R  8.45m
ne  1 10 20 m 3
T (0)  32keV
q95  3
Bt  5.7T
ASDEX Upgrade
R  1.65m
ne  1 10 20 m 3
T (0)  2.5keV
q95  3.3
Bt  2.5T
PhD Network, Garching, 8.10.2009
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Standard neoclassical theory does not work, ...
• near the axis
the banana width is assumed to be small
against the radial distance to the axis
• for very strong gradients, with gradient
length smaller than the banana width
• for high-Z impurities in strongly rotating
plasmas, with toroidal Mach numbers >> 1
- leads to asymmetries of the density on the
flux surface
• ...
PhD Network, Garching, 8.10.2009
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The End
PhD Network, Garching, 8.10.2009
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The kinetic equation
(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
 
d 6 N  f a  x , v , t  d 3 x d 3v
The kinetic equation is the Fokker-Planck equation


f a  f a ea    f a
v  
E  v  B    Cab  f a , f b 
t
x ma
v b
Collision operator
the time derivative along
the particle orbit
prescribed macroscopic
fields
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.
PhD Network, Garching, 8.10.2009
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