Transcript No Slide Title

EFTC-11 : Aix-en-Provence
NEOCLASSICAL TOROIDAL
ANGULAR MOMENTUM TRANSPORT
IN A
ROTATING IMPURE PLASMA
S. Newton
P. Helander
This work was funded jointly by the UK Engineering and Physical Sciences Research Council and by
the European Communities under the contract of Association between EURATOM and UKAEA
OVERVIEW
• Motivation
• Current observations
• Neoclassical transport
• Current predictions
• Impure, rotating plasma
• Calculating the transport
• Results
• Summary
MOTIVATION
• Internal Transport Barriers (ITBs) observed in tokamaks
- steep pressure & temperature gradients
- low radial transport
- believed to be caused by sheared electric field
• Toroidal velocity related to radial electric field
dVs
ps'
Er
B  ms ns
V
   s  es n...s E  Vs  B   R s
 ps  
Bp ns es Bp
dt
 Er (r) determined by angular momentum transport
CURRENT OBSERVATIONS
If turbulence is suppressed in an ITB
 neoclassical angular momentum transport
should play key role in ITB formation / sustainment
Bulk ion thermal diffusivity
- observed at neoclassical level in ITB core plasma
- prediction assumes bulk ions in low collisionality, ‘banana’ regime
Bulk ion viscosity determines angular momentum confinement
- measured toroidal viscosity order of magnitude higher than
neoclassical prediction
 angular momentum transport anomalous
NEOCLASSICAL TRANSPORT
• Gyroradius r << perpendicular length scale Lr
- follow the motion of the guiding centre
- determined by magnetic field structure and
Coulomb collisions
• Parallel friction  neoclassical cross-field diffusion
- classical transport from perpendicular friction
• Transit frequency wt = vT / qR
• Two extreme collisionality regimes
- n >> wt : Pfirsch-Schlüter regime
- n << wt : banana regime
NEOCLASSICAL TRANSPORT
Pfirsch-Schlüter
Banana
- parallel motion is diffusive
- trapped particle orbits form
- radial drift  step width
- step width ~  1/2 r p
D
PS
n q r
2
- trapped particle effects
dominate
2
DB n
D
1
~
3 / 2
q2r 2
 3/ 2
wt /n
Parallel friction  neoclassical cross-field diffusion
CURRENT PREDICTIONS
1971, Rosenbluth et al calculate viscosity in pure plasma:
- bulk ions in banana regime, slow rotation
- scales as nii q 2r 2 - expected for Pfirsch-Schlüter regime
1985, Hinton and Wong, 1987, Catto et al:
- extended to sonic plasma rotation:
- still no enhancement characteristic of banana regime
- transport is diffusive, driven by gradient of toroidal velocity
- plasma on a flux surface rotates as a rigid body
- angular velocity determined by local radial electric field
INTERPRETATION
Er
V 
Bp
ExB
• Trapped particles collide:
- change position, toroidal velocity determined by local field
- no net transfer of angular momentum
• Angular momentum transported by passing particles:
- same toroidal velocity as trapped particles due to friction
- typical excursion from flux surface ~ q r
 momentum diffusivity ~ nii q 2r 2
IMPURE ROTATING PLASMA
Typically Zeff > 1
- plasma contains heavy, highly charged impurity species
- mixed collisionality plasma
1976, Hirshman:
particle flux ~ Pfirsch-Schlüter regime
heat flux
~ banana regime
Rotating Plasma
- centrifugal force pushes particles to outboard side of flux surface
- impurity ions undergo significant poloidal redistribution
- variation in collision frequency around flux surface
1999, Fülöp & Helander: particle flux typical of banana regime
IMPURE ROTATING PLASMA
Zeff
MAST discharge
#8321
V ~ 300 km s-1
R[m]
Rotating Plasma
- centrifugal force pushes particles to outboard side of flux surface
- impurity ions undergo significant poloidal redistribution
- variation in collision frequency around flux surface
1999, Fülöp & Helander: particle flux typical of banana regime
IMPURE ROTATING PLASMA
Typically Zeff > 1
- plasma contains heavy, highly charged impurity species
- mixed collisionality plasma
1976, Hirshman:
particle flux ~ Pfirsch-Schlüter regime
heat flux
~ banana regime
Rotating Plasma
- centrifugal force pushes particles to outboard side of flux surface
- impurity ions undergo significant poloidal redistribution
- variation in collision frequency around flux surface
1999, Fülöp & Helander: particle flux typical of banana regime
CALCULATING THE TRANSPORT
Hinton & Wong: - transform kinetic equation to rotating frame
- consider effects occurring on different timescales
- expansion in d  ri / Lr
f  f 0  f1  f 2  ...
f1 ~ d f 0
• Cross-field transport second order in d
- evaluate using flux-friction relations
- relate flux to collisional moment of f1:
Angular momentum flux:
mi
     d 3 v mi R 2 v2 CiL  f1 
2e
• Linearised collision operator
~
• Separate classical and neoclassical contributions: f1  f1  f1
CALCULATING THE TRANSPORT
•
~
f1 determined by Hinton & Wong:
- valid for any species, independent of form of Ci
•
f1 obtained from the drift kinetic equation
- this can be cast as a variational problem
- requires an assumed trial function
• Alternative method of solution
- subsidary expansion of drift kinetic equation in
small ratio of ion collision to bounce frequency
- adopt a model for the collision operator
- the drift kinetic equation may be solved analytically
COLLISION OPERATOR
• Neglect ion-electron collisions
• Impurity concentration typically  nii ~ niz
• Explicit form of collision operator: Ci = Cii + Ciz
Cii : Kovrizhnykh model operator for self collisions
Ciz : disparate masses  analogous to electron - ion collisions


mi
Ciz  n iz ,   L f1   v  Vz ,  f 0 
Ti


• Parallel impurity momentum equation used to determine Vz
mz nz Vz   Vz  nz ze  p  R  nz zeVz  B
TRANSPORT MATRIX
Represent the fluxes in matrix form
    L11
  
 q    L21
  L
   31
L12
L22
L32
L13   d ln N iTi  dr 


L23   d (ln Ti ) dr 
L33   d ln w  dr 
 e ~ miw 2 R 2 

• ni  N i   exp    
2Ti 
 Ti
• Homogeneous non-rotating plasma:
- spin-up as a rigid body
dni dTi dw


0
dr
dr
dr
- centrifugal potential between flux surfaces drives transport
• Slow rotation: Ni  ni and Ni Ti  pi
TRANSPORT MATRIX
Represent the fluxes in matrix form
    L11
  
 q    L21
  L
   31
L12
L22
L32
L13   d ln N iTi  dr 


L23   d (ln Ti ) dr 
L33   d ln w  dr 
• Off-diagonal terms are non-zero due to the presence of impurities
~
• Each Lij is sum of classical, Lij , and neoclassical, Lij , contribution
• Restricted to subsonic rotation to calculate neoclassical terms
• Angular momentum transport coefficients L31, L32 and L33
• L33 usual measure of toroidal viscosity
RESULTS
r ( )
• Not restricted to circular flux surfaces
- minor radius r ( ) and inverse aspect
ratio  ( ) functions of poloidal angle

• Zeff = 1 recover: Braginskii - classical contributions to fluxes
Hinton & Wong - neoclassical part of L33
• Classical angular momentum flux ~
Enhanced transport
- larger outboard step size ri ~ 1/B
- larger angular momentum miw R2
No dominant driving force
R2
R2
ni 2  nz 2
B
B
NEOCLASSICAL COEFFICIENTS
i , n
L31 
...

i
pi'
Ti '
w' 

   L31  L32  L33 
2
pi
Ti
w
miw R

pi I 2  iz1
mi i2
2

b
2
1

f
nr


c
2
n
 nr  f nr 2  f
c
t
 b2
b2
1  f c 

n

- flux surface average
nz
n
nz
ft - ‘fraction of trapped particles’
Zeff
 
Zeff
R2
r  2
R
2
• Off-diagonal  only iz appears
• n within average  effect of redistribution






B2
b  2
B
2
NEOCLASSICAL COEFFICIENTS
Most experimentally relevant limit:
- conventional aspect ratio,    << 1
- strong impurity redistribution, n cos  ~ 1 , n  nz nz
L31 ~ L32 ~ n iz
q r
2
2
 3/ 2
 2 M z2  q 2 r 2
n iz 3 / 2
L33 ~  M i 
z 


• Enhancement of  -3/2 over previous predictions
- effectiveness of rotation shear as a drive increased by small factor
- radial pressure and temperature gradients dominate
 strong density and temperature gradients sustain strong Er shear
NEOCLASSICAL COEFFICIENTS
Numerical evaluation using magnetic surfaces of MAST -  = 0.14
Zeff  2
L31
0.015
Zeff
0.010
0.005
previous
level
Zeff
0.00024
0.1
0.2
0.3
0.4
ion Mach number
0.5
• increase with
impurity content
• increase with
Mach
number
• Transport
 1.5
as~impurity
10 times
redistribution
previous
increases
predictions
- weak effect
for small  :
1
n cos ~ 0.04
NEOCLASSICAL COEFFICIENTS
I
mi
3
2 2
L
miw R 2 Rzi||   2 
     d v mi R v Ci  f1  
eB
2e
• Large aspect ratio and strong impurity redistribution  ( 2)   2
- convective angular momentum transport dominates
• Angular momentum transport enhanced by enhanced particle flux
I

Rzi||
eB
• Small w - impurities rotate poloidally to minimise friction: Rzi||  0
• Large w - poloidal impurity rotation reduced due to centrifugal
localisation  enhanced friction
NEOCLASSICAL COEFFICIENTS
• Large L31 , L32  spontaneous toroidal rotation may arise:
w'
1 
p'
T' 
 L31  L32 

w
L33 
p
T
• Rotation direction depends on edge boundary condition
• L23 relates heat flux to toroidal rotation shear:
    L11
  
 q    L21
  L
   31
L12
L22
L32
L13   d ln pi  dr 


L23   d (ln Ti ) dr 
L33   d ln w  dr 
Co-NBI  shear
 heat pinch
Sub-neoclassical heat
transport…
SUMMARY
• Experimentally, angular momentum transport in regions of
neoclassical ion thermal transport has remained anomalous
• In a rotating plasma impurities will undergo poloidal redistribution
• Including this effect, general expressions for particle, heat and
angular momentum fluxes derived for mixed collisionality plasma
• At conventional aspect ratio, with impurities pushed towards
outboard side, angular momentum flux seen to increase by a
factor of  -3/2  now typical of banana regime
• Radial bulk ion pressure and temperature gradients are the
primary driving forces, not rotation shear  strong density and
temperature gradients sustain strongly sheared Er
• Spontaneous toroidal rotation may arise in plasmas with no
external angular momentum source
REFERENCES
[1] S. I. Braginskii, JETP (U.S.S.R) 6, 358 (1958)
[2] P. J. Catto et al, Phys. Fluids 28, 2784 (1987)
[3] T. Fülöp & P. Helander, Phys. Plasmas 6, 3066 (1999)
[4] C. M. Greenfield et al, Nucl. Fusion 39, 1723 (1999)
[5] P. Helander & D. J. Sigmar, Collisional Transport in
Magnetized Plasmas (Cambridge U. P., Cambridge, 2002)
[6] F. L. Hinton & S. K. Wong, Phys. Fluids 28, 3082 (1985)
[7] S. P. Hirshman, Phys. Fluids 19, 155 (1976)
[8] W. D. Lee et al, Phys. Rev. Lett. 91, 205003 (2003)
[9] M. N. Rosenbluth et al, Plasma Physics & Controlled Nuclear
Fusion Research, 1970, Vol. 1 (IAEA, Vienna, 1971)
[10] J. Wesson, Nucl. Fusion 37, 577 (1997),
P. Helander, Phys. Plasmas 5, 1209 (1998)