Transcript fus2005 5334

Modelling the Neoclassical Tearing Mode
Howard Wilson
EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxon OX14 3DB
This work was funded jointly by the UK Engineering and Physical Sciences Research Council and EURATOM
Outline
• Background to neoclassical tearing modes:
– Consequences: magnetic islands
– Drive mechanisms
– Bootstrap current and the neoclassical tearing mode
– Threshold mechanisms
– Key unresolved issues
• Neoclassical tearing mode calculation
– The mathematical details
• Summary
Magnetic islands in tokamak plasmas
• In a tokamak, field lines lie on nested, toroidal flux surfaces
– To a good approximation, particles follow field lines
 Heat and particles are well-confined
• Tearing modes are instabilities that lead to a filamentation of the
current density
– Current flows preferentially along some field lines
– The magnetic field acquires a radial component, so that magnetic islands
form, around which the field line can migrate
O-point
X-point
r=r2
2pR
r=r2
r
pR
r=r1
0
pr
Rf
2pr
rq
Poloidal direction
2pR
r
pR
r=r1
0
pr
Rf
2pr
rq
Poloidal direction
Neoclassical Tearing Modes arise from a filamentation
of the bootstrap current
• The bootstrap current exists due to a combination of a plasma
pressure gradient and trapped particles
• The particle energy, v2, and magnetic moment, m, are conserved
v||   v 2  2mB 
1/ 2
• Particles with low v|| are “trapped” in low B region:
– there are a fraction ~(r/R)1/2 of them
– they perform “banana” orbits
B
r
R
The bootstrap current mechanism
• Consider two adjacent flux surfaces:
High density
Low density
Apparent flow
• The apparent flow of trapped particles “kicks” passing particles
through collisions:
– accelerates passing particles until their collisional friction balances the
collisional “kicks”
– This is the bootstrap current
– No pressure gradient  no bootstrap current
– No trapped particles  no bootstrap current
The NTM drive mechanism
Consider an initial small “seed” island:
Perturbed flux surfaces;
lines of constant W
Poloidal
angle
• The pressure is flattened within the
island
Pressure
• Thus the bootstrap current is
removed inside the island
Pressure flattens across island
Minor radius
• This current perturbation amplifies
the magnetic island
Cross-field transport provides a threshold for growth
• In the absence of sources in the vicinity of the island, a model
transport equation is:
  0
  c    p  c|| || p
c  2 p  c||||2 p  0
1
 ~
w
kw
|| ~ q
Ls
Thin islands, field lines
along symmetry dn...||0
• For wider islands, c||||>>c p flattened
•For thinner islands such that c||||~c
1/ 2
 Ls 
wc ~  
 kq 
1/ 4
 c 
 
c 
 || 
 pressure gradient sustained
 bootstrap current not perturbed
Wider islands, field lines
“see” radial variations
Let’s put some numbers in (JET-like)
1/ 2
 Ls 
wc ~  
 kq 
1/ 4
 c 
 
c 
 || 
Ls~10m
kq~3m–1
c~3m2s–1
c||~1012m2s–1
~3mm
(1) This width is comparable to the orbit width of the ions
(2) It assumes diffusive transport across the island, yet the length scales are
comparable to the diffusion step size
(3) It assumes a turbulent perpendicular heat conductivity, and takes no
account of the interactions between the island and turbulence
• To understand the threshold, the above three issues must be addressed
 a challenging problem, involving interacting scales.
Electrons and ions respond differently to the island:
Localised electrostatic potential is associated with the island
• Electrons are highly mobile, and move rapidly along field lines
 electron density is constant on a flux surface (neglecting c)
• For small islands, the EB velocity dominates the ion thermal velocity:
kq w
v|||| ~ vth ,i
Ls
v E   i Ls 1 qi
~
~
kq kqi vth ,i
v||||
w r s w
vE   ~
~
wB
r
• For small islands, the ion flow is provided by an electrostatic potential
– this must be constant on a flux surface (approximately) to provide quasineutrality
• Thus, there is always an electrostatic potential associated with a magnetic
island (near threshold)
– This is required for quasi-neutrality
– It must be determined self-consistently
An additional complication: the polarisation current
• For islands with width ~ion orbit (banana) width:
– electrons experience the local electrostatic potential
– ions experience an orbit averaged electrostatic potential
 the effective EB drifts are different for the two species
 a perpendicular current flows: the polarisation current
Jpol
E×B
• The polarisation current is not divergence-free, and drives a current along the
magnetic field lines via the electrons
• Thus, the polarisation current influences the island evolution:
– a quantitative model remains elusive
– if stabilising, provides a threshold island width ~ ion banana width (~1cm)
– this is consistent with experiment
Summary of the Issues
• What provides the initial “seed” island?
– Experimentally, usually associated with another, transient, MHD event
• What is the role of transport in determining the threshold?
– Is a diffusive model of cross-field transport appropriate?
– How do the island and turbulence interact?
– How important is the “transport layer” around the island separatrix?
• What is the role of the polarisation current?
– Finite ion orbit width effects need to be included
– Need to treat v||||~vE·
• How do we determine the island propagation frequency?
– Depends on dissipative processes (viscosity, etc)
• Let us see how some of these issues are addressed in an analytic calculation
An Analytic Calculation
An analytic calculation: the essential ingredients
• The drift-kinetic equation
– neglects finite Larmor radius, but retains full trapped particle orbits
• We write the ion distribution function in the form:
 q 
f i  1  i  FMi  g i c, , q; v, v  
Ti 

where gi satisfies the equation:

Lines of
constant W

c
q
v|| g i
g i
g i
c B  
qi v d   g i
 v d  gi 
 k|| v||
 C ( gi )


g


i
2
 c ,q Rq q c ,
 W
B
mi
v
v
 
v|| A||  qi 
qi FMi 
T


v  


  *i 
 T d
mi 


c


i
c ,q


Self-consistent
electrostatic potential
  q  Iv||   v||  F dn T
*i
  Mi
  1  i 
Ti  Rq q  ci  n dc *i
 
Vector potential
associated with B
• Solved by identifying two small parameters:
w
bj

j 
r
w
bj=particle banana width
w=island width
r=minor radius
An analytic calculation: the essential ingredients
• The ion drift-kinetic equation:
vv|||| ggii
gi
g
c B  
qi v d   g i


 k|| v|| i  v d  gi 
 C ( gi )


g

i
2
Rq qq cc,,
 c ,q Rq
 W
B
mi
v
v
  q  Iv||   v||  FMi dn T*i
 
v || A||  qii
qi FMi 
T
ii



 

v
  *i 
d     1 
d

mi 
Tii  Rq q  ci  n dc *i
 
  c ,q c   Tii
Black terms are O(1)
Blue terms are O()
Red terms are O(i)
Pink terms are O(i)
We expand:
g i    im n g i( m,n )
m ,n
Order 0 solution
• To O(0), we have:
v g
 || i
Rq q
c ,
Iv||   v||  g i Iv||   v||  FMi dn *Ti
 
 


Rq q   ci  c Rq q   ci  n dc *i
 g i( 0, 0 ) ( c , ,q , v|| , v  )  g i
( 0, 0 )
(1, 0 )
g
 i
( c , , v|| , v  )
( 0, 0 )
T 
Iv||   g i
F


n
i 


 Mi
 hi
ci  c
n c i 
No orbit info, no island info
Orbit info, no island info
• The free functions introduce the effect of the island geometry, and are
determined from constraint equations [on the O() equations]
Order  solution
• To O(0), we have:
g i( 0, 0 )


v|| g i( 0,1)
g i( 0, 0) c B  
( 0, 0 )
( 0, 0 )

 k|| v||

C
(
g
)



g
i
i
2
Rq q c ,
 W
B
c ,q

v|| A|| 
qi FMi
T  



  *i 

 
mi
c  
c ,q

•Average over q coordinate (orbit-average…a bit subtle due to trapped ptcles):
 ( 0, 0 )   T*e  FMe dn
c  hW
 g e  
T
n dc
  i  FMi dn c  hW
*e
g i( 0,0) 

i
n dc
q

c  hW



m
 leading order density is a function of perturbed flux
 undefined as we have no information on cross-field transport
 introduce perturbatively, and average along perturbed flux surfaces:
hW  (W  1)
W
dW
2 2 1 Q
wc
Q(W) 
1
2p

W  cos d
Note: solution implies multi-scale interactions
• Solution for gi(0,0) has important implications:
 flatten density gradient inside island stabilises micro-instabilities
 steepen gradient outside could enhance micro-instabilities
 however, consistent electrostatic potential implies strongly sheared flow
shear, which would presumably be stabilising
unperturbed
Density
across X-pt
3
2
1
0
c/w
1
2
c
across O-pt
• An important role for numerical modelling would be to
 understand self-consistent interactions between island and m-turbulence
 model small-scale islands where transport cannot be treated perturbatively
 model the “transport layer” around the island separatrix
These are all neglected in the analytic approach
Order  equation provides another constraint equation,
with important physics
• Averaging this equation over q eliminates many terms, and provides an
important equation for gi(1,0)
  Rq  dh  g (1, 0 )
 i
 Rqk||  1 
~ dW  
v|| m
 

W
I FMi dn dh   T*i    v||  c
 
~
 n dc dW *i
c  ci   W
q

Rq
C ( g i(1, 0 ) )
v||


q

0
q
T 
Iv||   g i
F


n
i 


 Mi
 hi


ci  c
n c i 
( 0, 0 )
g
(1, 0 )
i
• We write
~
hi (W, )  H i W   H i (W, )
Provides
bootstrap
contribution
~
• We solve above equation for Hi(W) and H i W,  
 yields bootstrap and polarisation current
Provides
polarisation
contribution
Different solutions in different collisionality limits

 Rqk|| 

 Rq  dh  g i(1, 0 )
1 

~

v|| m dW  

W
I FMi dn dh   T*i    v||  c
 
~
 n dc dW *i
c  ci   W
q

Iv||   g i
FMi n  



 hi


ci  c
n c i 
( 0, 0 )
g
(1, 0 )
i
Rq
C ( g i(1, 0 ) )
v||
T
i


q

0
q
~
hi (W, )  H i W   H i (W, )
• Eqn for Hi(W) obtained by averaging along lines of constant W to eliminate red terms
 recall, bootstrap current requires collisions at some level
 bootstrap current is independent of collision frequency regime
~
• Equation for H i (W, ) depends on collision frequency
 larger polarisation current in collisional limit (by a factor ~q2/e1/2)
• A kinetic model is required to treat these two regimes self-consistently
 must be able to resolve down to collisional time-scales
 or can we develop “clever” closures?
Closing the system
• The perturbation in the plasma current density is evaluated from the distribution
functions
• The corresponding magnetic field perturbation is derived by solving Ampére’s
equation with “appropriate” boundary conditions ()
 B  J
• The island width is related to the magnetic field perturbation
 The “modified Rutherford” equation
2
2


1 dw
1
dn
w
1
dn
 qi  
 q
   Cbse1/ 2 q

C
 

pol 
r dt
s n dr w2  wc2
w
sn
dr
 w
  
Inductive
current
Equilibrium
current
gradients
Bootstrap
current
polarisation
current
The Modified Rutherford Equation: summary
dw
dt
Need to generate “seed” island
 additional MHD event
 poorly understood?
Stable solution
 saturated island width
 well understood?
w
Unstable solution
 Threshold
 poorly understood
 needs improved transport model
 need improved polarisation current
Summary
• A full treatment of neoclassical tearing modes will likely require a kinetic model
• A range of length scales will need to be treated
 macroscopic, associated with equilibrium gradients
 intermediate, associated with island and ion banana width
 microscopic, associated with ion Larmor radius and layers around separatrix
• A range of time scales need to be treated
 resistive time-scale associated island growth
 diamagnetic frequency time-scale associated with transport and/or island
propagation
 time-scales associated with collision frequencies
• In addition, the self-consistent treatment of the plasma turbulence and formation of
magnetic islands will be important for
• understanding the threshold for NTMs
• understanding the impact of magnetic islands on transport (eg formation of
transport barriers at rational surfaces)