Tutorial:Modelling the Neoclassical Tearing Mode
Howard Wilson
Department of Physics, University of York, Heslington, York, YO10 5DD
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
r
r=r1
r=r2
r
R0 r 2r
R
2R
X-point
O-point
Poloidal directionToroidal dire
ction
• 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
r
r=r1
r=r2
r
R0 r 2r
R
2R
Poloidal directionToroidal dire
ction
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, , are conserved
• 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
2/12|| 2vv B
r
The bootstrap current mechanism
• Consider two adjacent flux surfaces:
• 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
High density
Low density
Apparent flow
The NTM drive mechanism
• The pressure is flattened within the island
• Thus the bootstrap current is removed inside the island
• This current perturbation amplifies the magnetic island
Consider an initial small “seed” island:
Perturbed flux surfaces;lines of constant
Pressure flattens across island
Minor radius
Pre
ssur
e
Poloidalangle
Cross-field transport provides a threshold for growth
• In the absence of sources in the vicinity of the island, a model transport equation is:
• For wider islands, ||||>> p flattened
•For thinner islands such that ||||~
pressure gradient sustained bootstrap current not perturbed
0 |||| pp
02||||
2 pp
w
1~
Thin islands, field linesalong symmetry dn...||0
Wider islands, field lines“see” radial variations
sL
wk ~||
4/1
||
2/1
~
k
Lw s
Let’s put some numbers in (JET-like)
4/1
||
2/1
~
k
Lw s
(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.
Ls~10m ~3m2s–1
k~3m–1 ||~1012m2s–1
~3mm
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 )
• For small islands, the E velocity dominates the ion thermal velocity:
• For small islands, the ion flow is provided by an electrostatic potential– this must be constant on a flux surface (approximately) to provide quasi-neutrality
• 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
s
ith L
wk ,|||| v~v
r
vk
wB
k ithiE
,~~v
wsr
L
wisiE
1
~~v
v
||||
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 EB drifts are different for the two species a perpendicular current flows: the polarisation current
• 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
E×B
Jpol
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:
where gi satisfies the equation:
• Solved by identifying two small parameters:
,
||vig
Rq
,
ig
ig
k ||||v igB
Bc 2
)( igC
v ||||
,
*
A
cm
Fq Ti
i
Mii
id g v
i
TiMi
ci d
dn
n
F
Rq
I
*
*|||| vv 1
vv
v
id
i
i g
m
q
d
i
i
T
qv
i
i
T
q
v,v;,,1 iMi
i
ii gF
T
qf
r
ww
bjj
bj=particle banana widthw=island widthr=minor radius
Vector potentialassociated with B
Self-consistentelectrostatic potential
Lines ofconstant
vv
v
id
i
i g
m
q
d
i
i
T
qv
i
i
T
q
,
||vig
Rq
,
ig
ig
k ||||v igB
Bc 2
)( igC
v ||||
,
*
A
cm
Fq Ti
i
Mii
id g v
i
TiMi
ci d
dn
n
F
Rq
I
*
*|||| vv 1
An analytic calculation: the essential ingredients
• The ion drift-kinetic equation:
We expand:),(
,
nmi
n
nm
mii gg
,
||vig
Rq
Black terms are O(1)
,
ig
ig
k ||||v igB
Bc 2
)( igC
v ||||
,
*
A
cm
Fq Ti
i
Mii
Blue terms are O()
d
i
i
T
qv
id g v
i
TiMi
ci d
dn
n
F
Rq
I
*
*|||| vv 1
Red terms are O(i)
i
i
T
q
vv
v
id
i
i g
m
q
d
i
i
T
qv
i
i
T
q
Pink terms are O(i)
Order 0 solution
• To O(0), we have:
• The free functions introduce the effect of the island geometry, and are determined from constraint equations [on the O() equations]
,
||vig
Rqi
TiMi
ci
i
ci d
dn
n
F
Rq
Ig
Rq
I
*
*|||||||| vvvv
)v,v,,()v,v,,,( ||
)0,0(
||)0,0(
ii gg
i
i
TiMii
ci
i hn
n
FgIg
)0,0(
||)0,1( v
No orbit info, no island info
Orbit info, no island info
Order solution
• To O(), we have:
•Average over coordinate (orbit-average…a bit subtle due to trapped ptcles):
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:
,
)1,0(||v
ig
Rq
,
)0,0(
ig
)0,0(
||||vig
k )0,0(2 ig
B
Bc )( )0,0(igC
||||
,
*
v A
cm
Fq Ti
i
Mii
hd
dn
n
Fg Mi
i
Ti
i
)0,0(
dQQ
dwh cos
2
1)(
22)1(
1
hm
q
hd
dn
n
Fg Me
e
Te
e
*
*)0,0(
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
• An important role for numerical modelling would be to understand self-consistent interactions between island and -turbulence model small-scale islands where transport cannot be treated perturbatively
3 2 1 0 1 2
De
nsi
ty
/w
unperturbed
across X-pt
across O-pt
These are all neglected in the analytic approach
model the “transport layer” around the island separatrix
• Averaging this equation over eliminates many terms, and provides an important equation for gi
(1,0)
• We write
• We solve above equation for Hi() and yields bootstrap and polarisation current
Order equation provides another constraint equation, with important physics
cii
TiMii
d
dh
d
dn
n
FIg
d
dh
m
RqRqk ||
*
*)0,1(
||
||
v~~v
1
0)(v
)0,1(
||
igCRq
),(~
),( iii HHhProvidesbootstrapcontribution
Provides polarisationcontribution
i
i
TiMii
ci
i hn
n
FgIg
)0,0(
||)0,1( v
,~
iH
• Eqn for Hi() obtained by averaging along lines of constant to eliminate red terms recall, bootstrap current requires collisions at some level bootstrap current is independent of collision frequency regime
• Equation for depends on collision frequency larger polarisation current in collisional limit (by a factor ~q2/)
• 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?
Different solutions in different collisionality limits
cii
TiMii
d
dh
d
dn
n
FIg
d
dh
m
RqRqk ||
*
*)0,1(
||
||
v~~v
1
0)(v
)0,1(
||
igCRq
),(~
),( iii HHhi
i
TiMii
ci
i hn
n
FgIg
)0,0(
||)0,1( v
),(~ iH
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 ()
• The island width is related to the magnetic field perturbation The “modified Rutherford” equation
JB
wdr
dn
snwC
ww
w
dr
dn
nsC
dt
dw
ri
polbs
22
22
2/1 111
Inductivecurrent
Equilibriumcurrentgradients
Bootstrapcurrent
polarisationcurrent
The Modified Rutherford Equation: summary
w
dt
dw
Unstable solution Threshold poorly understood needs improved transport model need improved polarisation current
Stable solution saturated island width well understood?
Need to generate “seed” island additional MHD event poorly understood?
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)