1. Modelling the Neoclassical Tearing Mode
Tutorial:
Modelling the Neoclassical Tearing Mode
Howard Wilson
Department of Physics, University of York, Heslington, York, YO10 5DD

2. Background to neoclassical tearing modes:
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

3. 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 r
r=r1
r=r2 rq Rf pr
2pr pR
2pR
X-point
O-point r
r=r1
r=r2 rq Rf pr
2pr pR
2pR
Poloidal direction
Toroidal direction
Toroidal direction
Poloidal direction

4. 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
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

5. 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
Apparent flow
Low density
High density

6. 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
Thus the bootstrap current is removed inside the island
This current perturbation amplifies the magnetic island

7. 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, c||||>>c p flattened
For thinner islands such that c||||~c
pressure gradient sustained
bootstrap current not perturbed
Thin islands, field lines
along symmetry dn...||0
Wider islands, field lines
“see” radial variations

8. Let’s put some numbers in (JET-like)
Ls~10m c~3m2s–1
kq~3m– 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.

9. 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:
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

10. 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
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

11. What provides the initial “seed” island?
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

12. An Analytic Calculation

13. 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:
Lines of
constant W x q c
Self-consistent
electrostatic potential
Vector potential
associated with dB
rbj=particle banana width
w=island width
r=minor radius

14. An analytic calculation: the essential ingredients
The ion drift-kinetic equation:
Blue terms are O(D)
Black terms are O(1)
Pink terms are O(Ddi)
Red terms are O(di)
We expand:

15. Order D0 solution To O(D0), we have: No orbit info, no island info
The free functions introduce the effect of the island geometry, and are determined from constraint equations [on the O(D) equations]
No orbit info, no island info
Orbit info, no island info

16. Order D solution To O(Dd0), we have:
Average over q 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:

17. 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 m-turbulence
model small-scale islands where transport cannot be treated perturbatively
These are all neglected in the analytic approach
model the “transport layer” around the island separatrix
unperturbed
across X-pt
across O-pt c

18. Order Dd 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)
We write
We solve above equation for Hi(W) and
yields bootstrap and polarisation current
Provides
bootstrap
contribution
Provides
polarisation
contribution

19. Different solutions in different collisionality limits
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 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?

20. The island width is related to the magnetic field perturbation
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 (D)
The island width is related to the magnetic field perturbation
The “modified Rutherford” equation
Equilibrium
current
gradients
Bootstrap
current
Inductive
current
polarisation
current

21. The Modified Rutherford Equation: summary
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

22. 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)