Tutorial: Modelling the Neoclassical Tearing Mode Howard Wilson

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

2. Outline • Background toneoclassical 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 O-point X-point r=r2 2pR r pR Rf r=r1 pr 2pr 0 rq 2pR r=r2 r pR Rf r=r1 pr 2pr 0 Toroidal direction rq Poloidal direction • 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 Toroidal direction Poloidal direction

4. Neoclassical Tearing Modes arise from a filamentation of the bootstrap current B r R • 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

5. The bootstrap current mechanism High density Low density Apparent flow • 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

6. The NTM drive mechanism Poloidal angle Consider an initial small “seed” island: Perturbed flux surfaces; lines of constant W • 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–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.

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 Jpol E×B • 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

11. 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 Black terms are O(1) We expand: Blue terms are O(D) Red terms are O(di) Pink terms are O(Ddi) • The ion drift-kinetic equation:

15. Order D0 solution • To O(D0), we have: • 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 • model the “transport layer” around the island separatrix These are all neglected in the analytic approach • 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 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. 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)