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

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

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

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

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

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

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

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.

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

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

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

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: 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

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:

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

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:

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

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

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?

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

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

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)