Effects of 3D Magnetic Perturbations on Toroidal Plasmas

Small 3D magnetic perturbations have many interesting and useful effects on tokamak and quasi-symmetric stellarator plasmas. Plasma transport equations that include these effects, most notably on diamagnetic-level toroidal plasma rotation, have recently been developed. The 3D magnetic perturbations and their plasma effects can be classified according to their toroidal mode number n. Low n non-resonant fields induce a neoclassical toroidal viscosity (NTV) that damps toroidal rotation throughout the plasma toward an offset flow in the counter-current direction; recent tokamak experiments have generally confirmed and exploited these predictions by applying external low n non-resonant magnetic perturbations. Medium n toroidal field ripple produces similar effects plus possible ripple trapping NTV effects and direct ion losses in the edge. A low n (e.g., n= 1) resonant field is mostly shielded by the toroidally rotating plasma at and inside the resonant (rational) surface; if it is large enough it can stop plasma rotation at the rational surface, facilitate magnetic reconnection there and lead to a growing locked mode, which often causes a plasma disruption. Externally applied 3D magnetic perturbations usually have many components; in the plasma their lowest n (e.g., n=1) externally resonant components can be amplified by kink-type plasma responses, particularly at high β. Low n plasma instabilities (e.g., NTMs, RWMs) cause additional 3D magnetic perturbations in tokamak plasmas; tearing modes can bifurcate the topology and form magnetic islands. Finally, multiple resonant magnetic perturbations (RMPs) can, if not shielded by plasma flow effects, cause local magnetic stochasticity and influence Hmode edge pedestal transport. These various effects of 3D magnetic perturbations can be used to directly modify plasma toroidal rotation and indirectly plasma transport, e.g., for reducing anomalous transport and ELM control. The present understanding and modeling of these various effects, and key open issues for development of a predictive capability of them for ITER are discussed. 1 Magnetic Field Representation Tokamaks are two-dimensional (2D) axisymmetric magnetic systems to lowest order. But small 3D perturbations δB arise from externally applied fields and plasma instabilities. The B field magnitude in near-axisymmetric tokamaks can be written using the poloidal magnetic flux ψ (radial coordinate), straight-field-line poloidal angle θ and axisymmetric toroidal angle ζ as |B| = |B0(ψ, θ)| } {{ } 2D axisymm. + ∑ n,m δBn(ψ,m) cos (mθ − nζ − φm,n) } {{ } low m,n resonant, non-resonant + δBN (ψ, θ) cos(Nζ) } {{ } medium n, ripple + · · · . (1) The 3D magnetic perturbations and their effects on toroidal plasmas can be classified by their toroidal mode number n: low n (1 to 5) resonant (with magnetic field line pitch, q=m/n) and non-resonant fields, medium n (mainly due to ripple from N toroidal field coils) and high n (· · · , due to microturbulence). Fields in quasi-symmetric stellarators can be represented similarly. This theory-based overview paper concentrates on low and medium n perturbations. Plasma flows are discussed first. Then, key 3D theory elements and finally combined effects are discussed. 2 Plasma Toroidal Rotation And Transport Equations Key equations: Plasma transport equations for density, temperature and flows in tokamak plasmas that include 3D magnetic perturbation effects, including on diamagnetic-level plasma flows, have recently been developed [1]. These developments build on the fluid moment approach to stellarator plasma transport in which flows within a magnetic surface are obtained first [2], before the self-consistent radial electric field and net cross-field “radial” transport fluxes are determined. But they go one step further by assuming the 3D magnetic perturbations are gyroradius small compared to the axisymmetric (or stellarator quasi-symmetric) magnetic field. Then, various constraints on plasma flows are obtained on successive time scales [1]: 1) Radial ion force balance is enforced by compressional Alfvén waves on the μs time scale, which yields