A new drift kinetic theory for the plasma response to the neoclassical tearing mode (NTM) magnetic perturbation is presented. Small magnetic islands of width, w ≪ a (a is the tokamak minor radius) are assumed, retaining the limit w ∼ ρ bi (ρ bi is the ion banana orbit width) to include finite orbit width effects. When collisions are small, the ions/electrons follow streamlines in phase space; for passing particles, these lie in surfaces that reproduce the magnetic island structure but have a radial shift by an amount, proportional to ρ ϑ i / e , where ρ ϑ i / e is the ion/electron poloidal Larmor radius. This shift is associated with the curvature and ∇B drifts and is found to be in opposite directions for V ∥ ≶ 0 , where V ∥ is the component of velocity parallel to the magnetic field. The particle distribution function is then found to be flattened across these shifted or drift islands rather than the magnetic island. This results in the pressure gradient being sustained across the magnetic island for w ∼ ρ ϑ i and hence reduces the neoclassical drive for NTMs when w is small. This provides a physics basis for the NTM threshold, which is quantified. In Imada et al (2019 Nucl. Fusion 59 046016, and references therein), a 4D drift kinetic non-linear code has been applied to describe these modes. In the present paper, the drift island formalism is employed. Valid at low collisionality, it allows a dimensionality reduction to a 3D problem, simplifying the numerical task and efficiently resolving the collisional boundary layer across the trapped-passing boundary. An improved model is adopted for the magnetic drift frequency. This decreases the NTM threshold, compared to the results shown in Imada et al (2019 Nucl. Fusion 59 046016, and references therein), making it in quantitative agreement with experimental observations, with w c = 0.45 ρ ϑ i , where w c is the threshold magnetic island half-width, or 2.85ρ bi for the full threshold island width, predicted for our equilibrium.

A new drift kinetic theory for the response of ions to small magnetic islands in toroidal plasma is presented. Islands whose width w is comparable to the ion poloidal Larmor radius are considered, expanding the ion response solution in terms of , where r is the minor radius. In this limit, the ion distribution can be represented as a function of toroidal canonical momentum, . With effects of grad-B and curvature drifts taken into account, the ion distribution function is a constant on a ‘drift island’ structure, which is identical to the magnetic island but radially shifted by . The distribution is then flattened across the drift island, rather than the magnetic island. For small islands , the pressure gradient is maintained across the magnetic island, suppressing the bootstrap current drive for the neoclassical tearing mode (NTM) growth. As , the ions are largely unperturbed. However, the electrons respond to the electrostatic potential required for quasi-neutrality and this provides a stabilizing contribution to the NTM evolution. This gives a new physical understanding of the NTM threshold mechanism, with implications for the design of NTM control systems for future tokamaks such as ITER.

In [K Imada et al. Nucl. Fusion 59 (2019) 046016] a new 4D drift kinetic nonlinear theory, valid in the limit of a low beta, small inverse aspect ratio, circular cross section, toroidal geometry, to describe the plasma response to the neoclassical tearing mode (NTM) magnetic perturbation is derived. In [A V Dudkovskaia et al. Plasma Phys. Control. Fusion 63 (2021) 054001] this theory is reduced in a low collisionality limit, which allows a dimensionality reduction to a 3D problem to efficiently resolve the collisional dissipation layer in the vicinity of the trapped-passing boundary. [A V Dudkovskaia et al. Plasma Phys. Control. Fusion 63 (2021) 054001] adopts an improved model for the magnetic drift frequency, which reduces the threshold magnetic island half-width from $8.73 \rho_{b i}$, where $\rho_{b i}$ is the trapped ion banana orbit width, to $1.46 \rho_{b i}$, making it in a closer agreement with experimental observations for the large aspect ratio tokamak equilibrium. In the present paper, the theory is extended to a high beta, arbitrary tokamak geometry to capture the plasma shaping effects on the NTM threshold physics with the focus on the non-zero triangularity discharges that are known to have a strong impact on the plasma MHD stability. First, it is found that the higher triangularity plasma is more prone to NTMs which is in agreement with the $2/1$ tearing mode onset relative frequency measurements in DIII-D. Second, the NTM threshold dependence on the tokamak inverse aspect ratio obtained in [A V Dudkovskaia et al. Plasma Phys. Control. Fusion 63 (2021) 054001] is refined and extended to a finite aspect ratio limit. Third, the NTM threshold dependence on poloidal beta is obtained and benchmarked against the EAST threshold island width measurements.

Finite radial transport around magnetic islands is believed to play an important role in the threshold, spatial structure and temporal evolution of neoclassical tearing modes (NTMs). We report on novel measurements of NTMs with mode structure m/n = 2/1 on the MAST spherical tokamak (ST), which have allowed a direct evaluation of the effect of transport on island behaviour for the first time on an ST. Temperature profiles obtained with the upgraded Thomson scattering (TS) system on MAST have been used to constrain the solutions of a heat transport equation for the NTM magnetic island (Fitzpatrick 1995 Phys. Plasmas 2 825), allowing the determination of the critical width for temperature flattening across an island w c , an important parameter in the modified Rutherford equation (MRE) for NTM evolution. The measured value of w c = 0.7 ± 0.2 cm obtained for an ensemble of high β N MAST discharges has been used in an analysis of the MRE for 2/1 NTM growth and saturation on MAST. Using a probabilistic method for parameter and error estimation, which takes account of the experimental uncertainty on measured equilibrium parameters, it is found that the temporal evolution of island size is well described by marginally, classically unstable NTMs with strongly destabilizing bootstrap current and stabilizing curvature terms. Finally, further analysis of a β ramp-down discharge is presented, in which the measured w c value explains the observed threshold width well.

A new drift-kinetic theory of the ion response to magnetic islands in tokamak plasmas is presented. Small islands are considered, with widths w much smaller than the plasma radius r, but comparable to the trapped ion orbit width ρ bi. An expansion in w=r reduces the system dimensions from five down to four. In the absence of an electrostatic potential, the ions follow stream lines that map out a drift-island structure that is identical to the magnetic island, but shifted by an amount ∼ few ρ bi. The ion distribution function is flattened across these drift islands, not the magnetic island. For small islands, w ∼ ρ bi , the shifted drift islands result in a pressure gradient being maintained across the magnetic island, explaining previous simulation results [E. Poli et al., Phys. Rev. Lett. 88, 075001 (2002)]. To maintain quasineutrality an electrostatic potential forms, which then supports a pressure gradient in the electrons also. This influence on the electron physics is shown to stabilize small magnetic islands of width a few ion banana widths, providing a new threshold mechanism for neoclassical tearing modes-a key result for the performance of future tokamaks, including ITER.