Modelling neoclassical tearing modes in tokamak plasmas

Understanding the physics of the neoclassical tearing mode (NTM) onset and its stabilisation is one of the key issues in providing successful operation of future power plants. The latter, in turn, requires a well developed predictive theory of the tearing mode threshold in order to specify and optimise control schemes. A new drift kinetic theory is presented to calculate the plasma response to the NTM magnetic island. Small magnetic islands compared to the tokamak minor radius are assumed but island widths, $w$, comparable to the ion banana orbit width, $\rho_{b i}$, are treated accurately, retaining finite orbit width effects. To provide dimensionality reduction, streamlines, $S$, are derived that can be interpreted as a generalised radial coordinate. Adopting a low collisionality plasma, the distribution function is found to be constant on contours of constant $S$ when collisions are neglected. Proceeding to next order, and introducing collisions, the dependence of the particle distribution on $S$ and pitch angle, $\lambda$, is determined. $S$ contours reproduce the magnetic island geometry but have a radial shift of a few poloidal gyro-radii, $\rho_{\vartheta}$. This radial shift is found only for passing particles and is in opposite directions for ${ V }_{ \parallel }\gtrless 0$, $V_{\parallel}$ is the parallel component of velocity. The distribution function being flattened across these $S$ islands rather than the magnetic island restores the pressure gradient across a magnetic island of width $w \lesssim \rho_{\vartheta i}$, which provides a physics basis for the NTM threshold by suppressing the NTM drive. Collisions cannot be treated perturbatively near the trapped-passing boundary in pitch angle, and thus here a thin collisional boundary layer is identified. This layer matches the passing and trapped solutions outside the layer and being the dominant source of dissipation provides the island propagation frequency. The solution provides a threshold island width, $w_c$ (below which magnetic islands are healed), which arises from the passing particle dynamics, and the relevant parameter is the ion poloidal gyro-radius, $\rho_{\vartheta i}$: $w_c = 3\rho_{\vartheta i}$.