|Tokamak plasmas display finite macroscopic rotation in standard machine operation. Yet, the theoretical understanding of stationary states is lacking as compared to static configurations. In this work, we investigate the magnetohydrodynamic equilibrium and stability properties of high-beta tokamak plasmas flowing at a significant fraction of the sound speed.
In the equilibrium part, we introduce a new family of analytical solutions of the Grad-Shafranov-Bernoulli system with diffuse flows in both the toroidal and poloidal directions. Furthermore, our solution allows finite plasma shaping, making it suitable to model present-day tokamak devices. The solution strategy consists of a combination of a variational perturbative scheme in terms of the inverse aspect ratio, a boundary perturbation approach in terms of the triangularity and the Green's function method. While the equilibrium solution corresponding to a circular cross-section is given in closed-form, those for elliptical and D-shaped scenarios are provided in a series-form in terms of Mathieu functions, even so, they can accommodate experimentally relevant values for elongation and triangularity. All solutions show excellent performance when benchmarked against the code FLOW.
As an example of the applicability of our analytical equilibrium for the circular cross-section, we perform a linear stability analysis focusing on the development of ideal external kink modes and resistive wall modes for a purely toroidal velocity profile. The stability problem is expressed as a set of algebraic equations which incorporate a kink mode drive for instabilities and resistive wall effects, while pressure and shear-flow drives are captured at the eigenmode equation level. Solutions are found by a multidimensional shooting method for the coupled side-bands. Results are compared against a sharp-boundary model with a solid body rotation from the literature. Although, in general, results indicate that the qualitative character of the instabilities under study in the presence of a diffuse or a solid body rotation are similar, rotation has a stronger destabilizing effect in the former model. Arguably, this difference is due to a global shear-flow drive effect.