A Diophantine approach to evaluating the robustness of magnetic surfaces

The rotational transform or safety factor profile of a toroidally confined plasma is a key determinant of its stability characteristics with respect to magnetohydrodynamic (MHD) instabilities. In practice, however, we often rely on a combination of accumulated experience and trial-and-error to provide intuition as to which specific resonances to avoid or values of rotational transform to target in profile optimization and design. In this work, we recast the Energy Principle in terms of a resonance parameter to draw out explicit connections between the physics properties of ideal MHD equilibria (specifically, ideal linear stability) and the structure of numbers. With this, we use the theory of rational approximations to classify the ideal stability properties of magnetic surfaces with both rational and irrational rotational transform. It is often held that certain “low-order rationals” appearing in the rotational transform or safety factor profile are more dangerous than others from the perspective of MHD stability. By introducing a theoretical framework based on the structure of numbers, we place this notion on more rigorous mathematical footing, which can then be directly applied to rotational transform/safety factor profile design and optimization in tokamaks and stellarators alike.