Formally Reducing the Large Aspect Ratio Magnetohydrodynamic Equations*

In their ideal form, the magnetohydrodynamic (MHD) equations model the evolution of plasmas with varying density (ρ), velocity (v), and magnetic fields (B). For large aspect ratio tokamaks, a multi-scale analysis can simplify these complex systems by separating out irrelevant degrees of freedom. Previous researchers, such as Strauss, have developed reduced MHD (RMHD) models using velocity and magnetic field stream functions. In this work, we formalize the MHD reduction process initiated by Strauss through the framework of fast-slow systems. Our contributions include: a) refining and completing Strauss’s original arguments, and b) introducing a new division of quickly- and slowly-evolving dependent variables. We retain the stream-function description while incorporating a new representation for the density field, performing our analysis in both low- and high-plasma β and low-flow scaling regimes. Additionally, we identify the dynamics of these new variables as evolving coordinates of a manifold. Future work will explore how this geometric interpretation connects with Hamiltonian and symplectic structures.