Physics-constrained, low-dimensional models for MHD: First-principles and data-driven approaches

Modeling and control of plasmas is a notoriously challenging, yet vital topic in modern physics. The magnetohydrodynamic (MHD) equations, for example, involve coupling between fluid dynamics and electromagnetism and exhibit nonlinear, multi-scale spatio-temporal dynamics. This work develops a novel reduced-order modeling framework for compressible plasmas, leveraging decades of progress in first-principles and data-driven modeling of fluids. First, we introduce a dimensionally consistent reduction technique to approximate the plasma in terms of a low-dimensional set of energetic coherent structures, or modes. Next, we derive an analytic model by Galerkin projection of the compressible Hall-MHD equations onto these modes. Importantly, we explicitly constrain the structure of the Galerkin model to enforce conservation of energy with a power balance argument. This theoretical framework enables the development of sparse and interpretable nonlinear reduced-order models from data that are intrinsically connected to the underlying physics. We demonstrate this approach on data from high-fidelity numerical simulations, tuned to model a 3D, turbulent spheromak experiment. We find excellent agreement with a low-dimensional model that describes the evolution of dominant coherent structures in the plasma. This reduced-order modeling framework demonstrates promise for the prediction, estimation, and control in industrial and laboratory plasmas.