The structure of global conservation laws in Galerkin plasma models

Plasmas are highly nonlinear and multi-scale, motivating a hierarchy of models to understand and describe their behavior. However, there is a scarcity of plasma models of lower fidelity than magnetohydrodynamics (MHD). Galerkin models, obtained by projection of the MHD equations onto a truncated modal basis, can furnish this gap in the lower levels of the model hierarchy. In the present work, we develop low-dimensional Galerkin plasma models which preserve global conservation laws by construction. This additional model structure enables physics-constrained machine learning algorithms that can discover these types of low-dimensional plasma models directly from data. This formulation relies on an energy-based inner product which takes into account all of the dynamic variables. The theoretical results here build a bridge to the extensive Galerkin literature in fluid mechanics, and facilitate the development of physics-constrained reduced-order models from plasma data.