Exact Nonlinear Decomposition of Ideal-MHD Waves Using Eigenenergies. III. Gravity, Generalized Inhomogeneous Quasi-linear Partial Differential Equations, Mode Conversion, and Numerical Implementation

Abstract Precise tracking and measurement of the energy carried by the individual magnetohydrodynamic (MHD) modes has important implications and utility in both astrophysical and laboratory plasmas. Previously, this was only achievable in limited linear MHD cases in the β ≪ 1 or β ≫ 1 regimes. In a series of papers, of which this is the third, we introduced the Eigenenergy Decomposition Method (or EEDM) and derived exact analytical expressions for the modal energy components—called eigenenergies—of nonlinear 3D disturbances governed by the homogeneous ideal MHD equations. Here we extend the method to inhomogeneous ideal MHD by introducing a source term accounting for gravity, and we provide detailed guidelines for applying the decomposition scheme to any general inhomogeneous quasi-linear partial differential equations that possess a globally conserved quantity, beyond the realm of MHD. Furthermore, we show that the eigenenergies can be used to locate and measure nonlinear mode conversions, which is an additional feature of the method. Finally, we provide well-categorized context for the application of the method to simulations and discuss the possible numerical inaccuracies that may inevitably arise owing to discretization. This paper provides a more mature description of the method and its interpretation and is recommended as the starting point for readers unfamiliar with the method.