Mori-Zwanzig Modal Decomposition: Applications and Analysis
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We present the Mori-Zwanzig (MZ) Modal Decomposition (MZMD), a novel technique first developed in [Woodward, M, et al. arXiv:2311.09524 2023] for performing modal analysis of large scale spatio-temporal structures in complex dynamical systems, and show that it represents an efficient generalization of Dynamic Mode Decomposition (DMD). The MZ formalism provides a mathematical framework for constructing non-Markovian reduced-order models of resolved variables from high-dimensional dynamical systems, incorporating the effects of unresolved dynamics through the memory kernel and orthogonal dynamics. We present a formulation and analysis of the modes and spectrum from MZMD and compare them to DMD when applied to several different flows: a) a 2D flow over a cylinder for a range of Reynolds numbers and b) hypersonic laminar-turbulent boundary-layer transition flow over a range of geometries, where data are obtained from Direct Numerical Simulation (DNS). For each flow, we show that the addition of memory terms by MZMD improves the resolution of spatio-temporal structures, especially within the strongly nonlinear regions of the flow, such as the transitional/turbulent regime. The transitional regions contains features that arise due to nonlinear mechanisms, such as the generation of the so-called "hot" streaks on the surface of the flared cone. Furthermore, compared to DMD, MZMD improves future state prediction accuracy, while requiring nearly the same computational cost.
