Mori-Zwanzig Modal Decomposition

We introduce the Mori-Zwanzig (MZ) Modal Decomposition (MZMD), a novel technique 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 it to DMD when applied to a complex flow: a Direct Numerical Simulation (DNS) data-set of laminar-turbulent boundary-layer transition flow over a flared cone at Mach 6. We show that the addition of memory terms by MZMD improves the resolution of spatio-temporal structures within the transitional/turbulent regime, which 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. As a result, compared to DMD, MZMD improves future state prediction accuracy, while requiring nearly the same computational cost.