A Characteristic Dynamic Mode Decomposition

TL;DR

This paper introduces a dynamic mode decomposition method that effectively captures moving structures in complex data by transforming to a reference frame aligned with their group velocity, improving modal analysis.

Contribution

The paper proposes a characteristic dynamic mode decomposition that accounts for convecting phenomena, enabling better representation of traveling structures in data.

Findings

Reduces moving structures to few modes in a proper reference frame

Improves decay of singular values for convecting phenomena

Applicable to turbulent flows and coherent structures

Abstract

Temporal or spatial structures are readily extracted from complex data by modal decompositions like Proper Orthogonal Decomposition (POD) or Dynamic Mode Decomposition (DMD). Subspaces of such decompositions serve as reduced order models and define either spatial structures in time or temporal structures in space. On the contrary, convecting phenomena pose a major problem to those decompositions. A structure traveling with a certain group velocity will be perceived as a plethora of modes in time or space respectively. This manifests itself for example in poorly decaying singular values when using a POD. The poor decay is counter-intuitive, since a single structure is expected to be represented by a few modes. The intuition proves to be correct and we show that in a properly chosen reference frame along the characteristics defined by the group velocity, a POD or DMD reduces moving structures to a few modes, as expected. Beyond serving as a reduced model, the resulting entity can be used to define a constant or minimally changing structure in turbulent flows. This can be interpreted as an empirical counterpart to exact coherent structures. We present the method and its application to a head vortex of a compressible starting jet.