Analysis of Fluid Flows via Spectral Properties of the Koopman Operator

This article reviews theory and applications of Koopman modes in fluid mechanics. Koopman mode decomposition is based on the surprising fact, discovered in Mezić (2005), that normal modes of linear oscillations have their natural analogs—Koopman modes—in the context of nonlinear dynamics. To pursue this analogy, one must change the representation of the system from the state-space representation to the dynamics governed by the linear Koopman operator on an infinite-dimensional space of observables. Whereas Koopman in his original paper dealt only with measure-preserving transformations, the discussion here is predominantly on dissipative systems arising from Navier-Stokes evolution. The analysis is based on spectral properties of the Koopman operator. Aspects of point and continuous parts of the spectrum are discussed. The point spectrum corresponds to isolated frequencies of oscillation present in the fluid flow, and also to growth rates of stable and unstable modes. The continuous part of the spectrum corresponds to chaotic motion on the attractor. A method of computation of the spectrum and the associated Koopman modes is discussed in terms of generalized Laplace analysis. When applied to a generic observable, this method uncovers the full point spectrum. A computational alternative is given by Arnoldi-type methods, leading to so-called dynamic mode decomposition, and I discuss the connection and differences between these two methods. A number of applications are reviewed in which decompositions of this type have been pursued. Koopman mode theory unifies and provides a rigorous background for a number of different concepts that have been advanced in fluid mechanics, including global mode analysis, triple decomposition, and dynamic mode decomposition.