Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

Abstract

The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatiotemporal Koopman operator, which has a traveling-wave interpretation. The relationship leads to a generalization of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known. DOI: 10.1103/PhysRevFluids.1.032402This paper presents a unifying view of a range of methods used to characterize nonlinear solutions of the Navier-Stokes equations. Ordered by roughly decreasing complexity in their treatment of nonlinearity, these are invariant solutions, a generalized form of Koopman mode decomposition, dynamic mode decomposition, and resolvent mode decomposition. The instances in which the four methods of analysis coincide are identified here. We hope to provide a more rigorous basis on which to interpret these methods and to inform the potential user of the appropriate tool for a particular problem.Invariant solutions are exact (nonlinear) solutions of the Navier-Stokes equations that remain the same after certain symmetry operations are applied (such as reflection, rotation, or shifts in space or time). They are sometimes called exact coherent structures and are implicated in the formation of apparently repeating highly ordered spatiotemporal patterns in turbulence. These solutions are the subject of an active and long-standing area of research; see, for example, [1-3] for reviews. From the viewpoint of dynamical systems, turbulence may be understood in terms of a state trajectory visiting the neighborhood of various invariant solutions.Koopman modes are a general way of analyzing the dynamics of a nonlinear system [4][5][6][7]. The Koopman modes arise from the spectral analysis of the Koopman operator, which is an infinitedimensional operator that evolves functions of the system's state. Koopman mode decomposition is closely related to dynamic mode decomposition (DMD) [7][8][9][10][11][12], which is a popular way to * a.sharma@soton.ac.uk

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“…Effectively, then, mode structure is obtained by approximating the (action of the) resolvent operator. Sharma et al (2016) and Towne et al (2018) have noted a correspondence between resolvent modes, DMD, and spectral Proper Orthogonal Decomposition (SPOD) modes. In particular, Towne et al (2018) noted that resolvent modes are equivalent to SPOD modes when the nonlinear forcing is uncorrelated in space and time.…”

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