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Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

Published online by Cambridge University Press:  29 May 2018

Aaron Towne Open the ORCID record for Aaron Towne [Opens in a new window] ,
Oliver T. Schmidt Open the ORCID record for Oliver T. Schmidt [Opens in a new window]  and
Tim Colonius
Aaron Towne*
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
Oliver T. Schmidt
Affiliation:
California Institute of Technology, Pasadena, CA 91125, USA
Tim Colonius
Affiliation:
California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: atowne@stanford.edu
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Abstract

We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulent Flows

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Journal of Fluid Mechanics , Volume 847 , 25 July 2018 , pp. 821 - 867
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© 2018 Cambridge University Press 

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