A sparsity-promoting resolvent analysis for the identification of spatiotemporally-localized amplification mechanisms

Abstract: This work introduces a variant of resolvent analysis that identifies forcing and response modes that are sparse in both space and time. This is achieved through the use of a sparse principal component analysis (PCA) algorithm, which formulates the associated optimization problem as a nonlinear eigenproblem that can be solved with an inverse power method. We apply this method to parallel shear flows, both in the case where we assume Fourier modes in time (as in standard resolvent analysis) and obtain spatial localization, and where we allow for temporally-sparse modes through the use of a linearized Navier-Stokes operator discretized in both space and time. Appropriate choice of desired mode sparsity allows for the identification of structures corresponding to high amplification that are localized in both space and time. We report on the similarities and differences between these structures and those from standard methods of analysis. After validating this space-time resolvent analysis on statistically-stationary channel flow, we next implement the methodology on a time-periodic Stokes boundary layer, demonstrating the applicability of the approach to non-statistically-stationary systems.