Sparse space-time resolvent analysis

Resolvent analysis has emerged as a common tool for identifying linear amplification mechanisms in a broad range of fluid flows. However, it is in general suited only for the analysis of harmonic forcing and response structures in statistically-stationary systems. Here, we develop a generalized space-time extension of resolvent analysis that can be applied to time-varying systems. In addition, we combine this method with sparsity-promoting methods to identify resolvent modes that are localized in both space and time. Sparsity is achieved by modifying the standard optimization problem associated with the singular value decomposition to include an $L_1$-norm term. This results in a nonlinear eigenproblem, which can be solved using a generalized inverse power method. We first demonstrate this space space-time resolvent analysis on statistically-stationary turbulent channel flow, where it is shown that sparse modes can be identified in both space and time. Next, the method is used to identify time-localized amplification mechanisms in an oscillating turbulent Stokes boundary layer. We will discuss the application of this method to other time-evolving systems, and also comment on how the proposed method relates to harmonic resolvent and transient growth analysis.