On the shape of resolvent modes in shear-driven turbulence

The resolvent formulation of the Navier-Stokes equations gives a means for prediction of turbulent structures and statistics using the singular value decomposition of the resolvent operator based on the appropriate turbulent mean, following the framework developed by McKeon & Sharma (2010). This talk will discuss analytic approximations to the shape of resolvent modes in shear-driven turbulent flows. Such systems typically exhibit large spectral gaps in the singular values of their associated family of resolvent operators, which can make resolvent-based decompositions particularly efficient for reduced-complexity modeling. Here, we use results concerning the pseudospectra of scalar operators (e.g., Reddy et al. 1993, Trefethen 2005) to derive analytic approximations to components of leading pseudospectral (resolvent) modes. This provides a theoretical framework for understanding the origin of observed structures, and gives a method for mode estimation without the need for large numerical computations. We will discuss the implications of these findings for real-time estimation and control, and will further demonstrate the utility of this approach for modeling passive scalar dynamics.