Following marginal stability manifolds in quasilinear dynamical reductions of multiscale flows in two space dimensions

Abstract

A two-dimensional extension of a recently developed formalism for slow-fast quasilinear (QL) systems subject to fast instabilities is derived. Prior work has demonstrated that the emergent dynamics of these systems is characterized by a slow evolution of mean fields coupled to marginally stable, fast fluctuation fields. By exploiting this emergent behavior, an efficient fast-eigenvalue/slow-initial-value solution algorithm can be developed in which the amplitude of the fast fluctuations is slaved to the slowly evolving mean fields to ensure marginal stability (and temporal scale separation) is maintained. For 2D systems that are spatially-extended in one direction, the fluctuation eigenfunctions are labeled by their wavenumbers characterizing spatial variability in that direction, and the marginal mode(s) also must coincide with the fastest-growing mode(s) over all admissible wavenumbers. Here, we introduce two equivalent procedures for deriving an ordinary differential equation governing the slow evolution of the wavenumber of the fastest-growing fluctuation mode that simultaneously must be slaved to the mean dynamics to ensure the mode has zero growth rate. We illustrate the procedure in the context of a 2D model partial differential equation that shares certain attributes with the equations governing strongly stratified shear flows. The slaved evolution follows one or more marginal stability manifolds, which constitute select state-space structures that are not invariant under the full flow dynamics yet capture quasi-coherent states in physical space in a manner analogous to invariant solutions identified in, e.g., transitionally-turbulent shear flows. Accordingly, we propose that marginal stability manifolds are central organizing structures in a dynamical systems description of certain classes of multiscale flows where scale separation justifies a QL approximation of the dynamics.