  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.
