Integration of slow-fast quasilinear models of turbulent shear flows

The quasilinear (QL) reduction, which retains fluctuation-fluctuation nonlinearities only where they feed back onto mean fields, is often employed as a model reduction strategy. This approximation can be justified in the limit of temporal scale separation between the mean and fluctuation dynamics as arises in the asymptotic description of strongly stratified shear turbulence. Here, we utilize carefully constructed model problems to derive two important extensions to our recently introduced formalism for integrating slow-fast QL systems, which exploits the tendency of these systems to self-organize about a marginal stability manifold and slaves the amplitude of the (marginal) fluctuations to the slowly-evolving mean field. The first extension accommodates large-amplitude bursting events, in which temporal scale separation is transiently lost until marginal stability is re-established. The second extension yields a slow equation for the wavenumber of the marginal mode. Together, these extensions enable scale-selective adaptivity in both space and time. Our formalism is consistent with the idea that shear flow turbulence tracks low-dimensional state-space structures during slow evolutionary phases punctuated by intermittent bursting events.