  A multiscale reduced description of turbulent free shear flows in the
presence of strong stabilizing density stratification is derived via asymptotic
analysis of the Boussinesq equations in the simultaneous limits of small Froude
and large Reynolds numbers. The analysis explicitly recognizes the occurrence
of dynamics on disparate spatiotemporal scales, yielding simplified partial
differential equations governing the coupled evolution of slow large-scale
hydrostatic flows and fast small-scale isotropic instabilities and internal
waves. The dynamics captured by the coupled reduced equations is illustrated in
the context of two-dimensional strongly stratified Kolmogorov flow. A
noteworthy feature of the reduced model is that the fluctuations are
constrained to satisfy quasilinear (QL) dynamics about the comparably
slowly-varying large-scale fields. Crucially, this QL reduction is not invoked
as an ad hoc closure approximation, but rather is derived in a physically
relevant and mathematically consistent distinguished limit. Further analysis of
the resulting slow-fast QL system shows how the amplitude of the fast
stratified-shear instabilities is slaved to the slowly-evolving mean fields to
ensure the marginal stability of the latter. Physically, this marginal
stability condition appears to be compatible with recent evidence of
self-organized criticality in both observations and simulations of stratified
turbulence. lgorithmically, the slaving of the fluctuation fields enables
numerical simulations to be time-evolved strictly on the slow time scale of the
hydrostatic flow. The reduced equations thus provide a solid mathematical
foundation for future studies of three-dimensional strongly stratified
turbulence in extreme parameter regimes of geophysical relevance and suggest
avenues for new sub-grid-scale parameterizations.
