Exploiting self-organized criticality in strongly stratified turbulence
Résumé
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. Algorithmically, 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 parametrizations.