Stability conditions for gradient-transport models of turbulent density-stratified shear flow

The linear stability of certain two-dimensional disturbances in turbulent density-stratified shear flow is examined. The vertical, turbulent transports of momentum and buoyancy are modelled as gradient transports, and the interaction between velocity and momentum fields through the gradient Richardson number is included. It is argued that solutions should be stable, since the ensemble-averaged equations may no longer exhibit the small-scale instabilities related to turbulence. The stability conditions obtained constitute a generalization of those obtained by Kranenburg (1980a, b) for one-dimensional disturbances, and set certain bounds to the dependence of eddy viscosity and diffusivity on the Richardson number.