Asymptotic analysis of mixing in stratified turbulent flows, and the conditions for an inertial sub-range

In an important study, Maffioli et al. (J. Fluid Mech., Vol. 794 , 2016) used a scaling analysis to predict that in the weakly stratified flow regime Frₕ1 (Frₕ is the horizontal Froude number), the mixing coefficient Γ (defined as the ratio of the dissipation rates of potential to kinetic energy) scales as Γ~ O(Frₕ⁻²). Direct numerical simulations confirmed this result, and also indicated that for the strongly stratified regime Frₕ 1, Γ~ O(1). Furthermore, the study argued that Γ does not depend on the buoyancy Reynolds number Reb, but only on Frₕ. We present an asymptotic analysis to predict theoretically how Γ should behave for Frₕ1 and Frₕ1 in the limit Reb→∞. To correctly handle the singular limit Reb→∞ we perform the asymptotic analysis on the filtered Boussinesq-Navier-Stokes equations, and demonstrate the precise sense in which the inviscid scaling analysis of Billant & Chomaz (Phys. Fluids, vol. 13, 1645-1651, 2001) applies to viscous flows with Reb→∞. The analysis yields Γ~ O(Frₕ⁻²(1+Frₕ⁻²)) for Frₕ1 and Γ~ O(1+Frₕ²) for Frₕ 1, providing a theoretical basis for the numerical observation made by Maffioli et al, as well as predicting the sub-leading behavior. Our analysis also shows that the Ozmidov scale LO does not describe the scale below which buoyancy forces are sub-leading, which is instead given by O(Frₕ1/2 LO), and that the condition for there to be an inertial sub-range when Frₕ 1 is not Reb1, but the more restrictive condition Reb Frₕ-4/3.