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 Fr_h≫1 (Fr_h 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(Fr_h^-2). Direct numerical simulations confirmed this result, and also indicated that for the strongly stratified regime Fr_h≪ 1, Γ∼ O(1). Furthermore, the study argued that Γ does not depend on the buoyancy Reynolds number Re_b, but only on Fr_h. We present an asymptotic analysis to predict theoretically how Γ should behave for Fr_h≪1 and Fr_h≫1 in the limit Re_b→∞. To correctly handle the singular limit Re_b→∞ 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 Re_b→∞. The analysis yields Γ∼ O(Fr_h^-2(1+Fr_h^-2)) for Fr_h≫1 and Γ∼ O(1+Fr_h^2) for Fr_h≪ 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 L_O does not describe the scale below which buoyancy forces are sub-leading, which is instead given by O(Fr_h^1/2 L_O), and that the condition for there to be an inertial sub-range when Fr_h≪ 1 is not Re_b≫1, but the more restrictive condition Re_b≫ Fr_h^-4/3.