An EDQNM model to explore the behavior of stably stratified flows in the high Prandtl number regime

Recent direct numerical simulations (DNS) and theoretical results indicate that the Prandtl number (Pr) can have profound and surprising effects on the behavior of stably stratified turbulence, which is particularly important for oceanic flows where Pr=700. Explorations using DNS of flows with high Pr as well as high buoyancy Reynolds numbers (Re_b, and also often low Froude numbers, Fr) are not possible for the foreseeable future due to the extreme computational cost. As a result, crucial open questions remain regarding how stratified turbulence behaves when Pr is large. To obtain insights into such regimes we therefore use a theoretical model based on a spherically averaged Eddy-Damped Quasi-Normal Markovian (EDQNM) closure that is applied to the kinetic and potential energy spectrum equations for a Boussinesq system. This spectral model enables exploration of the impact of Pr, Re_b, Fr on all scales in the flow and for parameter regimes far beyond the reach of DNS.

At fixed Re_b and for Pr=1, the model reproduces the asymptotic Fr scaling predicted by recent analytical studies and observed using DNS. At fixed Re_b and Fr, the model captures the effect observed in DNS that increasing Pr leads to a decrease of the mixing efficiency, as well as a reversal of the buoyancy flux at high wavenumbers which increases in magnitude with increasing Pr. We then use the model to explore the behavior of stratified turbulence in the high Pr regime and consider how the behavior in this limit depends on both Re_b and Fr. We also test a recent prediction that even when Fr>>1, the density field at the smallest scales of stratified turbulence remains an active, rather than passive scalar, contrary to the expectation based on the standard theory of stratified turbulence.