In this paper, using a shell model, we simulate highly turbulent stably stratified flow for weak to moderate stratification at unitary Prandtl number. We investigate the energy spectra and fluxes of velocity and density fields. We observe that for moderate stratification, in the inertial range, the kinetic energy spectrum $E_u\left(k\right)$ and the potential energy spectrum $E_b\left(k\right)$ show dual scaling—Bolgiano-Obukhov scaling $[E_u\left(k\right)\sim k^{-11/5}$ and $E_b\left(k\right)\sim k^{-7/5}]$ for $k<k_B$, where $k_B$ is the Bolgiano wave number, and Kolmogorov scaling $(\sim k^{-5/3})$ for $k>k_B$. In addition, we find that the mixing efficiency ${\eta }_{\mathrm{mix}}$ varies as ${\eta }_{\mathrm{mix}}\sim \mathrm{Ri}$ for weak stratification, whereas ${\eta }_{\mathrm{mix}}\sim {\mathrm{Ri}}^{1/3}$ for moderate stratification, where $\mathrm{Ri}$ is the Richardson number.

• Received 29 November 2022
• Accepted 24 April 2023

DOI:https://doi.org/10.1103/PhysRevE.107.055106