Bolgiano-Obukhov spectrum and mixing efficiency in stably stratified turbulence

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 Eᵤ(k) and the potential energy spectrum Eb(k) show dual scaling---Bolgiano-Obukhov scaling [Eᵤ(k)~k^-11/5 and Eb(k)~k^-7/5] for k<kB, where kB is the Bolgiano wave number, and Kolmogorov scaling (~k^-5/3) for k>kB. In addition, we find that the mixing efficiency ηₘᵢₓ varies as ηₘᵢₓ~Ri for weak stratification, whereas ηₘᵢₓ~Ri1/3 for moderate stratification, where Ri is the Richardson number.