Revisiting Bolgiano–Obukhov scaling for moderately stably stratified turbulence

Abstract

According to the celebrated Bolgiano–Obukhov (Bolgiano, J. Geophys. Res., vol. 64 (12), 1959, pp. 2226–2229; Obukhov, Dokl. Akad. Nauk SSSR, vol. 125, 1959, p. 1246) phenomenology for moderately stably stratified turbulence, the energy spectrum in the inertial range shows a dual scaling: the kinetic energy follows (i) ${\sim}k^{-11/5}$ for $k<k_{B}$, and (ii) ${\sim}k^{-5/3}$ for $k>k_{B}$, where $k_{B}$ is the Bolgiano wavenumber. The $k^{-5/3}$ scaling, akin to passive scalar turbulence, is a direct consequence of the assumption that buoyancy is insignificant for $k>k_{B}$. We revisit this assumption, and using the constancy of kinetic and potential energy fluxes and simple theoretical analysis, we find that the $k^{-5/3}$ spectrum is absent. This is because the velocity field at small scales is too weak to establish a constant kinetic energy flux as in passive scalar turbulence. A quantitative condition for the existence of the second regime is also derived in the paper.