Energy spectrum of buoyancy-driven turbulence

Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux Πᵤ, we demonstrate that, for stably stratified flows, the kinetic energy spectrum Eᵤ(k)~k^-11/5, the potential energy spectrum E_θ(k)~k^-7/5, and Πᵤ(k)~k^-4/5 are consistent with the Bolgiano-Obukhov scaling. This scaling arises due to the conversion of kinetic energy to the potential energy by buoyancy. For weaker buoyancy, this conversion is weak, hence Eᵤ(k) follows Kolmogorov's spectrum with a constant energy flux. For Rayleigh-B\'enard convection, we show that the energy supply rate by buoyancy is positive, which leads to an increasing Πᵤ(k) with k, thus ruling out Bolgiano-Obukhov scaling for the convective turbulence. Our numerical results show that convective turbulence for unit Prandt number exhibits a constant Πᵤ(k) and Eᵤ(k)~k^-5/3 for a narrow band of wave numbers.