Energy spectra in a model for convective turbulence

The energy cascade in both hydrodynamic and hydromagnetic Boussinesq convection is investigated at large Rayleigh numbers, using a scalar model for turbulence. Depending on the relative importance of direct and inverse transfer, either we observe classical Kolmogorov ${\mathit{k}}^{\mathrm{-}5/3}$ spectra or, if there is a strong inverse transfer of kinetic energy, we find a ${\mathit{k}}^{\mathrm{-}7/5}$ spectrum for the temperature fluctuation and a ${\mathit{k}}^{\mathrm{-}11/5}$ spectrum for the kinetic energy (Bolgiano-Obukhov scaling). We derive dissipative cutoff wave numbers that are consistent with these spectra.