Dual effects of buoyancy and enstrophy transfer on scaling behavior of a shell model proposed for homogeneous turbulent convection

In this paper, we explain why the Bolgiano-Obukhov (BO) scaling behavior is unavailable by the SabraT model proposed for turbulent thermal convection in the range of 1 < δ < 2, which is extended from the Sabra model by coupling temperature with velocity in the equations of motion as an external forcing, i.e., buoyancy. Numerical studies show that SabraT model is mainly governed by the enstrophy budget equation, at which the buoyancy is not always relevant to the statistical properties and the effect of buoyancy is dependent on the parameter γ that measures the ratio of enstrophy to energy. When buoyancy is important, BO scaling is expected using theoretical arguments, such as dimensional analysis. Instead of BO scaling, a new γ-dependent scaling behavior is setup in the buoyancy relevant regime, which is found to equivalently deviate from the enstrophy cascade scaling and BO scaling. This deviation is mainly discussed by two dimensionless parameters, which respectively measure the deviation of the energy/enstrophy transfer flux rate and the injected energy/enstrophy due to buoyancy from dimensional analysis. The introduced buoyancy plays as a relative small perturbed forcing on the Sabra model without changing much its intrinsical statistical properties, i.e., dimensional analysis is not always validated in both Sabra and SabraT models.