Attractive fixed-point solution study of a shell model for turbulent convection

Different scaling behaviors, such as Kolmogorov （K41） scaling and Bolgiano and Obukhov （BO） scaling, have been reported in various shell models proposed for turbulent thermal convection. However, two coexistent subranges with K41 and BO scaling are not set up with Bolgiano scale interlaying between the largest scale and the dissipation scale. In this paper, we summarize fixed- point solution study of the Brandenburg model with small perturbation theory by introducing a small disturbance term as the impact of buoyancy. Three groups of fixed-point solutions with different locations of the so-called buoyancy scale, above/below which buoyancy is significant/insignifant. Both theoretical and numerical results show that a modified K41 scaling, instead of K41 and BO coexistent scaling, is set up even though buoyancy may be significant over the scaling range, which suggests that the buoyancy scale is not related exactly to the Bolgiano scale. Thus, a K41 and BO coexistent scaling behavior is not setup for the Brandenburg model.