Symmetry breaking, anomalous scaling, and large-scale flow generation in a convection cell

We consider a convection process in thin loops of different geometries. At Ra=Ra(')(cr) a first transition leading to the generation of corner vortices is observed. At higher Ra (Ra>Ra(cr)) a coherent large-scale flow, which persists for a very long time, sets up. The mean velocity nu mass flux m, and the Nusselt number Nu in this flow scale with Ra as nu proportional to m proportional to Ra0.45 and Nu proportional to Ra0.9, respectively, in a wide range of r=(Ra-Ra(cr))/Ra(cr) variation. The "normal" scaling nu proportional to sqrt[Ra] is detected as r-->0 and its range shrinks with decrease of the aspect ratio. The time evolution of the coherent flow is well described by the Landau amplitude equation with the appropriate selection of the Ra-dependent Landau constants. Analysis of the aspect ratio influence on the range of validity of anomalous scaling, observed in this paper, indicates the important role played by both thermal boundary conditions and geometry of the system.