Symmetries, invariants, and cascades in a shell model of turbulence

Reduced wave number models of turbulence, namely shell models, show cascade processes and anomalous scaling of correlators which might be analogous to what is observed in Navier-Stokes (NS) turbulence. The scaling properties of the shell models depend on the specific symmetries and invariants of the models. A shell model is investigated. It is argued that this model might have a closer resemblance than the standard Gledzer-Ohkitani-Yamada model to the NS turbulence. The shell model investigated here coincides with the Sabra model proposed by L'vov et al. [Phys. Rev. E. 58, 1811 (1998)] for a specific choice of the free parameters of their model. For this choice of parameters, besides the energy and the "helicity," the model has a cubic inviscid invariant.