Symmetries, invariants and cascades in a shell model of turbulence

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Reduced wavenumber models of turbulence, shell models, show cascade processes and anomalous scaling of correlators which might be analogous to what is observed in Navier-Stokes (N-S) turbulence. The scaling properties of the shell models depends on the specific symmetries and invariants of the models. A new shell model is investigated. It is argued that this model might have a closer resemblance than the standard GOY model to the N-S turbulence. The new shell model coincides with the Sabra model proposed by L'vov et al. lvov, for a specific choice of the free parameters of their model. The model has for this choice of parameters, besides the energy and the 'helicity', a cubic inviscid invariant.