Helical shell models for three dimensional turbulence

In this paper we study a new class of shell models, defined in terms of two complex dynamical variables per shell, transporting positive and negative helicity respectively. The dynamical equations are derived from a decomposition into helical modes of the velocity Fourier components of Navier-Stokes equations (F. Waleffe, Phys. Fluids A {\bf 4}, 350 (1992)). This decomposition leads to four different types of shell models, according to the possible non-equivalent combinations of helicities of the three interacting modes in each triad. Free parameters are fixed by imposing the conservation of energy and of a ``generalized helicity'' $H_\alpha$ in the inviscid and unforced limit. For $\alpha=1$ this non-positive invariant looks exactly like helicity in the Fourier-helical decomposition of the Navier-Stokes equations. Long numerical integrations are performed, allowing the computation of the scaling exponents of the velocity increments and energy flux moments. The dependence of the models on the generalized helicity parameter $\alpha$ and on the scale parameter $\lambda$ is also studied. PDEs are finally derived in the limit when the ratio between shells goes to one.