Helical shell models for three-dimensional turbulence

In this paper a class of shell models is studied, defined in terms of the interactions of two complex dynamical variables per shell, transporting positive and negative helicity, respectively. Following a decomposition into helical modes of the velocity Fourier components of Navier-Stokes equations [F. Waleffe, Phys. Fluids A 4, 350 (1992)], classification of the helical interactions of the three modes in each triad leads to four different types of shell models. 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 nonpositive 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. Partial differential equations are finally derived in the limit when the ratio between shells goes to one.