Cascades and statistical equilibrium in shell models of turbulence

We study the Gledzer-Okhitani-Yamada shell model simulating the cascade processes of turbulent flow. The model has two inviscid invariants governing the dynamical behavior. Depending on the choice of interaction coefficients, or coupling parameters, the two invariants are either both positive definite, analogous to energy and enstrophy of two-dimensional (2D) flow, or only one is positive definite and the other not, analogous to energy and helicity of 3D flow. In the 2D-like model the dynamics depend on the spectral ratio of enstrophy to energy. That ratio depends on wave number as $k^{\alpha }$. The enstrophy transfer through the inertial subrange can be described as a forward cascade for $\alpha <2\mathrm{}$ and diffusion in a statistical equilibrium for $\alpha >2\mathrm{}$. The $\alpha =2\mathrm{}$ case, corresponding to 2D turbulence, is a borderline between the two descriptions. The difference can be understood in terms of the ratio of typical time scales in the inertial subrange and in the viscous subrange. The multifractality of the enstrophy dissipation also depends on the parameter $\alpha$ and seems to be related to the ratio of typical time scales of the different shell velocities.