On various levels of deterministic toy models for the Richardson cascade in turbulence

The Desnyanski-Novikov shell model is a deterministic dynamical model for scalar velocities $v_t(n)$ defined on the one-dimensional-lattice $n=0,1,2,\mathrm{.}\mathrm{.}$ labelling the length-scales $l_n=l_0{2}^{-n}$, in order to describe the cascade of energy from the biggest scale where it is injected by some external forcing towards the smaller scales where it is dissipated by viscosity. We describe the generalization of this model in two directions : (i) the one-dimensional-lattice $n=0,1,2,\mathrm{.}\mathrm{.}$ labelling the length-scales $l_n=l_0 2^{-n}$ is replaced by a scale-spatial tree structure of nested cells in order to allow spatial heterogeneities between different coherent structures that are localized in different regions of the whole volume ; (ii) the scalar velocities $v_t(n)$ are replaced by 3D-vector velocities in order to take into account the vorticity in the dynamical equations and to include vortex-stretching effects.