_{n,0}\;\;\;,\;\;\; D_{n}=\nu k_{n}^{2}u_{n}$$ where $\nu$ is the kinematic viscosity. The nonlinear kernel $\vec N$ is quadratic in the $u_{n}$, with a coupling constant growing like $k_{n}$ in order to reproduce the hierarchy of characteristic times of the Navier-Stokes dynamics. It must also conserve the total kinetic energy $E=\displaystyle{1\over 2}\sum_{n=0}^{N} u_{n}^{2}$. If interactions between shells (which are always supposed to be local) do not extend beyond nearest neighbours, the most general expression for the $n^{th}$ component of $\vec N$ is $$N_{n}[\vec u] = \alpha Q^{2\over 3} [k_{n}u_{n-1}u_{n}-k_{n+1}u_{n+1}^2]
+\beta [k_{n}u_{n-1}^2-k_{n+1}u_{n}u_{n+1}]$$ (this formula remains valid on the two boundaries $n=0$ and $n=N$, provided $u_{-1}=u_{N+1}=0$ is assumed).\
The model appears like the linear superposition of the Obukhov-Gledzer (OG) and Novikov-Desniansky (ND) chains, with respective weights $\alpha Q^{2/3}$ and $\beta$. We shall assume $\alpha,\beta >0$ and, without loss of generality, $\alpha +\beta =1$. Since, on the average, the $u_{n}$ decrease like $k_{n}^{-1/3}$ according to Kolmogorov-scaling, it is convenient to introduce a new set of variables $\phi_{n}$ by the relation $$u_{n}=Q^{-{n\over 3}} \phi_{n}
\label{defphi}$$ The equations for $\vec \phi$ read $$\frac{d}{dt}\vec{\phi} = k_{0} Q^{2\over 3} \vec{N}[\vec{\phi}]+\vec{F}-\vec{D}$$ where the expression of the $n^{th}$ component of $\vec N$ is now $$N_{n}[\vec{\phi}]=Q^{2n\over 3}[(\alpha \phi_{n-1}\phi_{n}+\beta \phi_{n-1}^2)
 - (\alpha \phi_{n+1}^2+\beta \phi_{n}\phi_{n+1})]
