amplitude appears invariably at the end of the cascade, which carries back energy towards large scales. In the final state, energy oscillates back and forth between the first two shells, the other ones being inactive.\
Things are different, as soon as a finite amount of Novikov interactions is introduced. This is because the term $\phi_{n-1}^{2}$ in equation (\[nlphi\_n\]) favours energy transfer to small scales. At a static level, assuming a rapid decay of the spectrum on the ultraviolet side (i.e. $\phi_{n-1}>>
\phi_{n}>>\phi_{n+1}$), one has now to achieve the balance $$\beta \phi_{n-1}^{2} \sim {Q^{4n\over 3}\over R}\phi_{n}$$ This yields the following general solution $\phi_{n}=\displaystyle{{Q^{4(n+2)\over 3}\over \beta R}}
\mbox{exp}(-b\,2^{n})$ with $b>0$ otherwise arbitrary. More physically, we may rewrite $\phi_{n}$ in the dissipative range as $$\phi_{n}={Q^{4(n-n_{d}+2)\over 3}\over \beta}\:\mbox{exp}(-a\,2^{(n-n_{d})})
 \;\;\;\;\;\; \mbox{for} \;n>n_{d}
\label{pfdiss_n}$$ with $n_{d}$ given by (\[defn\_d\]). These considerations suggest the following procedure for computing the fixed point $\phi_{n}^{e}$ in the presence of dissipation. The condition of equilibrium for the $n^{th}$ shell being quadratic in $\phi_{n-1}$, it can be used to express $\phi_{n-1}$ in terms of $\phi_{n}$ and $\phi_{n+1}$. Going from the last shell with $\phi_{N}$ parameterized as in (\[pfdiss\_n\]) to the first one, and keeping at each step the positive root, one ends 