We first recall the fundamental importance for atmospheric predictability of the large number of (spatial) degrees of freedom, as emphasised by Lorenz (1969), who studied the spectral space-time decorrelation with a quasi-normal model of turbulence. Results obtained with somewhat more refined quasi-normal models (Leith, 1971; Leith and Kraichnan, 1972; Metais and Lesieur, 1986) converged to those obtained with straightforward phenomenological arguments (Schertzer and Lovejoy, 2004a). The latter also stressed the drastic role of intermittency that yields strongly non gaussian statistics with much more frequent extremes, i.e. the loss of predictability is highly inhomogeneous and occurs by puffs (Schertzer and Lovejoy, 2004b). To go beyond multifractal phenomenology, we study the intermittency of the predictability of the Scaling Gyroscope Cascade model (Chigirinskaya and Schertzer, 1997, Chigirinskaya et al, 1997, 1998) that has several attractive properties: this dynamical model is indeed deduced from the Navier-Stokes equations by preserving only a certain type of direct interactions, the resulting indirect interactions being built dynamically along the tree-structure of the cascade. It therefore preserves most properties of the original equations, although being much more tractable. In particular, it gives the possibility to use the Multiplicative Ergodic Theorem to obtain rigorous results on the intermittency of turbulence.