We have previously shown that the Lévy fractional Brownian field family accounts for a complete statistical and analytical description of non-Kolmogorov wavefront phase [Opt. Lett. 33(6), 572 (in press, 2008)]. This is a non-stationary process having zero mean and stationary increments; then, replicating the well-known properties of the turbulent phase. Opposite to traditional models relying in the stationary (spectral) approximation of the phase, that ultimately leads to non-physical divergences. Our model avoids these pitfalls and gives exact analytical results to many observable quantities: Strehl ratio, angle-of-arrival variance, seeing and Zernike coefficients, and also, a generalized DIMM theory. Nevertheless, some coefficients are slightly below (~ 5-10%) when compared to other estimates in the occurrence of Kolmogorov turbulence. In the present work we show that this is due to the mono-fractal nature of this model; that is, the absence of inner- and outer-scales. To address this issue we introduce a gaussian stochastic process whose realizations are multi-fractals: the multi-scale Lévy fractional Brownian field.