The interference of logarithmic oscillations characteristic to statistical fractal and ultrametric structures is analyzed. We introduce an ultrametric model consisting in a hierarchy of branches for which the probability distribution xi(n) of the total number, n, of branches has a long tail of the inverse power law type xi(n)- A(ln n)n(-(1 + h)) as n --> infinity where H is a fractal exponent characteristic for the ultrametric structure and A(ln n) is a periodic function of ln n. A comparison is made with a statistical fractal derived by means of the Shlesinger-Hughes stochastic renormalization approach. We consider a positive random variable, X, selected from a narrow unimodal probability density with finite moments. A process of stochastic amplification of the random variable is introduced which leads to a probability density of the amplified variable with a long tail: ($) over bar P(X) dX similar to dX X(-(1 + H)) B[ln (X/X(m))] as X --> infinity where H is another fractal exponent, X(m) is a cutoff value of the random variable X and B[ln (X/X(m))] is a periodic function of ln (X/X(m)). Finally, the interaction between these two types of logarithmic oscillations is analyzed by assuming that the stochastic amplification of the random variable X takes place on an ultrametric structure. The final probability density of X, P*(X) dX, displays a phenomenon of amplitude modulation P*(X) dX similar to d[ln (X/X(m))] [ln (X/X(m))](-(1 + H)) Xi as X --> infinity where Xi = Xi((1)) + Xi((2)) is made up of two additive contributions: a periodic function Xi((1))[ln (X/X(m))] of ln (X/X(m)) and a superposition Xi((2)) {ln (X/X(m)) ln [ln (X/X(m))]} of periodic functions of ln (X/X(m)) modulated by much slower periodic functions in in [ln (X/X(m))]. The model is applied for the analysis of recycle flows in porous media. We assume that a hierarchical structure of pores exists which corresponds to an ultrametric space and that the recycle flow leads to a stochastic amplification of the residence time of fluid elements in the system. We show that the probability density of the residence time has an asymptotic behavior similar to that of P*(X) dX and investigate the possibilities of measurement of multiple logarithmic oscillations by means of tracer experiments. A physical interpretation of multiple logarithmic oscillations is given in the case of how systems: they are generated by two delayed feedback processes occurring in two different logarithmic time scales, ln t and ln ln t. The fast delayed feedback process in ln t is given by the recycle flows corresponding to a given level of the porous structure whereas the slow delayed feedback process in ln ln t is due to the exchange of fluid among the different levels of the hierarchical structure of pores.
