The Kolmogorov scaling law of turbulences has been considered the most
important theoretical breakthrough in the last century. It is an essential
approach to analyze turbulence data present in meteorological, physical,
chemical, biological and mechanical phenomena. One of its very fundamental
assumptions is that turbulence is a stochastic Gaussian process in small
scales5. However, experiment data at finite Reynolds numbers have observed a
clear departure from the Gaussian. In this study, by replacing the standard
Laplacian representation of dissipation in the Navier-Stokes (NS) equation with
the fractional Laplacian, we obtain the fractional NS equation underlying the
Levy stable distribution which exhibits a non-Gaussian heavy trail and
fractional frequency power law dissipation. The dimensional analysis of this
equation turns out a new scaling of turbulences, called the Levy-Kolmogorov
scaling, whose scaling exponent ranges from -3 to -5/3 corresponding to
different Levy processes and reduces to the limiting Kolmogorov scaling -5/3
underlying a Gaussian process. The truncated Levy process and multi-scaling due
to the boundary effect is also discussed. Finally, we further extend our model
to reflecting the history-dependent fractional Brownian motion.