Lévy-Kolmogorov scaling of turbulence

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 scales. 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 Lévy 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 Lévy-Kolmogorov scaling, whose scaling exponent ranges from -3 to -5/3 corresponding to different Lévy processes and reduces to the limiting Kolmogorov scaling -5/3 underlying a Gaussian process. The truncated Lévy 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.