Part-I: A Fractional Transport Model for Scalar Turbulence

The stochastic dynamics of coherent motions in turbulent flows involves with merging and filamentation of coherent vortices, which can induce a variant of non-Markovian random processes in the underlying scalar transport of particles with distinguished features of trappings (memory effects) and long jumps (nonlocal effects). Such anomalies leads to intermittent signals and non-Gaussian statistics, which in turn requires a proper (fractional) mathematical framework for better understanding of such scalar transports. One approach to model heavy-tailed behavior of turbulent mixing of a passive scalar is to derive the fractional scalar transport equation from the Kinetic theory, following the derivation of the fractional Navier-Stokes equations. We propose and derive the scalar transport equation with fractional Laplacian in the context of an isotropic turbulent flow, incorporating Lévy distribution and explore the subsequent impressions of the corresponding fractional exponent on the dynamics of coherent vortices via numerical simulation.