A generalized Fick's law to describe non-local transport effects

Abstract

Fick's law is extensively used as a model for turbulent diffusion processes. It requires separation of scales between those of the process driving the diffusion mechanism and the scale of variation of the mean quantity being diffused. This makes the model unsuitable for the description of non-local transport processes like those occurring in some turbulent flows.A generalized Fick's law is proposed using a fractional derivative operator which accounts for non-local phenomena in virtue of its integral nature. This generalization is suggested as a model for two typical phenomena, like those observed in the convective boundary layer (CBL), which cannot be reduced to a local formulation: the inadequacy in the flux-gradient relationship when considering bottom-up dispersion (in particular, the counter-gradient transport) and the vertical drift of the location of the maximum concentration even in the absence of the mean velocity field.The solution of the generalized diffusion equation qualitatively reproduces the above described features, supporting the fractional derivative description of turbulent transport in complex flows. A quantitative approach requires extensive investigation in order to deal with the details of real cases.