# A new parameterization for the concentration flux using the fractional calculus to model the dispersion of contaminants in the Planetary Boundary Layer

## TL;DR

This paper introduces a fractional calculus-based parameterization for modeling contaminant dispersion in the Planetary Boundary Layer, demonstrating superior fit to experimental data over traditional models.

## Contribution

It develops a fractional differential equation model for concentration flux that outperforms classical models and suggests a link between fractional order and turbulence structure.

## Key findings

- Fractional model fits experimental data better than Gaussian models.
- Constant parameter fractional model outperforms some variable-parameter models.
- The fractional order relates to the physical turbulence structure.

## Abstract

In the present work, we propose a new parameterization for the concentration flux using fractional derivatives. The fractional order differential equation in the longitudinal and vertical directions is used to obtain the concentration distribution of contaminants in the Planetary Boundary Layer. We solve this model and we compare the solution against both real experiments and traditional integer order derivative models. We show that our fractional model gives very good results in fitting the experimental data, and perform far better than the traditional Gaussian model. In fact, the fractional model, with constant wind speed and a constant eddy diffusivity, performs even better than some models found in the literature where it is considered that the wind speed and eddy diffusivity are functions of the position. The results obtained show that the structure of the fractional order differential equation is more appropriate to calculate the distribution of dispersed contaminants in a turbulent flow than an integer-order differential equation. Furthermore, a very important result we found it is that there should be a relation between the order $\alpha$ of the fractional derivative with the physical structure of the turbulent flow.