A deformed derivative model for turbulent diffusion of contaminants in the atmosphere

TL;DR

This paper introduces a deformed derivative-based advection-diffusion model for atmospheric contaminant diffusion, demonstrating improved data fitting over classical models and comparable performance to Caputo fractional derivatives, with easier solvability.

Contribution

The study proposes a novel model using Hausdorff deformed derivatives for atmospheric diffusion, showing its effectiveness and computational advantages over traditional and other fractional derivative models.

Findings

Hausdorff derivative model fits experimental data better than classical models.

Hausdorff and Caputo models show similar performance in experiments.

Deformed derivative models are easier to solve and effectively describe atmospheric diffusion.

Abstract

In the present work, we propose an advection-diffusion equation with Hausdorff deformed derivatives to stud the turbulent diffusion of contaminants in the atmosphere. We compare the performance of our model to fit experimental data against models with classical and Caputo fractional derivatives. We found that the Hausdorff equation gives better results than the tradition advection-diffusion equation when fitting experimental data. Most importantly, we show that our model and the Caputo fractional derivative model display a very similar performance for all experiments. This last result indicates that regardless of the kind of non-classical derivative we use, an advection-diffusion equation with non-classical derivative displaying power-law mean square displacement is more adequate to describe the diffusion of contaminants in the atmosphere than a model with classical derivatives. Furthermore, since Hausdorff derivatives can be related to several deformed operators, and since differential equations with the Hausdorff derivatives are easier to solve than equations with Caputo and other non-local fractional derivatives, our result highlights the potential of deformed derivative models to describe the diffusion of contaminants in the atmosphere.