A numerical approach to the fractional Laplacian operator with applications to quasi-geostrophic flows

Abstract

The quasi-geostrophic models have been very successful for the study of oceanic and atmospheric dynamics in the mid-to-high latitude region of the earth where the Coriolis effect is significant. The governing equation of a quasi-geostrophic model is a transport equation with a fractional dissipation term. Although fractional operators have become a topic of great interest in the research community, the numerical discretization of such operators is very challenging due to their non-local behavior. In this work, we propose the numerical approximations of the bounded fractional Laplacian on a finite element discretization. In particular, we rely on the Riesz method and use a semi-analytical technique to approximate all the integrals involving the interaction between the inner local and the outer region. We test the implemented algorithm with numerical benchmarks, and we apply it to quasi-geostrophic flows. All the presented simulations are computationally expensive, due to the non-local behavior of the fractional Laplacian. For this reason, a parallel implementation of the numerical code has been developed.