Metastability for the dissipative quasi-geostrophic equation and the non-local enhancement

TL;DR

This paper investigates metastability in the 2-D dissipative quasi-geostrophic equation, revealing a non-local enhancement phenomenon that amplifies diffusion effects through shear-diffusion mechanisms, with implications for understanding dissipative dynamics.

Contribution

It introduces the concept of non-local enhancement in the dissipative quasi-geostrophic equation and provides rigorous analysis of enhanced dissipation rates.

Findings

Proved linear enhanced dissipation for small viscosity.

Discovered non-local re-enhancement of diffusion.

Quantified dissipation rates in the linearized setting.

Abstract

In this paper, we study the metastability for the 2-D linearized dissipative quasi-geostrophic equation with small viscosity $\nu$ around the quasi steady state $\theta_{sin}=e^{-\nu t}\sin y$. We proved the linear enhanced dissipation and obtained the dissipation rate. Moreover, the new non-local enhancement phenomenon was discovered and discussed. Precisely we showed that the non-local term re-enhances the enhanced diffusion effect by the shear-diffusion mechanism.