Long time dynamics and anomalous dissipation of energy in viscous forced active scalar equations

TL;DR

This paper investigates the long-term behavior of viscous active scalar equations with fractional diffusion, demonstrating the absence of anomalous energy dissipation and establishing the existence of a unique global attractor.

Contribution

It introduces a unified framework for analyzing long-time dynamics of active scalar equations with fractional Laplacian and damping, proving key properties like energy dissipation absence and attractor existence.

Findings

No anomalous energy dissipation in long-time averages

Existence of a unique global attractor

Application to geophysical fluid dynamics models

Abstract

We study an abstract family of advection-diffusion equations within the framework of the fractional Laplacian. The system involves two independent diffusion parameters: one introduced via a damping operator acting on the scalar unknown and the other as the coefficient of the fractional Laplacian. We establish existence and convergence results in specific parameter regimes and limits. In particular, we demonstrate the absence of anomalous energy dissipation for long-time averaged solutions. Moreover, we investigate the long time dynamics and prove the existence of a unique global attractor. These results are then applied to two specific classes of active scalar equations in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.