Determining modes for the surface Quasi-Geostrophic equation

TL;DR

This paper introduces a concept of a determining wavenumber for the surface quasi-geostrophic (SQG) equation, analyzing its dependence on forcing and its implications for understanding the flow's degrees of freedom.

Contribution

It defines a new determining wavenumber for the SQG equation and studies its behavior across different regimes, revealing bounds in subcritical and critical cases.

Findings

Determining wavenumber has a uniform upper bound in subcritical and critical cases.

The wavenumber can become unbounded in supercritical cases.

Provides a measure of the flow's degrees of freedom.

Abstract

We introduce a determining wavenumber for the surface quasi-geostrophic (SQG) equation defined for each individual trajectory and then study its dependence on the force. While in the subcritical and critical cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, which in some sense measure the number of degrees of freedom of the flow, or resolution needed to describe a solution to the SQG equation.