Dynamic Transitions of Quasi-Geostrophic Channel Flow

TL;DR

This paper investigates the dynamic transitions in quasi-geostrophic channel flows, extending previous bifurcation results by analyzing stability and identifying a key parameter that determines whether the flow undergoes smooth or abrupt changes.

Contribution

It provides an explicit expression for a non-dimensional number that controls transition types and analyzes the stability of bifurcated solutions in quasi-geostrophic flows.

Findings

Flow exhibits either continuous or catastrophic transition depending on parameter .

Numerical evaluation suggests catastrophic transitions are more likely in realistic conditions.

Extended previous bifurcation results by addressing stability of periodic solutions.

Abstract

The main aim of this paper is to describe the dynamic transitions in flows described by the two-dimensional, barotropic vorticity equation in a periodic zonal channel. In \cite{CGSW03}, the existence of a Hopf bifurcation in this model as the Reynolds number crosses a critical value was proven. In this paper, we extend the results in \cite{CGSW03} by addressing the stability problem of the bifurcated periodic solutions. Our main result is the explicit expression of a non-dimensional number $\gamma$ which controls the transition behavior. We prove that depending on $\gamma$, the modeled flow exhibits either a continuous (Type I) or catastrophic (Type II) transition. Numerical evaluation of $\gamma$ for a physically realistic region of parameter space suggest that a catastrophic transition is preferred in this flow.