Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

TL;DR

This paper studies how zonal flows in a two-layer quasi-geostrophic ocean model transition from steady to oscillatory states as wind forcing increases, revealing bifurcation scenarios including supercritical Hopf and double Hopf bifurcations with complex attractor structures.

Contribution

It provides a mathematical analysis of flow transitions in a two-layer ocean model, identifying bifurcation types and describing the topological structure of resulting attractors.

Findings

First transition is a supercritical Hopf bifurcation leading to periodic solutions.

Double Hopf bifurcation results in an attractor homeomorphic to S^3.

Flow exhibits stable, unstable, and quasi-periodic solutions depending on parameters.

Abstract

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that two modes become critical and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to $S^3$. The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.