Dynamic Transitions of Quasi-Geostrophic Channel Flow

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

Intrinsic unpredictability of strong El Ni\~no events

The El Ni\~no-Southern Oscillation (ENSO) is a mode of interannual variability in the coupled equatorial ocean/atmosphere Pacific. El Ni\~no describes a state in which sea surface temperatures in the eastern Pacific increase and upwelling of colder, deep waters diminishes. El Ni\~no events typically peak in boreal winter, but their strength varies irregularly on decadal time scales. There were exceptionally strong El Ni\~no events in 1982-83, 1997-98 and 2015-16 that affected weather on a global scale. Widely publicized forecasts in 2014 predicted that the 2015-16 event would occur a year earlier. Predicting the strength of El Ni\~no is a matter of practical concern due to its effects on hydroclimate and agriculture around the world. This paper presents a new robust mechanism limiting the predictability of strong ENSO events: the existence of an irregular switching between an oscillatory state that has strong El Ni\~no events and a chaotic state that lacks strong events, which can be induced by very weak seasonal forcing or noise.Comment: 4 pages, 6 figure

Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation

The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances. Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I. This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the H\"ormander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, is essential to understand the effect of noise and the phenomenon of phase diffusion. In addition, it is shown that the spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation, revealing useful scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical system approach. This approach is not limited to low-dimensional systems and the reduction method presented in part I is applied to a stochastic model relevant to climate dynamics in part III

Stability of the global ocean circulation: The connection of equilibria within a hierarchy of models

We address the problem of the multiple equilibria of the thermohaline circulation in a hierarchy of models. The understanding of the relation between bifurcation diagrams of box models, two-dimensional models and those of a global ocean general circulation model, is facilitated through analysis of the equilibrium solutions of a three-dimensional Atlantic-like sector model with an open southern channel. Using this configuration, the subtle effects of the wind-stress field, the effects of continental asymmetry and the asymmetry in the surface freshwater flux can be systematically studied. The results clarify why there is an asymmetric Atlantic circulation under a near equatorially-symmetric buoyancy forcing. They also lead to an explanation of the hysteresis regime that is found in models of the global ocean circulation. Both explanations are crucial elements to understanding the role of the ocean in past and future climate changes

The origin of low-frequency variability of double-gyre wind-driven flows

Bifurcation analysis on flows in a two-layer shallow-water model is used to clarify the dynamical origin of low-frequency variability of the double-gyre wind-driven ocean circulation. In many previous model studies, generic low-frequency variations appear to be associated with distinct regimes, characterized by the level of kinetic energy of the mean flow. From these transient flow computations, the current view is that these regimes, and transitions between them, arise through a complex nonlinear interaction between the mean flow and its high-frequency instabilities (the eddies). On the contrary, we demonstrate here, for a particular (but relevant) case, that the origin of these high- and low-energy states is related to the existence of low-frequency instabilities of steady-state flows. The low-frequency modes have distinct spatial patterns and introduce preferential patterns oscillating on interannual to decadal time scales into the flow. In addition, these lowfrequency modes are shown to be robust to the presence of (idealized) topography; the latter may even have a destabilizing effect