550 research outputs found
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 which controls the transition
behavior. We prove that depending on , the modeled flow exhibits either
a continuous (Type I) or catastrophic (Type II) transition. Numerical
evaluation of 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
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