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

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

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

Modes of internal thermohaline variability in a single-hemispheric ocean basin

In this study, the internal modes of variability of the thermohaline ocean circulation in an idealized single-hemispheric basin are investigated. The modes are determined systematically by performing linear stability analyses of fully three-dimensional thermohaline flows. Relevant for the large-scale low-frequency variability of the flows are two kinds of oscillatory modes. The modes in one of these classes, which have an interdecadal time scale, are truly three-dimensional phenomena and are characterized by westward propagating temperature anomalies, while modes in the other class have a centennial time scale. The latter modes are shown to persist as internal modes of variability of two-dimensional flows and are therefore related to overturning (or loop) oscillations. The physical mechanism of propagation of these centennial modes is described and the relevance of each class of internal modes is investigated through analysis of stochastically forced transient flows

Imperfections of the North Atlantic wind-driven ocean circulation: Continental geometry and windstress shape

Multiple equilibria of the wind-driven gyres have been found in idealized quasi-geostrophic and shallow water models. In this paper we demonstrate that multiple equilibria persist within a reduced gravity shallow water model under quite realistic continental geometry and windstress forcing for the North Atlantic. Multiple mean flow patterns of the Gulf Stream exist and differ with respect to their separation behavior along the North American coast. The origin of these equilibria is investigated by determining the structure of steady solutions within a hierarchy of equivalent barotropic ocean models using continuation techniques. Within each model, the magnitude of lateral friction is used as a control parameter. It is shown that symmetry breaking, found in a quasi-geostrophic model for a rectangular ocean basin with idealized wind forcing is at the origin of two different mean states of the Gulf Stream. The steady states found become unstable only to a small number of oscillatory modes, which either have intermonthly or interannual periods. The modes of variability remain strongly related through the hierarchy of models indicating that their physics is not strongly dependent on the shape of the continents but is controlled by internal ocean dynamics

Derivation of Delay Equation Climate Models Using the Mori-Zwanzig Formalism

Models incorporating delay have been frequently used to understand climate variability phenomena, but often the delay is introduced through an ad-hoc physical reasoning, such as the propagation time of waves. In this paper, the Mori-Zwanzig formalism is introduced as a way to systematically derive delay models from systems of partial differential equations and hence provides a better justification for using these delay-type models. The Mori-Zwanzig technique gives a formal rewriting of the system using a projection onto a set of resolved variables, where the rewritten system contains a memory term. The computation of this memory term requires solving the orthogonal dynamics equation, which represents the unresolved dynamics. For nonlinear systems, it is often not possible to obtain an analytical solution to the orthogonal dynamics and an approximate solution needs to be found. Here, we demonstrate the Mori-Zwanzig technique for a two-strip model of the El Nino Southern Oscillation (ENSO) and explore methods to solve the orthogonal dynamics. The resulting nonlinear delay model contains an additional term compared to previously proposed ad-hoc conceptual models. This new term leads to a larger ENSO period, which is closer to that seen in observations.Comment: Submitted to Proceedings of the Royal Society A, 25 pages, 10 figure

Regimes of low-frequency variability in a three-layer quasi-geostrophic ocean model

The temporal variability of the midlatitude double-gyre wind-driven ocean circulation is studied in a three-layer quasi-geostrophic model over a broad range in parameter space. Four different types of flow regimes are found, each characterized by a specific time-mean state and spatio-temporal variability. As the lateral friction is decreased, these regimes are encountered in the following order: the viscous antisymmetric regime, the asymmetric regime, the quasi-homoclinic regime and the inertial antisymmetric regime. The variability in the viscous and the inertial antisymmetric regimes (at high and low lateral friction, respectively) is mainly caused by Rossby basin modes. Low-frequency variability, i.e.on interannual to decadal time-scales, is present in the asymmetric and quasi-homoclinic regime and can be related to relaxation oscillations originating from low-frequency gyre modes. The focus of this paper is on the mechanisms of the transitions between the different regimes. The transition from the viscous antisymmetric regime to the asymmetric regime occurs through a symmetry-breaking pitchfork bifurcation. There are strong indications that the quasi-homoclinic regime is introduced through the existence of a homoclinic orbit. The transition to the inertial antisymmetric regime is due to the symmetrization of the time-mean state zonal velocity field through rectification effects