The large-scale, near-surface flow of the mid-latitude oceans is dominated by the presence of a larger, anticyclonic and a smaller, cyclonic gyre. The two gyres share the eastward extension of western boundary currents, such as the Gulf Stream or Kuroshio, and are induced by the shear in the winds that cross the respective ocean basins. This physical phenomenology is described mathematically by a hierarchy of systems of nonlinear partial differential equations (PDEs). We study the low-frequency variability of this wind-driven, double-gyre circulation in mid-latitude ocean basins, subject to time-constant, purely periodic and more general forms of time-dependent wind stress. Both analytical and numerical methods of dynamical systems theory are applied to the PDE systems of interest. Recent work has focused on the application of non-autonomous and random forcing to double-gyre models. We discuss the associated pullback and random attractors and the non-uniqueness of the invariant measures that are obtained. The presentation moves from observations of the geophysical phenomena to modeling them and on to a proper mathematical understanding of the models thus obtained. Connections are made with the highly topical issues of climate change and climate sensitivity.

# Dynamical systems

The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem. Discrete and Continuous Dynamical Systems - A. 2017;37 (1) :189-228.Abstract

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A Mathematical Theory of Climate Sensitivity: A Tale of Deterministic & Stochastic Dynamical Systems. 11th AIMS Conf. on Dynamical Systems, Differential Equations & Applications, Honoring Peter Lax’s 90th Birthday, Orlando, FL, July 2016. 2016.Abstract

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Development at the wildland urban interface and the mitigation of forest-fire risk. Proceedings of the National Academy of Sciences. 2007;104 (36) :14272–14276.Abstract

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Lecture 2: Toward a Mathematical Theory of Climate Sensitivity. Workshop on Mathematics of Climate Change, Related Hazards and Risks, CIMAT, Guanajuato, Mexico. 2013.Abstract

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Toward a Mathematical Theory of Climate Sensitivity. International Congress on Industrial and Applied Mathematics (ICIAM), Vancouver. 2011.Abstract

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The Complex Physics of Climate Change: Nonlinearity and Stochasticity. Workshop on Critical Transitions in Complex Systems, Imperial College London, United Kingdom [Internet]. 2012. Conference websiteAbstract

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Lecture 3 : The Coupled Dynamics of Climate and Economics. Workshop on Mathematics of Climate Change, Related Hazards and Risks, CIMAT, Guanajuato, Mexico. 2013.Abstract

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A delay differential model of ENSO variability, Part 2: Phase locking, multiple solutions, and dynamics of extrema. Nonlinear Processes in Geophysics. 2010;17 (2) :123–135.

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Advanced Data Assimilation in Strongly Nonlinear Dynamical Systems. Journal of Atmospheric Sciences. 1994;51 :1037–1056.

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Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation. Nonlinear Processes in Geophysics. 1995;2 :241–268.Abstract

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Empirical model reduction and the modelling hierarchy in climate dynamics and the geosciences. Stochastic physics and climate modelling. Cambridge University Press, Cambridge. 2009 :35–72.Abstract

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Global Hopf Bifurcation in a Simple Climate Model. Siam Journal on Applied Mathematics [Internet]. 1983;43 (5) :1019–1041. Publisher's VersionAbstract

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Cryothermodynamics: the chaotic dynamics of paleoclimate. Physica D. 1994;77 (1-3) :130–159.

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