The non-autonomous behavior of a low-order ocean model and its pullback attractors
Stefano Pierini  1, 2, *, @  , Michael Ghil  3, 4, @  , Mickael Chekroun  4, @  
1 : DiST, University of Naples Parthenope
Naples -  Italy
2 : CoNISMa
Rome -  Italy
3 : École Normale Supérieure and PSL Research University
École normale supérieure [ENS] - Paris
Paris -  France
4 : University of California, Los Angeles
* : Corresponding author

A low-order quasigeostrophic ocean model [1] captures several key features of intrinsic low-frequency variability of the oceans' wind-driven circulation [2]. This double-gyre model is used here as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. Our study of this prototype model relies on the theoretical framework of non-autonomous dynamical systems and of their pullback attractors (PBAs), namely the time-dependent invariant sets that attract all trajectories initialized in the remote past [3,4]. Ensemble simulations help us explore these PBAs.

The chaotic PBAs are first analyzed subject to periodic forcing [5]. Distinct forms of sensitivity to the initial state are identified and found to correspond to distinct types of system behavior. The system is then forced by a synthetic aperiodic forcing [6]. The existence of a global PBA is rigorously demonstrated. We then assess the convergence of trajectories to this PBA by computing the probability density function (PDF) of trajectory localization in the model's phase space. The model sensitivity with respect to forcing amplitude and the dependence of the attracting sets on the choice of the ensemble of initial states are then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states, whereas the trajectories in the latter converge to small portions of the global PBA. This complex behavior requires, therefore, separate PDFs for chaotic and nonchaotic trajectories. Finally, the extension of [5] to the case of random dynamical systems is outlined.

(1) Pierini, S., 2011. J. Phys. Oceanogr., 41, 1585-1604.

(2) Ghil, M., 2017. Discr. Cont. Dyn. Syst. A, 37, 189–228.

(3) Ghil, M., M. D. Chekroun, and E. Simonnet, 2008. Physica D, 237, 2111–2126.

(4) Chekroun, M. D., E. Simonnet, and M. Ghil, 2011. Physica D, 240, 1685–1700.

(5) Pierini, S., 2014. J. Phys. Oceanogr., 44, 3245-3254.

(6) Pierini, S., M. Ghil and M. D. Chekroun, 2016. J. Climate, 29, 4185-4202.

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