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    Bifurcation diagram of our idealized ocean model (1) in the autonomous case, i.e., with in Eq. (3). (a) The range of the variable is plotted vs the wind stress intensity γ. (b),(c) The limit cycles corresponding to the blue and red vertical segments in (a) are shown here projected onto the planes and , respectively.

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    Ensemble behavior of forced solutions of the double-gyre ocean model. (a) Time dependence of the total forcing , for . (b),(c) Evolution of 644 initial states emanating from the subset in the plane for (b) and (c) . (d),(e) Corresponding time series of . The set is given by .

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    Evolution in time of as a function of the initial time , for (left) and (right) , for different initial times but with the same distribution of IDs.

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    Evolution in time of as a function of the amplitude ε of the time-dependent part of the forcing, , for (left) and (right) . The amplitudes are (a),(f) , (b),(g) , (c),(h) , (d),(i) , and (e),(j) . The total forcing amplitude is reported at the bottom of each panel (the black areas correspond to ).

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    Maps of FTLEs for the present model. Maps of at (a),(d) yr, (b),(e) yr, and (c),(f) yr, for (top) and (bottom) . In the color bars to the right, warm colors indicate instability, i.e., , and cool colors the opposite, while heavy black lines in the panels correspond to .

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    Mean normalized distance for 15 000 trajectories starting in the initial set : (a) and (b) . The boxes A–D and 1–4 indicate subdomains of from which IDs are taken for our analysis (see text).

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    Evolution of ensembles of 420 trajectories each that emanate from small subsets of : (a),(b) and IDs in boxes A and B of Fig. 6a, respectively, and (c),(d) and IDs in boxes C and D of Fig. 6b, respectively.

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    Superposition of two time series of starting from nearby IDs in . The initial points for are (a) and (b) , while for they are (c) and (d) . The lower graphs in (a)–(d) marked with a prime are the corresponding normalized distances . The two nearby trajectories of (a)–(d) are included in those of Figs. 7a–d, respectively. The horizontal black lines mark the value .

  • View in gallery

    (a) Intersection with the () plane at yr of 15 000 trajectories (gray dots) emanating from the whole set , and from boxes 1 and 2 (cyan dots, insets) and A (red dots) of Fig. 6a for ; (b) as in (a), but from boxes 3 and 4 (cyan) and C (red) of Fig. 6b for . The insets in both panels zoom in on the way that the red dots associated with the chaotic orbits and the cyan dots associated with the nonchaotic orbits are embedded into the global PBA (gray dots).

  • View in gallery

    PDF of localization of the 15 000 trajectories of Figs. 6 and 9 projected onto the () plane at yr, for (a),(c),(e) and (b),(d),(f) . The PDF is computed in (a) and (b) from all 15 000 trajectories emanating from ; in (c) and (d) only the chaotic trajectories emanating from boxes A and C, respectively, are used; and in (e) and (f) only the nonchaotic trajectories emanating from boxes 1–2 and 3–4, respectively, are used (cf. Fig. 9).

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Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

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  • 1 Universitá di Napoli Parthenope, Napoli, Italy
  • | 2 Ecole Normale Supérieure, Paris, France, and University of California, Los Angeles, Los Angeles, California
  • | 3 University of California, Los Angeles, Los Angeles, California
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Abstract

A low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used 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. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is 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 scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.

Corresponding author address: Stefano Pierini, Dipartimento di Scienze e Tecnologie, Universitá di Napoli Parthenope, Centro Direzionale, Isola C4, 80143 Napoli, Italy. E-mail: stefano.pierini@uniparthenope.it

Abstract

A low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used 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. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is 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 scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.

Corresponding author address: Stefano Pierini, Dipartimento di Scienze e Tecnologie, Universitá di Napoli Parthenope, Centro Direzionale, Isola C4, 80143 Napoli, Italy. E-mail: stefano.pierini@uniparthenope.it

1. Introduction and motivation

Weather and climate predictability have been investigated for several decades, starting from the pioneering work of Lorenz (1963), by relying on the concepts and methods of autonomous dissipative nonlinear dynamical systems (Ghil and Childress 1987, and references therein). For such systems, any initial volume in phase space contracts in time, on the average, thus eventually reducing to a time-invariant set of zero volume called an attractor. In the autonomous case, an attractor can be a fixed point, a limit cycle, an invariant torus, or a strange attractor (Eckmann and Ruelle 1985; Ott 2002; Tél and Gruiz 2006).

Investigating a system’s attractors, in particular the strange attractors that describe the statistical properties of deterministic chaos, is fundamental in this context (Eckmann and Ruelle 1985) and has helped considerably in the study of multiple weather regimes and of their predictability on time scales of weeks to months (Ghil and Robertson 2002, and references therein). On the other hand, when studying the longer time scales—interannual, interdecadal, and longer—that are associated with climate change, it becomes necessary to take into consideration the time dependence of both anthropogenic and natural forcing (Martinson et al. 1995; Mann et al. 1998; Chang et al. 2015). The proper framework for studying changes in the system’s internal variability, and not just in its mean properties, when subject to variable forcing is that of nonautonomous and random dynamical systems (Drótos et al. 2015; Ghil 2015, 2016).

In the latter framework, the natural generalization of the attractors is provided by pullback attractors (PBAs). A global PBA is defined in the mathematical literature as a time-dependent set in the system’s phase space X that is invariant under its governing equations—together with the equally time-dependent, invariant measure supported on this set—and to which all trajectories in X starting in the distant past converge (Arnold 1998; Rasmussen 2007; Kloeden and Rasmussen 2011; Carvalho et al. 2012). As we shall see later, such a global PBA might include two or more local attractors, which only attract trajectories from certain subsets of X. These concepts are clarified and discussed further in the present paper’s appendix. When the equations or the forcing include stochastic processes, a PBA is called a random attractor. In the physical literature, Romeiras et al. (1990) introduced the somewhat vaguer concept of a snapshot attractor.

The concepts and methods of nonautonomous and random dynamical systems, including pullback attraction, were introduced into climate dynamics by Ghil et al. (2008) and Chekroun et al. (2011), and they were also pursued vigorously by T. Tél and his group (Bódai et al. 2011; Bódai and Tél 2012; Bódai et al. 2013). Several recent studies have explored the properties of PBAs for low-order climate models using ensemble simulations (Bódai et al. 2011, 2013; Bódai and Tél 2012; De Saedeleer et al. 2013; Pierini 2014a; Drótos et al. 2015).

Drótos et al. (2015) used the simple conceptual climate model of Lorenz (1984), forced by a periodic seasonal component and a linear decrease in the imposed equator-to-pole temperature difference. These authors argued that the PBAs—or snapshot attractors, using the simpler concept introduced by Romeiras et al. (1990)—and their natural probability distributions are the only tools with which mathematically sound statements can be made at a given point in time within a changing climate. Their claim was based on the finding that the time-dependent, chaotic attractor of the Lorenz (1984) model was independent of the initial states, or initial data (IDs) for short, within the parameter ranges and for the ensembles of IDs they considered. This claim appears to be fully justified for the Lorenz (1984) model, which seems to possess a unique global PBA in those parameter ranges.

The ensemble simulations of a periodically forced quasigeostrophic ocean model by Pierini (2014a) also provided evidence of ID independence in chaotic regimes, but dependence on the IDs did emerge for particular parameter ranges, for which chaotic basins of attraction coexist with nonchaotic basins; trajectories starting in the latter converge onto small subsets of the PBA. Thus, the fundamental problem of the dependence of the natural probability distribution of a changing climate on initial states needs further investigation.

In this paper, we carry out such an investigation. The four-variable quasigeostrophic model of the ocean’s wind-driven, double-gyre circulation formulated by Pierini (2011) and used in the periodically forced simulations of Pierini (2014a) is studied here via ensemble simulations subject to a deterministic aperiodic forcing that mimics time dependence dominated by interdecadal climate variability. Two reference cases separated by a tipping point are considered: above the tipping point, large-amplitude relaxation oscillations shape the behavior of the autonomous system, whereas below it small-amplitude oscillations are dominant. In this second range the system is excitable, because an appropriate time-dependent forcing can excite the relaxation oscillations (e.g., Pikovsky and Kurths 1997; Pierini 2011, 2012). Various properties of these two reference cases are analyzed, such as the time of convergence from arbitrary IDs to PBAs and the sensitivity of the latter to forcing amplitude, as well as the dependence of the local attractors on the choice of the initial ensemble of states.

The paper is organized as follows. In section 2, the model is described and basic aspects of the autonomous system’s solutions and of the two reference ensemble simulations under aperiodic forcing are discussed. In section 3, the time of convergence from arbitrary IDs to the PBAs is estimated. In section 4, the sensitivity of PBAs with respect to the forcing amplitude is investigated. Section 5 is devoted to the dependence of PBAs on the choice of the ensemble of IDs: chaotic and nonchaotic basins of attraction are analyzed in detail. In section 6, the main results are summarized and discussed, and an appendix provides the rigorous mathematical background.

2. Model description

a. Model formulation

The mathematical model used in this study was developed by Pierini (2011) to analyze fundamental dynamical features of the decadal variability in the Kuroshio Extension (Dijkstra and Ghil 2005, and references therein). This model is derived through projection of the governing equations onto an orthonormal basis and subsequent low-order truncation, following the approach of Platzman (1960), Saltzman (1962), Lorenz (1963), and many others since, such as Lorenz (1982), Legras and Ghil (1985), Ghil and Childress (1987), Olbers (2001), and Crucifix (2012). S. Pierini and colleagues had already investigated this decadal variability with a relatively realistic primitive equation ocean model (Pierini 2006; Pierini et al. 2009; Pierini 2010, 2014b).

The model is derived via a severe truncation of the spectral representation of the evolution equation of potential vorticity in the quasigeostrophic, reduced-gravity approximation, in a rectangular domain. The Cartesian orthonormal basis onto which the streamfunction is projected is composed of two trigonometric functions in the zonal and two in the meridional direction; the zonal ones incorporate, moreover, the exponential factor proposed by Jiang et al. (1995) to account for westward intensification of midlatitude surface currents. Here is the vector of horizontal coordinates and t is time.

The nonlinear coupled ordinary differential equations for the variables , are given by
e1
Here denotes the vector of expansion coefficients of the streamfunction, will be used as the vector of forcing terms , and the bilinear terms are given by
e2

The analytical expressions of the Jacobian coefficients , the linear ones , and the forcing terms were derived by Pierini (2011); moreover, all the parameter values used here are the same as adopted in that study and in Pierini (2014a). Pierini (2012) used the model governed by Eqs. (1) and (2) to systematically analyze abrupt climate changes seen as noise-induced transitions in excitable systems, while S. Vannitsem and colleagues (Vannitsem 2014a,b; Vannitsem and De Cruz 2014) used it to represent the ocean component in their low-order ocean–atmosphere coupled models. Pierini (2014a) applied periodic forcing to model (1) and explored the PBAs of the forced system. The present study can thus be seen as the extension of the latter study to an aperiodic forcing case.

In this study, the forcing is defined as
e3
Here is the double-gyre wind stress curl used in Pierini (2011), with γ its dimensionless intensity, while is a fixed realization of an Ornstein–Uhlenbeck process, normalized to have unit variance and an autocorrelation time . This fixed realization is smoothed with the sliding window of width , and the resulting aperiodic forcing is weighted by the dimensionless parameter ε.

b. Model behavior

In the autonomous case, there is no t dependence and we let . The bifurcation diagram in this case is plotted in Fig. 1a. This diagram is obtained by forward integration of the model, starting from vanishing initial data and dropping a sufficiently long initial segment of the trajectories. The time needed for the trajectories to land on the model’s attractor is discussed in the appendix.

Fig. 1.
Fig. 1.

Bifurcation diagram of our idealized ocean model (1) in the autonomous case, i.e., with in Eq. (3). (a) The range of the variable is plotted vs the wind stress intensity γ. (b),(c) The limit cycles corresponding to the blue and red vertical segments in (a) are shown here projected onto the planes and , respectively.

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

The value corresponds to the blue limit cycle shown in Figs. 1b and 1c, in the planes and , respectively. This limit cycle lies above the supercritical Hopf bifurcation that occurs at (cf. Pierini 2011) and resembles a linear, harmonic oscillation.

The value , however, corresponds to a much larger and more complex limit cycle, plotted in red in Figs. 1b and 1c. This limit cycle is associated with a relaxation oscillation that can be excited by an appropriate noise or deterministic time-dependent forcing when the system lies below a homoclinic bifurcation (e.g., Simonnet et al. 2005; Pierini 2011). The sudden growth in the size and complexity of the closed, periodic trajectory at corresponds to such a global bifurcation.

Pierini (2011, 2012, 2014a) did, in fact, study already the present model in the presence of both stochastic and deterministically periodic forcing, and found that such a homoclinic bifurcation does occur in it and that this bifurcation gives rise to what is currently termed a tipping point by the climate community (e.g., Lenton et al. 2008). Such a dynamical mechanism is a typical paradigm of abrupt climate change associated with an intrinsically nonlinear mode of variability (Lorenz 1963; Ghil and Childress 1987; Crucifix 2012), and it is sometimes called coherence resonance (Pikovsky and Kurths 1997). When coherence resonance does occur, the system displays strong dependence on forcing amplitude, as we will see in section 4.

In the nonautonomous case, in Eq. (3), and we use the fixed aperiodic forcing obtained with autocorrelation time and a sliding window of width . Figure 2a shows a realization with yr and . This choice provides an idealized aperiodic forcing that mimics North Pacific multidecadal variability (Chao et al. 2000; Chang et al. 2015) for the minimal representation of the Kuroshio Extension provided by (1). On the other hand, the relaxation oscillations of the autonomous model display decadal variability as well, in agreement with the oceanic phenomenon that inspired this simple model. The dimensional time variable used throughout this paper refers to this multidecadal interpretation, although the present model’s properties are likely to be shared by a wide class of systems of climate relevance.

Fig. 2.
Fig. 2.

Ensemble behavior of forced solutions of the double-gyre ocean model. (a) Time dependence of the total forcing , for . (b),(c) Evolution of 644 initial states emanating from the subset in the plane for (b) and (c) . (d),(e) Corresponding time series of . The set is given by .

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

Apart from this oceanographic interpretation, the forcing in Eq. (3) extends the analysis of Pierini (2014a) from purely periodic to deterministically aperiodic. This choice disrupts the cyclostationarity and cycloergodicity properties of PBAs and makes the forced model’s behavior more realistic.

The evolution of the model solutions subject to the forcing of Fig. 2a is shown in Figs. 2b and 2c for two reference cases, and , respectively; the solutions of the corresponding autonomous system (i.e., with ) are plotted in Figs. 1b and 1c. In the present figure, N trajectories emanate from N IDs uniformly distributed at time on a reference subset of the plane, which is shown in Figs. 2b and 2c here, while the coordinates of the initial states (not shown) are chosen the same way as in Pierini (2014a).

Figures 2b and 2c provide a first representation of the sets that approximate the corresponding PBAs; the small number of trajectories is in fact chosen for the sake of graphical clarity. The correct identification and characterization of the PBAs requires, however, an analysis of the PDFs that evolve along the trajectories and of the convergence to the appropriate invariant time-dependent sets. This convergence is investigated in the next section.

3. Convergence to PBAs

As mentioned in the introduction, PBAs are the extension of the attractors of dissipative autonomous dynamical systems to the nonautonomous case. When time-dependent forcing is present, the trajectories originating from the set of IDs at time τ tend to a subset of phase space at time , but while depends on τ in general, in the limit of this dependence vanishes.

The limit is the pullback limit and it provides the natural generalization of the forward asymptotics associated with autonomous systems (Arnold 1998; Rasmussen 2007; Kloeden and Rasmussen 2011; Carvalho et al. 2012; Ghil et al. 2008; Chekroun et al. 2011); see the appendix herein. In practical terms, though, a question arises: How large should be for to be virtually independent of τ?

An effective way to answer this question is to compute the PDF of the localization of the trajectories that emanate from . To obtain a function that provides a clear graphical representation of the invariant measure on the attracting set, the following parameter is computed:
e4
where is the PDF of localization of the variable using the trajectories shown in Figs. 2b and 2c. To make independent of the length of the signal (this is important to make the panels of Fig. 3 comparable; see below), is decomposed into n cells and is scaled in such a way that its integral over each cell equals 1. In our case yr and .
Fig. 3.
Fig. 3.

Evolution in time of as a function of the initial time , for (left) and (right) , for different initial times but with the same distribution of IDs.

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

In Figs. 2d and 2e, we plot for the trajectories shown in Figs. 2b and 2c, respectively. The variability in induced by the forcing of Fig. 2a is impressive when compared with Figs. 4a and 4f, which are discussed in section 4 and were obtained with . Of particular interest is the remarkable reduction of the range of variability of found at yr and at yr. This variability reduction is most evident in the case with (Fig. 2d) and it corresponds to a partial synchronization of the trajectories. As we will see in section 4, both synchronization episodes occur when the total forcing decreases well below unity.

Fig. 4.
Fig. 4.

Evolution in time of as a function of the amplitude ε of the time-dependent part of the forcing, , for (left) and (right) . The amplitudes are (a),(f) , (b),(g) , (c),(h) , (d),(i) , and (e),(j) . The total forcing amplitude is reported at the bottom of each panel (the black areas correspond to ).

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

To estimate the rate of converge to the PBA, five cases are considered in Fig. 3, for both and . The trajectories emanate from the same distribution of IDs but at the different initial times yr. Comparison of with shows that the two distributions coincide when t in the former is greater than yr. We can therefore conclude that the set to which the trajectories converge approximates well the PBA after roughly 15 yr from the beginning of the integration, and that this important result holds at any time during the nonstationary process.

4. Sensitivity of PBAs to the amplitude of the forcing

Model robustness is important in making any inferences about the role of time-dependent forcing (Ghil 2015, and references therein). We thus proceed to analyze in this section the sensitivity of the present model’s PBAs with respect to the amplitude ε of the time-dependent component of the forcing, while in the next section dependence on IDs will be analyzed.

Figure 4 shows the dependence of on ε, with this amplitude increasing from (i.e., the autonomous case) in the top two panels to (the value used in Fig. 2) in the bottom two panels, while the overall intensity γ of the total wind stress equals 0.96 in the left column and 1.1 in the right column, as in Figs. 2 and 3. The PBAs corresponding to the blue and red limit cycles visible in Fig. 1b are shown here in Figs. 4a and 4f. It is interesting to notice that, for , any point of evolves toward the same limit cycle, while the phase of each orbit on its limit cycle does depend on the IDs. Note that the limit cycles in Figs. 1b and 1c were obtained, like the bifurcation diagram in Fig. 1a, from vanishing initial data, .

Inspecting the panels of the left column, which correspond to , from top to bottom, shows a dramatic transition from the small limit cycles, in the top two rows, to the highly variable ones of the bottom two rows. To better understand this behavior, the total time-dependent forcing amplitude is plotted at the bottom of each panel. Each transition from small- to large-amplitude oscillations is preceded by an increase of above the critical level ; consider, for instance, the one already noticed in Figs. 2d and 4e. More generally, coherence resonance is activated when , provided this has occurred at least 10–30 yr in advance; note that even very small positive excursions above yield the same behavior, as shown for example in Fig. 4c.

This result agrees with Pierini (2012), who noticed that model (1), when subjected to stochastic forcing, behaves approximately like an adiabatically changing autonomous dynamical system, as far as coherence resonance is concerned. Likewise, when decreases well below unity, the system tends to produce small-amplitude limit cycles, according to the bifurcation diagram of Fig. 1a: this corresponds to the synchronization events already discussed.

The same behavior is found for , in the right column of Fig. 4, but now the relaxation oscillations are typically present, since they are self-sustained and since is the norm. The consequence of the fluctuations of on can again be easily interpreted by taking into account the diagram of Fig. 1a. Three synchronization events are present for (Fig. 4j), after intervals during which . In general, if , the changes in the PBA are less dramatic.

5. PBA dependence on the choice of initial ensemble

In section 3, we have investigated the evolution of the same set of IDs starting from different initial times τ; here we study the evolution of different initial states starting from the same initial time . This second analysis is fundamental in a climate perspective, as it concerns the dynamical origin of the PBAs and of the invariant measures supported on them, as well as the dependence of predictability on IDs.

In the periodic-forcing case, Pierini (2014a) provided numerical evidence that the present ocean model’s PBA can be obtained from any set of IDs, provided its evolution is chaotic; see Figs. 3e and 3f therein. Drótos et al. (2015) have recently stressed that the chaotic PBAs of the Lorenz (1984) atmospheric model are independent of IDs and have suggested that this is likely to be the typical situation in climate models.

PBA dependence on IDs may, however, be more complex. In his systematic study of the present model’s PBAs as a function of the forcing period, Pierini (2014a) identified periods for which a chaotic basin of attraction—which converges onto the global PBA—coexists with nonchaotic basins. Thus, for example, a chaotic basin can coexist with another basin whose phase-space flow converges onto a single periodic trajectory lying on the global PBA; see, for example Figs. 6a–c of Pierini (2014a). It is therefore natural to investigate whether a similar behavior may be present also in the aperiodic-forcing case studied in the present paper. In the following, we use two complementary mathematical tools to explore this possibility.

a. Finite-time Lyapunov exponents

The typical approach to the investigation of chaotic versus nonchaotic behavior is to evaluate the finite-time Lyapunov exponents (FTLEs) of the system (Nese 1989). FTLEs have been widely used in physical oceanography to help identify localized coherent structures and material surfaces (Haller 2001; D’Ovidio et al. 2009).

If two nearby points—whose distance according to Eulerian metrics in phase space is at time and at time t—diverge exponentially in time, the Lyapunov exponent
e5
is independent of t and it is positive; if, on the contrary, the trajectories converge exponentially, then is still independent of t and it is negative. In the presence of a bound on the region of phase space occupied by all possible model trajectories, divergence of trajectories indicates that the behavior is chaotic (e.g., Ghil and Childress 1987).

To test whether chaotic behavior may coexist with nonchaotic trajectories, we computed maps of for trajectories that originate from a homogeneous distribution of at times , and 60 yr. These maps are plotted in Fig. 5 for both and .

Fig. 5.
Fig. 5.

Maps of FTLEs for the present model. Maps of at (a),(d) yr, (b),(e) yr, and (c),(f) yr, for (top) and (bottom) . In the color bars to the right, warm colors indicate instability, i.e., , and cool colors the opposite, while heavy black lines in the panels correspond to .

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

The first thing to note in all six panels is the existence of extensive regions in which that coexist with regions where , so the chaotic nature of the flow is far from being independent on IDs for the model (1), subject to the aperiodic forcing of Fig. 2a. Another notable feature is that exhibits strong time dependence, for both (upper panels) and (lower panels).

The remarkable decrease of the range of as t increases from yr to yr, both in Figs. 5a–c and Figs. 5d–f, is to be expected, due to the saturation of the exponential behavior, but the spatial distribution depends strongly on time as well. This dependence clearly indicates that complex temporal behavior lies behind these maps, which cannot, therefore, provide an unambiguous indication of the flow’s chaoticity depending on . (Such complexities are further discussed in connection with Fig. 8 below.)

b. Mean normalized distance

A more robust identification of chaotic and nonchaotic regions within the set of initial states can be obtained by introducing and evaluating the metric σ, defined by
e6
where the normalized distance , given by , is averaged over the whole forward time integration of the available trajectories. The maps of σ in Fig. 6 reveal large chaotic regions where on average (warm colors) but also nonchaotic regions where (blue) and thus initially close trajectories do remain close on average.
Fig. 6.
Fig. 6.

Mean normalized distance for 15 000 trajectories starting in the initial set : (a) and (b) . The boxes A–D and 1–4 indicate subdomains of from which IDs are taken for our analysis (see text).

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

To check the character of the two types of evolution, we plot in Fig. 7 many trajectories that leave two small subsets of for which —namely the sets A and C in Fig. 6, for and , respectively—to compare with those that leave the sets B and D of Fig. 6, for which .

Fig. 7.
Fig. 7.

Evolution of ensembles of 420 trajectories each that emanate from small subsets of : (a),(b) and IDs in boxes A and B of Fig. 6a, respectively, and (c),(d) and IDs in boxes C and D of Fig. 6b, respectively.

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

Figures 7a and 7c confirm the extreme sensitivity to IDs that lie in boxes A and C: a comparison of these two panels with Figs. 2b and 2c, respectively, shows that the attracting sets to which trajectories started in A and B converge are dense in the PBAs obtained by starting from the whole . The corresponding PDFs, as represented by (not shown) coincide in fact with those of Figs. 2d and 2e and the same result is obtained by starting from any small region of for which . Figures 7b and 7d also confirm that the trajectories leaving from boxes B and D converge to time-dependent sets in the plane whose measure is much smaller than that of either B or D.

c. Discussion

It is worth exploring in greater detail the evolution of some of the trajectories plotted in Fig. 7. We display in the top graphs of Figs. 8a–d the time series of two initially nearby trajectories selected among those of Figs. 7a–d, respectively; each pair of trajectories has lines, one in blue and one in red, both drawn from the same panel of Fig. 7. The corresponding normalized distances are shown in the bottom graphs of Figs. 8a–d marked with a prime.

Fig. 8.
Fig. 8.

Superposition of two time series of starting from nearby IDs in . The initial points for are (a) and (b) , while for they are (c) and (d) . The lower graphs in (a)–(d) marked with a prime are the corresponding normalized distances . The two nearby trajectories of (a)–(d) are included in those of Figs. 7a–d, respectively. The horizontal black lines mark the value .

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

In the chaotic cases, represented by both graphs in Figs. 8a, and 8c, the values of range from to , up to order ; the former very small values occur over the same time intervals—already apparent in Figs. 2b′ and 2c′—in which synchronization sets in. In the nonchaotic cases, like those of Figs. 8b, 8d, the values of lie typically below unity, although an initial divergence is noticeable in Fig. 8d′; this slow divergence may be due to a transient growth of polynomial rather than exponential nature.

Inspection of the bottom graphs in Figs. 8a–d confirms the validity of the mean normalized distance σ of Eq. (6) as a sharper diagnostic tool than the FTLEs. Longer integrations (not shown) simply extend in time the ensemble of trajectories shown in Figs. 7a and 7b, but the chaotic versus nonchaotic character of the trajectories—as diagnosed by σ—is preserved.

To further analyze the structure of the model’s PBAs, the intersection of many trajectories with the (, ) plane at a reference time yr is shown in Figs. 9a and 9b for and , respectively. The gray dots in both panels arise from 15 000 IDs evenly distributed in and are associated with the filamentary patterns that characterize chaotic attractors (e.g., Chekroun et al. 2011; Drótos et al. 2015).

Fig. 9.
Fig. 9.

(a) Intersection with the () plane at yr of 15 000 trajectories (gray dots) emanating from the whole set , and from boxes 1 and 2 (cyan dots, insets) and A (red dots) of Fig. 6a for ; (b) as in (a), but from boxes 3 and 4 (cyan) and C (red) of Fig. 6b for . The insets in both panels zoom in on the way that the red dots associated with the chaotic orbits and the cyan dots associated with the nonchaotic orbits are embedded into the global PBA (gray dots).

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

The red dots arise from the small boxes A and C that are located in the chaotic regions of Figs. 6a and 6b, respectively. These red dots, too, are distributed all over the corresponding attractors in the two panels, as expected from the previous analysis; in fact, only the 420 red dots that correspond to the trajectories of Figs. 7a and 7c have been plotted here, to avoid complete overlapping with the gray dots.

On the contrary, the cyan dots that arise from the large nonchaotic boxes 1, 2, 3, and 4 of Figs. 6a and 6b converge onto small subsets of the PBAs. This convergence onto a smaller-dimensional set is very similar to what Pierini (2014a) found for some forcing periods that yielded periodic PBAs; for example, see Figs. 6a–c therein.

For in Fig. 9b, the similarity with the result of Pierini (2014a) is striking because all the trajectories originating from boxes 3 and 4 of Fig. 6b converge toward two distinct points. It is worth noting that the blue boxes 3 and 4 of Fig. 6b are separated by a chaotic ridge that apparently acts as a barrier separating two distinct basins of attraction, each converging to a single point. In this case the asymptotic evolution is independent of initial data. On the other hand, for (Fig. 9a) the attracting cyan sets are small segments, so the evolution of the nonchaotic trajectories does depend, although very weakly, on IDs.

Graphs like those of Fig. 9 cannot provide quantitative information on the probability of localization of trajectories on the model’s global PBA. Such information can be obtained, though, from the three-dimensional PDFs plotted in Fig. 10.

Fig. 10.
Fig. 10.

PDF of localization of the 15 000 trajectories of Figs. 6 and 9 projected onto the () plane at yr, for (a),(c),(e) and (b),(d),(f) . The PDF is computed in (a) and (b) from all 15 000 trajectories emanating from ; in (c) and (d) only the chaotic trajectories emanating from boxes A and C, respectively, are used; and in (e) and (f) only the nonchaotic trajectories emanating from boxes 1–2 and 3–4, respectively, are used (cf. Fig. 9).

Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1

For the subcritical case in the left column of the figure, the distribution is highly inhomogeneous and concentrates in two areas labeled 1 and 2 in Fig. 10a; these concentrations extend over very small portions of the filament and result from large contributions of both chaotic (Fig. 10c) and nonchaotic (Fig. 10e) trajectories, with a larger contribution from the latter. In the supercritical case (right column), the same situation holds, with two peaks contained in the small area labeled 3.

In summary, in the small regions of phase space in which the invariant sample measure on our model’s global PBA peaks in a filamentary pattern, both chaotic and nonchaotic trajectories coexist. The PDF peaks in Figs. 10a and 10b are clearly an indication of enhanced overall predictability, and recall those observed in Figs. 6 and 7 of Chekroun et al. (2011) for a highly idealized model of El Niño–Southern Oscillation with seasonal forcing.

6. Summary and conclusions

In this study, a low-order quasigeostrophic model of the wind-driven ocean circulation has been used as a prototype of an unstable, nonlinear, nonautonomous dynamical system of climatic relevance. We have studied this double-gyre model’s PBAs (i.e., those time-dependent invariant sets that attract all trajectories initialized in the remote past) along with the invariant sample measures that live on these sets. PBAs (Arnold 1998; Ghil et al. 2008; Chekroun et al. 2011) and their close cousins, snapshot attractors (Romeiras et al. 1990; Bódai et al. 2011, 2013; Bódai and Tél 2012; Drótos et al. 2015), have only recently been recognized as the natural tools for investigating basic features of a changing nonequilibrium climate, and the theory of nonautonomous dynamical systems as the proper mathematical framework for such investigations.

We chose here an aperiodic forcing dominated by interdecadal variability to mimic the effects of multidecadal climate changes (Chang et al. 2015) on the midlatitude double-gyre circulation; see section 2. The present study generalizes therewith a set of results previously obtained in the presence of periodic forcing (Pierini 2014a). We recall that, in the absence of time-dependent forcing [i.e., when in Eq. (3)], a tipping point at separates small-amplitude, nearly harmonic oscillations from large-amplitude relaxation oscillations.

In section 3, the convergence of ensembles of trajectories to PBAs was assessed in order to estimate the time required to identify the PBAs. To do so, we considered—in addition to projections of the trajectories from the full, four-dimensional phase space onto a plane spanned by the and variables—the suitably scaled, time-dependent statistic ; see Eq. (4) and Fig. 2. A key result of this section is that a period of about 15 years suffices for trajectories to converge to the PBA.

The sensitivity of PBAs with respect to the amplitude ε of the time-dependent portion of the forcing was studied in section 4. Below the tipping point , the range of variability within the PBA increases dramatically as ε increases, and so does the intermittency; see left column of Fig. 4. For , the changes in both range and intermittency are less striking (cf. right column of Fig. 4). The difference in behavior is clearly associated with the solutions of the autonomous system: a small-amplitude limit cycle for versus a higher-amplitude, more complex and self-sustained relaxation oscillation for .

Finally, the dependence of the attracting sets on the choice of the ensemble of initial states is investigated in section 5. Our investigation relies, on the one hand, on the classical FTLEs to measure the rate of divergence of trajectories, but it introduces also a novel metric, namely the time mean σ of the normalized distance between two initially nearby trajectories. The use of and σ [cf. Eq. (6)] is required by the extreme complexity of the evolution of the distance between the trajectories: this evolution is far from purely exponential and it involves large variations over different time scales and episodes of synchronization induced by the forcing, as seen in Fig. 8.

In our model, the statistics of an ensemble of trajectories in a chaotic regime does depend on the ensemble of states chosen in the remote past. Figures 6, 7, and 9, in particular, illustrate very well that the model’s global PBA—whose existence is rigorously proven in the appendix—contains both chaotic and nonchaotic trajectories.

Both of these types of trajectories have attractor basins that seem to be separated by fractal boundaries: the former fill the global PBA, while the latter form small and smooth subsets of the PBA. This complex, interwoven pattern of rapidly divergent and nondivergent trajectories shows that a time-independent PDF, as often used in predictability studies, is not sufficient to describe the model’s long-term behavior or to characterize its predictability.

These intriguing results have been obtained with a highly idealized nonlinear model subject to deterministic aperiodic forcing. As usual in this kind of approach, they raise as many questions as they answer about the broader issues of climate modeling and climate predictability. The two most urgent questions for the climate sciences are these:

  • To which extent is this simple model’s behavior characteristic of a wide class of nonautonomous dynamical systems, whether deterministic or stochastic?
  • How relevant is this class to climate modeling and climate predictability studies?
Again as usual, these questions require one to climb the rungs of a hierarchy of successively more complex and detailed climate models (Ghil 2001; Dijkstra and Ghil 2005; Ghil 2015, and references therein). Naturally, introducing noise in the forcing may substantially modify the system’s behavior: a random dynamical system extension of the present study is in progress.

Acknowledgments

SP and MG would like to thank the TEMASAV Project of the Regione Campania of Italy (POR Campania FSE 2007/2013) and the coordinator, Giancarlo Spezie, for having supported exchange visits between the Environmental Research and Teaching Institute (CERES-ERTI) in Paris and the Department of Science and Technology (DiST) of the University of Naples Parthenope, thus promoting this collaboration. The research herein was supported by Grants N00014-12-1-0911 and N00014-16-1-2073 from the Multidisciplinary University Research Initiative (MURI) of the Office of Naval Research and by the U.S. National Science Foundation Grant OCE 1243175 (MDC and MG). The authors wish to thank two anonymous reviewers, whose detailed and constructive comments helped improve the manuscript.

APPENDIX

Existence of a Global Pullback Attractor

We present in this appendix rigorous mathematical results on the existence of pullback attractors (PBAs) in ordinary differential equation (ODE) models—like the one given by Eqs. (1) and (2) here—and, in particular, the conditions on the forcing term for such an existence to be guaranteed. General results are known from the specialized literature on the existence of pullback attractors or related invariant manifolds (Kloeden and Rasmussen 2011; Carvalho et al. 2012; Chekroun et al. 2015a,b). For the sake of conciseness and clarity, however, we provide below the main elements of such an existence theory, while emphasizing the energy estimates involved; see also Kondrashov et al. (2015, Theorem 3.1 and Corollary 3.2).

In particular, these existence results are shown to apply to Eqs. (1) and (2) considered in the main text. We mention also that the approach presented below can be adapted to the infinite-dimensional setting and thus to the original two-dimensional quasigeostrophic partial differential equation model, by working in the appropriate function spaces to define the corresponding solutions.

The less mathematically inclined reader can skip appendix section a and proceed directly to section b.

a. Rigorous mathematical background

In what follows X denotes the Euclidean space , endowed with its natural inner product , and denotes the closed ball in X centered at zero, with time-dependent radius .

Theorem 1

Consider the following evolution equation for ,
ea1
for which the initial value problem with possesses a unique global solution for any
Assume that the bilinear term B is energy preserving, that is,
ea2
and that there exists such that, for all ,
ea3
Suppose furthermore that belongs to —the space of X-valued, locally square-integrable functions—and that
ea4
with
ea5
for some such that .
Then the propagator U associated with (A1) possesses a unique pullback attractor that pullback attracts any time-dependent set such that , with that satisfies the growth condition
ea6
Remark 1

Inequality (A3) states that the linear terms in Eq. (1) include dissipative effects.

Remark 2
  1. We recall that a propagator on X is a two-parameter family of continuous mappings , such that is the identity operator in X, and that the “multiplication rule”
    ea7
    applies. It is trivial to show that when a nonautonomous ODE system, such as Eq. (A1), possesses a unique global solution for any , then a propagator U is well defined and is actually given by , where denotes the solution at time t of Eq. (A1) that emanates from at time .
  2. A time-dependent set is said to pullback attract the time-dependent set if
    ea8
    where denotes the Hausdorff semidistance in X (Carvalho et al. 2012; Chekroun et al. 2011).

Proof

Let us multiply Eq. (A1) by . Energy conservation by the quadratic term [cf. Eq. (A2)] then yields
ea9
By using the dissipativity condition of Eq. (A3) on and Young’s inequality [Brézis 2010, ch. 2; see Eq. (2) therein and footnote on p. 92] on , we obtain that, for all , there exists a constant such that
ea10
Now by choosing ε sufficiently small for to hold, we obtain, after integration over ,
ea11
If with that satisfies the slow-growth condition of Eq. (A6), then the first term on the right-hand side converges to zero as , ensuring thus the existence of a dissipation time such that, for all ,
ea12

We have thus proved that for , ensuring in turn the desired pullback dissipation. The existence and uniqueness proceed then from this dissipation property and the theory of pullback attractors [e.g., Carvalho et al. (2012)]. ∎

b. Interpretation

We provide here some interpretations of the application of theorem 1 in the context of our idealized climate model governed by Eqs. (1) and (2). First, the dissipativity parameter α of condition (A3) above is given here by
ea13
where denotes the spectrum of the linear operator L. For the parameter regime presented in the main text, we find .

The result of theorem 1 states thus that, for any forcing for which σ defined in (A5) is strictly positive, the double-gyre ocean model of Eq. (1) possesses a pullback attractor in the sense of theorem 1. The constant σ in Eq. (A11) is a measure of the rate of pullback dissipation. This constant is very small, on the order of , which helps explain, as we shall see, the complexity of the model’s phase portrait.

The identity (A8) means that attracts—with respect to the Hausdorff semidistance and in a pullback sense—any time-dependent set of initial states at time τ, as they evolve with the action of the propagator forward in the time t and as the pullback time τ is sent further and further away into the past.

However, the Hausdorff semidistance is not a distance in the classical sense. For instance, implies that the set is included in the set , but does not guarantee the converse. So if the global attractor turns out to be the union of a strange part and of some topologically simpler parts, such as fixed points or periodic orbits, it is definitely possible to have pullback attraction toward the global PBA , while single trajectories land in different regions of that are either chaotic or not. We are then back, in a pullback setting, to the notion of local attractors that compose the global attractor and that can be of topologically very different natures. In the autonomous context, such coexistence of topologically distinct local attractors is well known in the climate sciences (Ghil and Childress 1987; Dijkstra and Ghil 2005; Simonnet et al. 2005, 2009, and references therein), while it is also observed in the numerical results of Figs. 5 and 6 herein for the present ocean model.

As shown in these figures, the boundaries between the corresponding local basins of attractions, as measured with the metrics of section 5, are seemingly intricate and strongly suggest that these objects have fractal features. An attractor basin of chaotic solutions being separated by a fractal boundary from the attractor basin of a fixed point has been documented in a quadratic autonomous model governed by four ODEs that arises in population dynamics; see Fig. 5a in Roques and Chekroun (2011).

In that ecological model, however, no abutting of two different attractor basins of chaotic regions on the global attractor is present. A key difference between the ocean model studied herein and the population dynamics model of Roques and Chekroun (2011) lies in the quadratic terms here being energy preserving [cf. Eq. (A2)], while this is not the case in population dynamics.

There is, therefore, a Hamiltonian skeleton here that may help explain the difference between the two phase portraits: The pullback dissipation being very small argues for the asymptotic trajectories—in spite of the dissipative nature of the dynamics—lying quite close to this skeleton. It is thus not surprising to recover interleaved chaotic islands, as shown in Fig. 6. In the autonomous context, the effects of small dissipation have been studied by Ghil and Wolansky (1992), Feudel and Grebogi (1997), and Seoane et al. (2007), among others.

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