Michael Ghil

Mickaël D Chekroun

Eric Simonnet

Climate dynamics and fluid mechanics: Natural variability and related uncertainties

Keywords: Climate change, physical oceanography, dynamical systems, bifurcations, structural stochastic stability, Arnol'd tongues

The purpose of this review-and-research paper is twofold: (i) to review the role played in climate dynamics by fluid-dynamical models; and (ii) to contribute to the understanding and reduction of the uncertainties in future climate-change projections. To illustrate the first point, we review recent theoretical advances in studying the winddriven circulation of the oceans. In doing so, we concentrate on the large-scale, winddriven flow of the mid-latitude oceans, which 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. The boundary currents and eastward jets carry substantial amounts of heat and momentum, and thus contribute in a crucial way to Earth's climate, and to changes therein.

Changes in this double-gyre circulation occur from year to year and decade to decade. We study this low-frequency variability of the wind-driven, double-gyre circulation in mid-latitude ocean basins, via the bifurcation sequence that leads from steady states through periodic solutions and on to the chaotic, irregular flows documented in the observations. This sequence involves local, pitchfork and Hopf bifurcations, as well as global, homoclinic ones.

The natural climate variability induced by the low-frequency variability of the ocean circulation is but one of the causes of uncertainties in climate projections. The range of these uncertainties has barely decreased, or even increased, over the last three decades. Another major cause of such uncertainties could reside in the structural instability-in the classical, topological sense-of the equations governing climate dynamics, including but not restricted to those of atmospheric and ocean dynamics.

We propose a novel approach to understand, and possibly reduce, these uncertainties, based on the concepts and methods of random dynamical systems theory. The idea is to compare the climate simulations of distinct general circulation models (GCMs) used in climate projections, by applying stochastic-conjugacy methods and thus perform a stochastic classification of GCM families. This approach is particularly appropriate given recent interest in stochastic parametrization of subgrid-scale processes in GCMs.

As a very first step in this direction, we study the behavior of the Arnol'd family of circle maps in the presence of noise. The maps' fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification.

Introduction and motivation

Charney et al. [START_REF] Charney | Carbon Dioxide and Climate: A Scientific Assessment[END_REF] carried out the first attempt to obtain a consensus estimate of the equilibrium sensitivity of climate to changes in atmospheric CO 2 concentrations. The result was the now famous range of 1.5 K to 4.5 K for a doubling of this concentration.

As the relatively new science of climate dynamics evolved through the 1980s and 1990s, it became quite clear -from observational data, both instrumental and paleoclimatic, as well as model studies -that Earth's climate never was and is unlikely to ever be in equilibrium. The three successive IPCC reports (1991 [START_REF]Climate Change, The IPCC Scientific Assessment[END_REF], 1996, and 2001 [START_REF] Houghton | Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC)[END_REF]) concentrated therefore, in addition to estimates of equilibrium sensitivity, on estimates of climate change over the 21st century, based on several scenarios of CO 2 increase over this time interval, and using up to 18 general circulation models (GCMs) in the fourth IPCC Assessment Report (AR4) [START_REF] Solomon | Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the IPCC[END_REF].

The GCM results of temperature increase over the coming 100 years have stubbornly resisted any narrowing of the range of estimates, with results for T s in 2100 as low as 1.4 K or as high as 5.8 K, according to the Third Assessment Report. The hope in the research leading up to the AR4 was that a set of suitably defined "better GCMs" would exhibit a narrower range of year-2100 estimates, but this does not seem to have been the case.

Figure 1: Frequency distributions of global mean, annual mean, near-surface temperature (T g ) for 2,017 GCM simulations, and doubled CO 2 (from [START_REF] Stainforth | Uncertainty in predictions of the climate response to rising levels of greenhouse gases[END_REF]).

The difficulty in narrowing the range of estimates for either equilibrium sensitivity of climate or for end-of-the-century temperatures is clearly connected to the complexity of the climate system, the multiplicity and nonlinearity of the processes and feedbacks it contains, and the obstacles to a faithful representation of these processes and feedbacks in GCMs. The practice of the science and engineering of GCMs over several decades has amply demonstrated that any addition or change in the model's "parametrizations"i.e., of the representation of subgrid-scale processes in terms of the model's explicit, large-scale variables -may result in noticeable changes in the model solutions' behavior.

As an illustration, Fig. 1 shows the sensitivity of an atmospheric GCM, which does not include a dynamical ocean, to changes in its model parameters. Several thousand simulations were performed as part of the "climateprediction.net" experiment [START_REF] Allen | Do-it-yourself climate prediction[END_REF], using perturbations in several parameters of the Hadley Centre's HadAM3 model [START_REF] Pope | The impact of new physical parameterisations in the Hadley Centre climate model -HadAM3[END_REF], coupled to a passive, mixedlayer ocean model. The figure clearly illustrates a wide range of responses to CO 2 doubling, from 2 K to more than 11 K [START_REF] Stainforth | Uncertainty in predictions of the climate response to rising levels of greenhouse gases[END_REF].

The last IPCC report [START_REF] Solomon | Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the IPCC[END_REF] has investigated climate change as a result of various scenarios of CO 2 increase as well, for a set of 18 distinct GCMs. The best estimate of the temperature change at the end of the 21st century from AR4 is about 4.0 • C for the worst scenario of greenhouse-gas increase, namely A1F1, roughly speaking a future world with a very rapid economic growth. The likely range of end-of-century global temperatures is of 2.4-6.4 • C in this case, and comparably large ranges of uncertainties obtain for all other scenarios as well [START_REF] Solomon | Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the IPCC[END_REF].

An essential contributor to this range of uncertainty is natural climate variability [START_REF]Natural Climate Variability on Decade-to-Century Time Scales[END_REF] of the coupled ocean-atmosphere system. As mentioned already in [START_REF] Ghil | Hilbert problems for the geosciences in the 21st century[END_REF], most GCM simulations do not exhibit the observed interdecadal variability of the oceans' buoyancy-driven, thermohaline circulation [START_REF] Dijkstra | Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach[END_REF]. Another striking example of low-frequency, interannual-andinterdecadal variability is provided by the near-surface, wind-driven ocean circulation [START_REF] Dijkstra | Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach[END_REF][START_REF] Dijkstra | Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño[END_REF]. The influence of strong thermal fronts -like the Gulf Stream in the North Atlantic or the Kuroshio in the North Pacific -on the mid-latitude atmosphere above is severely underestimated. Typical spatial resolutions in the century-scale GCM simulations of [START_REF]Climate Change, The IPCC Scientific Assessment[END_REF][START_REF] Houghton | Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC)[END_REF][START_REF] Solomon | Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the IPCC[END_REF][START_REF] Allen | Do-it-yourself climate prediction[END_REF][START_REF] Pope | The impact of new physical parameterisations in the Hadley Centre climate model -HadAM3[END_REF][START_REF] Stainforth | Uncertainty in predictions of the climate response to rising levels of greenhouse gases[END_REF] are of the order of 100 km at best, whereas resolutions of 20 km and less would be needed to really capture the strong mid-latitude ocean-atmosphere coupling just above the oceanic fronts [START_REF] Feliks | Low-frequency variability in the mid-latitude atmosphere induced by an oceanic thermal front[END_REF][START_REF] Feliks | Low-frequency variability in the mid-latitude baroclinic atmosphere induced by an oceanic thermal front[END_REF].

The purpose of this paper is twofold. First, we describe in Section 2 the most recent theoretical results regarding the internal variability of the mid-latitude wind-driven circulation, viewed as a problem in nonlinear fluid mechanics. These results rely to a large extent on the deterministic theory of dynamical systems [START_REF] Arnol | Geometrical Methods in the Theory of Differential Equations[END_REF][START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF]. Second, we address in Section 3 the more general issue of uncertainties in climate change projections. Here we rely on concepts and methods from random dynamical systems theory [START_REF] Arnold | Random Dynamical Systems[END_REF] to help understand and possibly reduce these uncertainties. Much of the material in the latter section is new, and it is supplemented by some rigorous mathematical results in Appendix A. A summary and an outlook on future work follow in Section 4.

Natural variability of the wind-driven ocean circulation 2.1 Observations

To a first approximation, the main oceanic currents are driven by the mean effect of the winds. The trade winds near the equator blow mainly from east to west and are called also the tropical easterlies. In mid-latitudes, the dominant winds are the prevailing westerlies, and towards the poles the winds are easterly again. Three of the strongest near-surface, mid-and-high-latitude currents are the Antarctic Circumpolar Current, the Gulf Stream in the North Atlantic, and the Kuroshio Extension east off Japan. The Antarctic Circumpolar Current, sometimes called the Westwind Drift, circles eastward around Antarctica; see Fig. 2.

Figure 2: A map of the main oceanic currents: warm currents in red and cold ones in blue.

The Gulf Stream is an oceanic jet with a strong influence on the climate of eastern North America and of western Europe. Actually, the Gulf Stream is part of a larger, gyre-like current system, which includes the North Atlantic Drift, the Canary Current and the North Equatorial Current. It is also coupled with the slow pole-to-pole thermohaline or overturning circulation, with cold denser water masses sinking in the subpolar North Atlantic and lighter water rising over much wider areas of the low and southern latitudes. From Mexico's Yucatan Peninsula, the Gulf Stream flows north through the Florida Straits and along the East Coast of the United States. Near Cape Hatteras, it detaches from the coast and begins to drift off into the North Atlantic towards the Grand Banks near Newfoundland.

The Coriolis force is responsible for the so-called Ekman transport, which deflects water masses orthogonally to the near-surface wind direction and to the right [START_REF] Gill | Atmosphere-Ocean Dynamics[END_REF][START_REF] Ghil | Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics[END_REF][START_REF] Pedlosky | Geophysical Fluid Dynamics[END_REF]. In the North Atlantic, this Ekman transport creates a divergence and a convergence of nearsurface water masses, respectively, resulting in the formation of two oceanic gyres: a smaller, cyclonic one in subpolar latitudes, the other larger and anticyclonic in the subtropics. This type of double-gyre circulation characterizes all mid-latitude ocean basins, including the South Atlantic, as well as the North and South Pacific.

The double-gyre circulation is intensified as the currents approach the East Coast of North America due to the β-effect. This effect is mainly due to the variation of the Coriolis force with latitude, while the oceans' bottom topography also contributes to it. The former, planetary β-effect is of crucial importance in geophysical flows and induces free Rossby waves propagating westward [START_REF] Gill | Atmosphere-Ocean Dynamics[END_REF][START_REF] Ghil | Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics[END_REF][START_REF] Pedlosky | Geophysical Fluid Dynamics[END_REF].

The currents along the western shores of the North Atlantic and of the other midlatitude ocean basins exhibit boundary-layer characteristics and are commonly called western boundary currents (WBCs). The northward-flowing Gulf Stream and the southward-flowing Labrador Current extension meet near Cape Hatteras and yield a strong eastward jet. The formation of this jet and of the intense recirculation vortices near the western boundary, to either side of the jet, is mostly driven by internal, nonlinear effects. Figure 3 illustrates how these large-scale wind-driven oceanic flows self-organize, as well as the resulting eastward jet. Different spatial and time scales contribute to this self-organization, mesoscales eddies playing the role of the synoptic-scale systems in the atmosphere. Warm and cold rings last for several months up to a year and have a size of about 100 km; two cold rings are clearly visible in Fig. 3. Meanders involve larger spatial scales, up to 1000 km, and are associated with interannual variability. The characteristic scale of the jet and gyres is of several thou-sand kilometers and they exhibit their own intrinsic dynamics on time scales of several years to possibly several decades.

A striking feature of the wind-driven circulation is the existence of two well-known North-Atlantic oscillations, with a period of about 7 and 14 years, respectively. Data analysis of various climatic variables, such as sea surface temperature (SST) over the North Atlantic or sea level pressure (SLP) over western Europe [START_REF] Wunsch | The interpretation of short climate records, with comments on the North Atlantic and Southern Oscillations[END_REF][START_REF] Da Costa | The 7.7 year North Atlantic oscillation[END_REF] and local surface air temperatures in Central England [START_REF] Plaut | Interannual and interdecadal variability in 335 Years of Central England temperatures[END_REF], as well as of proxy records, such as tree rings in Britain, travertine concretions in southeastern France [START_REF] Dubar | Approche climatique de la période romaine dans l'est du Var : recherche et analyse des composantes périodiques sur un concrétionnement centennal (Ier-IIe siècle apr. J.-C[END_REF], and Nile floods over the last millennium or so [START_REF] Kondrashov | Oscillatory modes of extended Nile River records (A.D. 622-1922[END_REF], all exhibit strikingly robust oscillatory behavior with a 7-yr period and, to a lesser extent, with a 14-yr period. Variations in the path and intensity of the Gulf Stream are most likely to exert a major influence on the climate in this part of the world [START_REF] Stommel | The Gulf Stream: A Physical and Dynamical Description[END_REF]. This is why theoretical studies of the low-frequency variability of the double-gyre circulation are important.

Given the complexity of the processes involved, climate studies have been most successful when using not just a single model but a full hierarchy of models, from the simplest "toy" models to the most detailed GCMs [START_REF] Ghil | Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy[END_REF]. In the following, we describe one of the simplest models of the hierarchy used in studying this problem.

Curl of the wind stress

A simple model of the double-gyre circulation

The simplest model that includes many of the mechanisms described above is governed by the barotropic quasi-geostrophic (QG) equations. The term geostrophic refers to the fact that large-scale rotating flows tend to run parallel to, rather than perpendicular to constantpressure contours; in the oceans, these contours are associated with the deviation from rest of the surfaces of equal water mass, due to Ekman pumping. Geostrophic balance implies in particular that the flow is divergence-free. The term barotropic, as opposed to baroclinic, has a slightly different meaning in geophysical fluid dynamics (GFD) than in engineering fluid mechanics: it means that the model describes a single fluid layer of constant density and therefore the solutions do not depend on depth [START_REF] Gill | Atmosphere-Ocean Dynamics[END_REF][START_REF] Ghil | Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics[END_REF][START_REF] Pedlosky | Geophysical Fluid Dynamics[END_REF].

We consider an idealized, rectangular basin geometry and simplified forcing that mimics the distribution of vorticity contribution by the winds, as sketched to the right of Fig. 3.

In our idealized model, the amounts of subpolar and subtropical vorticity injected into the basin are equal and the rectangular domain Ω = (0, L x ) × (0, L y ) is symmetric about the axis of zero wind stress curl. The barotropic two-dimensional (2-D) QG equations in this idealized setting are:

q t + J(ψ, q) -ν∆ 2 ψ + µ∆ψ = -τ sin 2πy L y , q = ∆ψ -λ -2 R ψ + βy.
(2.1)

Here q and ψ are the potential vorticity and streamfunction, respectively, and the Jacobian J corresponds to the advection of potential vorticity by the flow, J(ψ, q) = ψ x q y -ψ y q x = u•∇q, where u = (-ψ y , ψ x ), x points east and y points north. The physical parameters are the strength of the planetary vorticity gradient β, the Rossby radius of deformation λ -2 R , the eddy-viscosity coefficient ν, the bottom friction coefficient µ, and the wind-stress intensity τ . We use here free-slip boundary conditions ψ = ∆ 2 ψ = 0; the qualitative results described below do not depend on the particular choice of homogeneous boundary conditions.

We consider (2.1) as an infinite-dimensional dynamical system and study its bifurcation sets as the parameters change. Two key parameters are the wind stress intensity τ and the eddy viscosity ν. An important property of (2.1) is its mirror symmetry in the y = L y /2 axis. This symmetry can be expressed as invariance with respect to the discrete Z 2 group S:

S [ψ(x, y)] = -ψ(x, L y -y); (2.2)
any solution of (2.1) is thus accompanied by its mirror-conjugated solution. Hence, in generic terms, the prevailing bifurcations are of either the symmetry-breaking or the saddle-node or the Hopf type.

Bifurcations in the double-gyre problem

The historical development of a comprehensive nonlinear theory of the double-gyre circulation is interesting on its own, having seen substantial progress in the last 15 years. One can distinguish four main steps.

Symmetry-breaking bifurcations

The first step was to realize that the first generic bifurcation of this QG model was a genuine pitchfork bifurcation that breaks the system's symmetry as the nonlinearity becomes large enough [START_REF] Jiang | The nonlinear behavior of western boundary currents in a wind-driven, double-gyre, shallow-water model in[END_REF][START_REF] Jiang | Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model[END_REF][START_REF] Cessi | Symmetry-breaking multiple equilibria in quasigeostrophic, winddriven flows[END_REF]. The situation is shown in Fig. 4. When the forcing is weak or the dissipation is large, there is only one steady solution, which is antisymmetric with respect to the mid-axis of the basin. This solution exhibits two large gyres, along with their typical, βinduced WBCs. Away from the western boundary, such a near-linear solution (not shown) is dominated by Sverdrup balance between wind stress curl and the meridional mass transport [START_REF] Gill | Atmosphere-Ocean Dynamics[END_REF][START_REF] Sverdrup | Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific[END_REF].

As the wind stress increases, the near-linear Sverdrup solution develops an eastward jet along the mid-axis, which penetrates farther into the domain. This more intense, and hence more nonlinear solution is still antisymmetric about the mid-axis, but loses its stability for some critical value of the wind-stress intensity (indicated by "Pitchfork" in Fig. 4). A pair of mirror-symmetric solutions emerges and is characterized by a rather different vorticity distribution; the streamfunction fields associated with the two stable steady-state branches are plotted to the upper-left and right of Fig. 4. In particular, the jet in such a solution exhibits a large meander, reminiscent of the one seen in Fig. 3 just downstream of Cape Hatteras; note that the colors in Fig. 4 have been chosen to facilitate the comparison with The streamfunction field is plotted for a steady-state solution associated with each of the three branches; positive values in red and negative ones in blue (after [START_REF] Simonnet | Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation[END_REF]). Fig. 3. These asymmetric flows are characterized by one gyre being stronger in intensity than the other and therefore the jet is deflected either to the southeast or to the northeast.

✁ ✁ ✁ ✁ ✁ ✁ ✂ ✂ ✂ ✂ ✂ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ☎ ☎ ☎ ☎ ☎ ☎ ✆ ✆ ✆ ✆ ✆ ✆ ✝ ✝ ✝ ✝ ✝ ✝ ✞ ✞ ✞ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠

Gyre modes

The next step was taken in part concurrently with [START_REF] Jiang | The nonlinear behavior of western boundary currents in a wind-driven, double-gyre, shallow-water model in[END_REF][START_REF] Jiang | Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model[END_REF] and in part shortly after [START_REF] Speich | Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation[END_REF][START_REF] Dijkstra | Temporal variability of the wind-driven quasigeostrophic double gyre ocean circulation: basic bifurcation diagrams[END_REF][START_REF] Sheremet | Eigenanalysis of the two-dimensional wind-driven ocean circulation problem[END_REF] the first one. It involved the study of time-periodic instabilities through Hopf bifurcation from either an antisymmetric or an asymmetric steady flow. Some of these studies concentrated on the wind-driven circulation formulated for the stand-alone, single gyre [START_REF] Sheremet | Eigenanalysis of the two-dimensional wind-driven ocean circulation problem[END_REF][START_REF] Pedlosky | Ocean Circulation Theory[END_REF].

The idea was to develop a full generic picture of the time-dependent behavior of the solutions in more turbulent regimes, by classifying the various instabilities in a comprehensive way. However, it quickly appeared that one kind of asymmetric instabilities, called gyre modes [START_REF] Jiang | Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model[END_REF][START_REF] Speich | Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation[END_REF], was prevalent across the full hierarchy of models of the double-gyre circulation; furthermore, these instabilities trigger the lowest nonzero frequency present in these models. At first, it was puzzling that these modes appear after the first pitchfork bifurcation, never before, and it took several years to really understand their genesis: gyre modes appear by two eigenvalues merging, one associated with a symmetric eigenfunction and responsible for the pitchfork bifurcation, the other associated with an antisymmetric eigenfunction [START_REF] Simonnet | Spontaneous generation of low-frequency modes of variability in the wind-driven ocean circulation[END_REF]; this merging is referred to as M in Fig. 4. Such a phenomenon is not a bifurcation stricto sensu: one has topological C 0 equivalence before and after the eigenvalue merging, but not from the C 1 point of view. Still, the phenomenon is quite common in small-dimensional dynamical systems with symmetry, as exemplified by the unfolding of codimension-2 bifurcations of Bogdanov-Takens type [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF]. In particular, the fact that gyre modes trigger the lowestfrequency of the model is due to the frequency of these modes growing quadratically from zero until nonlinear saturation. Of course, these modes, in turn, become unstable shortly after the merging, through a Hopf bifurcation, indicated by "Hopf" in Fig. 4.

Global bifurcations

The importance of these gyre modes was further confirmed recently through an even more puzzling discovery. Several authors realized, independently of each other, that the lowfrequency dynamics of their respective double-gyre models was driven by intense relaxation oscillations of the jet [START_REF] Simonnet | Successive bifurcations in a shallowwater ocean model[END_REF][START_REF] Meacham | Low-frequency variability in the wind-driven circulation[END_REF][START_REF] Chang | Transition to aperiodic variability in a wind-driven double-gyre circulation model[END_REF][START_REF] Nadiga | Global bifurcation of Shilnikov type in a double-gyre ocean model[END_REF][START_REF] Simonnet | Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part I: Steady-state solutions[END_REF][START_REF] Simonnet | Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part II: Time-dependent solutions[END_REF][START_REF] Simonnet | Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation[END_REF]. These relaxation oscillations, already described in [START_REF] Jiang | Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model[END_REF][START_REF] Speich | Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation[END_REF], were now attributed to homoclinic bifurcations, with a global character in phase space [START_REF] Ghil | Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics[END_REF][START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF]. In effect, the QG model reviewed here undergoes a genuine homoclinic bifurcation (see Fig. 4), which is generic across the full hierarchy of double-gyre models. Moreover, this global bifurcation is associated with chaotic behavior of the flow due to the Shilnikov phenomenon [START_REF] Nadiga | Global bifurcation of Shilnikov type in a double-gyre ocean model[END_REF][START_REF] Simonnet | Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation[END_REF], which induces horseshoes in phase space.

The connection between such homoclinic bifurcations and gyre modes was not immediately obvious, but Simonnet et al. [START_REF] Simonnet | Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation[END_REF] emphasized that the two were part of a single, global dynamical phenomenon. The homoclinic bifurcation indeed results from the unfolding of the gyre modes' limit cycles. This familiar dynamical scenario is again well illustrated by the unfolding of a codimension-2 Bogdanov-Takens bifurcation, where the homoclinic orbits emerge naturally. We deal, once more, with the lowest-frequency modes, since homoclinic orbits have an infinite period. Due to the genericity of this phenomenon, it was natural to hypothesize that the gyre-mode mechanism, in this broader, global-bifurcation context, gave rise to the observed 7-yr and 14-yr North-Atlantic oscillations. Although this hypothesis may appear a little farfetched, in view of the simplicity of the double-gyre models analyzed in detail so far, it poses an interesting question.

Quantization and open questions

The chaotic dynamics observed in the QG models after the homoclinic bifurcation is eventually destroyed as the nonlinearity and the resolution both increase. As one expects the real oceans to be in a far more turbulent regime than those studied so far, some authors proposed different mechanisms for low-frequency variability in fully turbulent flow regimes [START_REF] Berloff | The turbulent oscillator: A mechanism of low-frequency variability of the wind-driven ocean gyres[END_REF][START_REF] Kravtsov | Dynamical origin of low-frequency variability in a highly nonlinear mid-latitude coupled model[END_REF]. It turns out, though, that -just as gyre modes could be reconciled with homoclinic-driven dynamics, -the latter can also be reconciled with eddy-driven dynamics, via the so-called quantization of the low-frequency dynamics [START_REF] Simonnet | Quantization of the low-frequency variability of the double-gyre circulation[END_REF].

Primeau [START_REF] Primeau | Multiple equilibria and low-frequency variability of the wind-driven ocean circulation[END_REF] showed that, in large basins comparable in size with the North Atlantic, there is not only one but a set of successive pitchfork bifurcations. One supercritical pitchfork bifurcation, associated with the destabilization of antisymmetric flows, is followed generically by a subcritical one, associated this time with a stabilization of antisymmetric flows (modulo high-frequency instabilities) [START_REF] Simonnet | Quantization of the low-frequency variability of the double-gyre circulation[END_REF]. As a matter of fact, this phenomenon appears to be a consequence of the spectral behavior of the 2-D Euler equations [START_REF] Simonnet | On the unstable spectrum of the linearized 2-D Euler equations in bounded domains[END_REF], and hence of the closely related barotropic QG model in bounded domains.

Remarkably, this scenario repeats itself as the nonlinearity increases, but now higher wavenumbers are involved in physical space. Simonnet [START_REF] Simonnet | Quantization of the low-frequency variability of the double-gyre circulation[END_REF] showed that this was also the case for gyre modes and the corresponding dynamics induced by global bifurcations: the low-frequency dynamics is quantized as the jet stream extends further eastward into the basin, due to the increased forcing and nonlinearity. Figure 5 families of regimes can be identified, the colored bands correspond to (supercritical) regimes driven by the gyre modes, the others to (subcritical) regimes driven by the eddies. Note that this scenario is also robust to perturbing the problem's symmetry. The theory appears therewith to be fairly complete for barotropic, single-layer models of the double-gyre circulation. In baroclinic models, with two or more active layers of different density, baroclinic instabilities [START_REF] Dijkstra | Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach[END_REF][START_REF] Feliks | Low-frequency variability in the mid-latitude baroclinic atmosphere induced by an oceanic thermal front[END_REF][START_REF] Gill | Atmosphere-Ocean Dynamics[END_REF][START_REF] Ghil | Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics[END_REF][START_REF] Pedlosky | Geophysical Fluid Dynamics[END_REF][START_REF] Stommel | The Gulf Stream: A Physical and Dynamical Description[END_REF][START_REF] Pedlosky | Ocean Circulation Theory[END_REF][START_REF] Simonnet | Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part II: Time-dependent solutions[END_REF][START_REF] Berloff | The turbulent oscillator: A mechanism of low-frequency variability of the wind-driven ocean gyres[END_REF][START_REF] Kravtsov | Dynamical origin of low-frequency variability in a highly nonlinear mid-latitude coupled model[END_REF] surely play a fundamental role, as they do in the observed dynamics of the oceans. However, it is not known to what extent baroclinic instabilities can destroy gyre-mode dynamics. The difficulty lies in a deeper understanding of the so-called rectification process [START_REF] Katsman | The rectification of the wind-driven ocean circulation due to its instabilities[END_REF], which arises from the nonzero mean effect of the baroclinic component of the flow.

Roughly speaking, rectification drives the dynamics far away from any steady states. In this situation, dynamical systems theory cannot be used as an explanation of complex, observed behavior resulting from successive bifurcations that are rooted in a simple steady state. Other tools from statistical mechanics and nonequilibrium thermodynamics should, therefore, be considered [START_REF] Robert | Statistical equilibrium states for two-dimensional flows[END_REF][START_REF] Trefethen | Hydrodynamic stability without eigenvalues[END_REF][START_REF] Bouchet | Emergence of intense jets and Jupiter's great red spot as maximum entropy structures[END_REF][START_REF] Farrel | Structural stability of turbulent jets[END_REF]. Combining these tools with those of the successivebifurcation approach may eventually lead to a more general and complete physical characterization of gyre modes in realistic models.

Climate-change projections and random dynamical systems

As discussed in Section 1, the climate system's internal low-frequency variability is not the only cause for the uncertainties in projecting its future evolution. In this section, we address more generally these uncertainties and present a novel approach for treating them. To do so, we start with some simple ideas about deterministic vs. stochastic modeling.

Motivation

Many physical phenomena can be modeled by deterministic evolution equations. Dynamical systems theory is essentially a geometric approach for studying the asymptotic properties of solutions to such equations in phase space. Pioneered by H. Poincaré [START_REF] Poincaré | Sur les équations de la dynamique et le problème des trois corps[END_REF], this theory took major strides over the last fifty years. The extent to which a particular dynamical system or class of such systems could adequately reflect those of the natural phenomena of interest is still an open question, though. Some concept of model robustness seems of the essence in lending credence to the physical relevance of a mathematical or numerical result. In this context, Andronov and Pontryagin [START_REF] Andronov | Systèmes grossiers[END_REF] took a major step toward classifying dynamical systems, by introducing the concept of structural stability. This topological concept essentially stipulates that there exist a one-to-one continuous change of variables, called homeomorphism, that transforms the phase portrait of a system into that of another one. Closely related to this concept is the notion of hyperbolicity introduced by Smale [START_REF] Smale | Differentiable dynamical systems[END_REF]. A system is hyperbolic if, (very) loosely speaking, its limit set can be continuously decomposed into contracting and expanding invariant sets; see [START_REF] Katok | Introduction to the Modern Theroy of Dynamical Systems[END_REF] for more rigorous definitions.

A very simple example is the phase portrait in the neighborhood of a fixed point of saddle type. In this case, the Hartman-Grobman theorem states that the dynamics in this neighborhood is structurally stable. The converse statement, i.e. whether structural stability implies hyperbolicity, is still an open question; it is true in the C 1 setting for diffeomorphisms that satisfy Axiom A and the strong transversality condition [START_REF] Robbin | A structural stability theorem[END_REF][START_REF] Robinson | Structural stability of C 1 diffeomorphisms[END_REF][START_REF] Mañé | A proof of the C 1 -stability conjecture[END_REF][START_REF] Palis | A global perspective for non-conservative dynamics[END_REF]. The concepts and tools of bifurcation theory are well grounded in the setting of hyperbolic dynamics. Problems with hyperbolicity and bifurcations arise, however, when one deals with much more complicated limit sets.

Hyperbolicity was introduced initially to help pursue the "dream" to find an open and dense set of dynamical systems that are structurally stable: i.e. systems that even when perturbed slightly preserve the same (homeomorphic) dynamics. Smale conjectured that hyperbolic systems form an open and dense set in the space of all C 1 dynamical systems. If this conjecture were true then, by density and robustness, hyperbolicity would be typical of dynamics. Unfortunately, though, this conjecture is only true for one-dimensional dynamics and flows on disks and surfaces [START_REF] Peixoto | Structural stability on two-dimensional manifolds[END_REF]. Smale himself found several counterexamples to his conjecture [START_REF] Smale | Structurally stable systems are not dense[END_REF]. One famous example is due to Newhouse [START_REF] Newhouse | The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms[END_REF] and another one is given by the Lorenz attractor [START_REF] Guckenheimer | Structural stability of Lorenz attractors[END_REF]. Newhouse was able to generate open sets of nonhyperbolic diffeomorphisms using homoclinic tangencies. For the physicist, it is even more striking that the famous Lorenz attractor [START_REF] Lorenz | Deterministic nonperiodic flow[END_REF] is structurally unstable. Families of Lorenz attractors, classified by topological type, are not even countable [START_REF] Williams | The structure of Lorenz attractors[END_REF].

Nonhyperbolic chaos appears, therefore, to be a severe obstacle to any "easy" classification of dynamic behavior. As mentioned by Palis [START_REF] Palis | A global perspective for non-conservative dynamics[END_REF], Kolmogorov already suggested at the end of the sixties that "the global study of dynamical systems could not go very far without the use of new additional mathematical tools, like probabilistic ones." Once more, Kolmogorov showed prophetic insight, and nowadays the concept of stochastic stability is an important tool in the study of genericity and robustness for dynamical systems. To replace the failed program of classifying dynamical systems based on structural stability and hyperbolicity, Palis [START_REF] Palis | A global perspective for non-conservative dynamics[END_REF] formulated what is now called the global conjecture: systems having only finitely many attractors (chaotic or periodic sinks) -such that (i) the union of their basins has total Lebesgue probability, and (ii) they are stochastically stable in their basins of attraction -are dense in the C r , r ≥ 1 topology. A system is stochastically stable if its Sinai-Ruelle-Bowen (SRB) measure [START_REF] Sinai | Gibbs measures in ergodic theory[END_REF] is stable with respect to stochastic perturbations; the SRB measure is given by lim n→∞ 1 n i δ zi , with z i being the successive iterates of the dynamics, and it corresponds intuitively to allowing the entire phase space to flow onto the attractor [START_REF] Eckmann | Ergodic theory of chaos and strange attractors[END_REF].

Stochastic stability favors metric approaches and is fundamentally based on ergodic theory. We would like to consider a more geometric approach, which can provide a coarser, more robust classification of GCMs and their climate-change projections. We propose, therefore, such a geometric approach, based on concepts from the rapidly growing field of random dynamical systems (RDS), as developed by L. Arnold [START_REF] Arnold | Random Dynamical Systems[END_REF] and his "Bremen group," among others. RDS in general describe the behavior of dynamical systems subject to external stochastic forcing. The tools of RDS theory have been developed to help study the geometric properties of stochastic differential equations (SDEs). In some sense, they are the stochastic counterpart of the geometric theory of ordinary differential equations (ODEs). This approach provides a rigorous mathematical framework for a stochastic form of robustness, while the more traditional, topological concepts do not seem to be appropriate.

RDS, random attractors, and robust classification

A central idea in developing stochastic parametrizations for GCMs is that these are aimed at compensating for our lack of knowledge in the best way possible [START_REF] Lin | Influence of a stochastic moist convective parameterization on tropical climate variability[END_REF][START_REF] Lin | Considerations for stochastic convective parameterization[END_REF][START_REF] Lin | Toward stochastic deep convective parameterization in general circulation models[END_REF][START_REF] Palmer | The prediction of uncertainty in weather and climate forecasting[END_REF]73,[START_REF] Stevens | Stochastic effects in the representation of stratocumulus-topped mixed layers[END_REF]. Since such parametrizations deal typically with small space scales, the underlying assumption is that the associated time scales are also much shorter than the scales of interest and therefore the time correlation of the phenomena being parametrized is negligibly small. Bearing this in mind, stochastic parametrizations essentially transform a deterministic autonomous system into a nonautonomous one, subject to random forcing.

The nonautonomous character of a dynamical system immediately raises a technical difficulty. Indeed, the classical notion of attractor is no longer valid since any object in phase space is "moving" with time and the natural concept of forward asymptotics is meaningless. One needs therefore another concept of attractor. In the deterministic nonautonomous framework, this concept corresponds to the pullback attractor [START_REF] Langa | Stability, instability, and bifurcation phenomena in non-autonomous differential equations[END_REF] that we present below.

Framework and objectives

Before defining the notion of pullback attractor, let us recall some basic important facts about nonautonomous dynamical systems. Consider the nonautonomous ODE ẋ = f (t, x) on a Banach space X. Rigorously speaking, we cannot associate a dynamical system acting on X with a nonautonomous ODE; nevertheless, in the case of unique solvability of the initial-value problem, we can introduce a two-parameter family of operators {S(t, s)} t≥s acting on X with s, t ∈ R, such that S(t, s)x(t) = x(s), t ≥ s, where x(t) is the solution of the Cauchy problem with initial data x(s). This family of operators satisfies S(s, s) = Id X , for all s ∈ R and S(t, τ ) • S(τ, s) = S(t, s) for all t ≥ τ ≥ s, s ∈ R. This family of operators is called a "process" by Sell [START_REF] Sell | Non-autonomous differential equations and dynamical systems[END_REF]. It extends the classical notion of resolvent from autonomous to nonautonomous linear ODEs.

We can now define the pullback attractor as simply the family of invariant sets {A(t)} that satisfy for every t ∈ R :

lim s→-∞ dist (S(t, s)x 0 , A(t)) = 0, for all x 0 ∈ X.
The idea of "pullback" attraction does not involve running time backwards; it corresponds instead to the idea that measurements in an experiment performed at present time t, started some time in the past, at a time s < t: the experiment has been running for long enough, and we are thus looking now at an "attracting state." Note that there exists several ways of defining a pullback attractor -the one retained here is a local one (cf. [START_REF] Langa | Stability, instability, and bifurcation phenomena in non-autonomous differential equations[END_REF] and references therein); for further information on nonautonomous dynamical systems in general, see [START_REF] Berger | On the gap between random dynamical systems and continuous skew products[END_REF].

We shall use next the concept of pullback attractor to define a random attractor, but need first to define an RDS. We denote by T one of the following sets: N, Z, R + , R. Let (X, B) be a measurable phase space, and (Ω, F, P, (θ(t)) t∈T ) be a metric dynamical system i.e. a flow in the probability space (Ω, F, P), such that (t, ω) → θ(t)ω is measurable and θ(t) : Ω → Ω is measure preserving, θ(t)P = P. Let ϕ : T × Ω × X → X, (t, ω, x) → ϕ(t, ω)x, be a mapping with the following two properties: (R 1 ): ϕ(0, ω) = Id X , and (R 2 ) (the cocycle property): For all s, t ∈ T and all ω ∈ Ω,

ϕ(t + s, ω) = ϕ(t, θ(s)ω) • ϕ(s, ω).
If ϕ is measurable, then it is called a measurable RDS over θ. If in addition, X is a topological space (respectively a Banach space), and ϕ satisfies (t, ω) → ϕ(t, ω)x continuous (resp.

C k , 1 ≤ k ≤ ∞) for all (t, ω) ∈ T × Ω, then ϕ is called a continuous (resp. C k ) RDS over the flow θ. If so, then (ω, x) → Θ(t)(x, ω) := (θ(t)ω, ϕ(t, ω)x),
is a (measurable) flow on Ω × X, the so-called skew-product of θ and ϕ. In the sequel, we shall use the terms "RDS" or "cocycles" synonymously. The choice of a metric dynamical system θ is a crucial step in this set-up and is dictated by external constraints specific to the problem at hand; as far as stochastic classification is concerned, it does play an important role [START_REF] Cong | Topological Dynamics of Random Dynamical Systems[END_REF].

In this stochastic context, the concept of random attractor is a natural and straightforward extension of the definition of pullback attractor, in which Sell's [START_REF] Sell | Non-autonomous differential equations and dynamical systems[END_REF] process is replaced by a cocycle, cf. Fig. 6, and A now depends on the realization ω of the noise, so that we have a family of random attractors A(ω), cf. Fig. 7. Roughly speaking, for a fixed realization of the noise, one "rewinds" the noise back to t → -∞ and lets the experiment evolve (forward in time) towards a possibly attracting set A(ω); the driving system θ enables one to do this rewinding without changing the statistics, cf. Figs. 6 and7.

Figure 6: Random dynamical systems (RDS) viewed as a flow on the bundle X × Ω = "dynamical space" × "probability space." For a given position x and realization ω, the RDS ω A( )

ϕ is such that Θ(t)(x, ω) = (θ(t)ω, ϕ(t, ω)x) is a flow on the bundle. {θ( )ω}x t X {ω}x X θ( )ω t A( ) ω ϕ( ,ω) t A( )= ω t θ( )
Ω ω Pullback attraction to A( ) ω 2 θ(-τ )ω θ(-τ )ω 1 θ(-τ )ω) B( θ(-τ )ω) B( 1 2
Figure 7: Schematic diagram of a random attractor A(ω), where ω ∈ Ω is a fixed realization of the noise. To be attracting, ∀B ∈ B, with B ⊂ 2 X being a collection of subsets of X, one must have lim t→+∞ dist(B(θ(-t)ω), A(ω)) = 0 with B(θ(-t)ω) := ϕ(t, θ(-t)ω)B; to be invariant, one must have ϕ(t, ω)A(ω) = A(θ(t)ω). This definition depends strongly on B; see [START_REF] Crauel | A uniformly exponential attractor which is not a pullback attractor[END_REF] for more details.

Other notions of attractor can be defined in the stochastic context, in particular based on the original SDE; see [START_REF] Crauel | Attractors for random dynamical systems[END_REF] or [START_REF] Crauel | A uniformly exponential attractor which is not a pullback attractor[END_REF] for a discussion on this topic. The present definition, though, will serve us well.

Having defined the concepts of RDS and random attractors, we now introduce the notion of stochastic conjugacy or equivalence. This notion enables one to rigourously compare two distinct RDS and it is defined as follows: two cocycles ϕ 1 (ω, t) and ϕ 2 (ω, t) are conjugated if and only if there exists a random homeomorphism h ∈ Homeo(X) and an invariant set such that h(ω)(0) = 0 and

ϕ 1 (ω, t) = h(θ(t)ω) -1 • ϕ 2 (ω, t) • h(ω). (3.1)
Stochastic equivalence extends classic topological conjugacy to the bundle space X × Ω, where there exists a one-to-one (stochastic) change of variables that transforms one phase portrait into the other.

Returning now to our main objective, suppose for instance that one is presented with results from two distinct GCMs, say two probability distributions functions (PDFs) of the temperature or precipitation in a given area. These two PDFs are generated, typically, by an ensemble of each GCM's simulations, as described in the introduction, and they are likely to differ in their spatial pattern. To ascertain the physical significance of this discrepancy, one needs to know how each GCM result varies as either a parametrization or a parameter value are changed.

In order to consider the difficult question of why GCM responses might differ, one idea is to investigate the structure of the space of all GCMs. We mean therewith the space of all deterministic GCMs, when their stochastic parametrizations are switched off. We know, by now, from experience with GCM results over several decades -including the four IPCC assessment reports [START_REF]Climate Change, The IPCC Scientific Assessment[END_REF][START_REF] Houghton | Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC)[END_REF][START_REF] Solomon | Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the IPCC[END_REF] and the climateprediction.net exercise [START_REF] Allen | Do-it-yourself climate prediction[END_REF][START_REF] Pope | The impact of new physical parameterisations in the Hadley Centre climate model -HadAM3[END_REF][START_REF] Stainforth | Uncertainty in predictions of the climate response to rising levels of greenhouse gases[END_REF] -that there is enormous scatter in this space; see also [START_REF] Held | The gap between simulation and understanding in climate modeling[END_REF][START_REF] Mcwilliams | Irreducible imprecision in atmospheric and oceanic simulations[END_REF]. Our question, therefore, is: can we achieve a genuine, robust classification of GCMs when stochastic parametrizations are used and for a given level of the noise?

As mentioned in Section 3.1, such a classification is not feasible by restricting ourself to deterministic systems and topological concepts. As one switches on stochastic parametrizations [START_REF] Lin | Influence of a stochastic moist convective parameterization on tropical climate variability[END_REF][START_REF] Lin | Considerations for stochastic convective parameterization[END_REF][START_REF] Lin | Toward stochastic deep convective parameterization in general circulation models[END_REF][START_REF] Palmer | The prediction of uncertainty in weather and climate forecasting[END_REF]73,[START_REF] Stevens | Stochastic effects in the representation of stratocumulus-topped mixed layers[END_REF], the situation might change, and hopefully improve, dramatically. Roughly speaking, as the level of the noise becomes very large, it is expected that the models' deterministic behavior will be completely destroyed, and all results would cluster into one huge, diffuse clump. We would like, therefore, to investigate how a classification based on stochastic equivalence evolves as the level of the noise or the stochastic parametrizations change. As the noise tends to zero, do we recover the "granularity" of the set of all deterministic dynamical systems? This idea is schematically represented in Fig. 8: for a given level of the noise, we expect the space of all GCMs to be decomposed into a possibly finite number of classes. Within one of these classes, all the GCMs are topologically equivalent in the stochastic sense defined above; see Eq. (3.1).

Serious difficulties might arise in this program, due to the presence of nonhyperbolic chaos in climate models. Several researches have pointed out that even the characteristics of noisy hyperbolic chaos may strongly depend on the intensity of the noise and its statistics [START_REF] Anishchenko | Effect of noise on the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors[END_REF][START_REF] Anishchenko | Influence of noise on statistical properties of nonhyperbolic attractors[END_REF][START_REF] Anishchenko | Peculiarities of the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors in the presence of noise[END_REF][START_REF] Grebogi | Unexpected robustness-againstnoise of a class of nonhyperbolic chaotic attractors[END_REF].

Such issues go well beyond the setting of this paper and are left for further investigation. Much more modestly, we will study here whether, in certain very simple cases, the conjectural view of Fig. 8 might be relevant for some dynamical systems that are "metaphors" of climate dynamics. The following subsection is dedicated to the study of such a metaphorical object, namely the Arnol'd circle map.

The stochastically perturbed circle map

To go further than our pictorial view of stochastic classification for GCMs in Fig. 8, we study now the effect of noise on a family of circle diffeomorphisms. This toy model exhibits two interesting features for our purpose. The first one is that the two-parameter family {F τ,ǫ } defined by Eq. (3.2) below exhibits an infinite number of topological classes (cf. chap 3 of [START_REF] Arnol | Geometrical Methods in the Theory of Differential Equations[END_REF]). The second feature of great interest is the frequency-locking behavior observed

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✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ 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✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✟ ✟ ✟ ✟ ✟ ✟ 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✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✒ ✒ ✓ ✓ ✔ ✔ ✔ ✔ ✕ ✕ ✕ ✕ ✖ ✖ ✖ ✗ ✗ ✗ ✘ ✘ ✘ ✘ ✙ ✙ ✙ ✙ ✚ ✚ ✚ ✚ ✚ ✚ ✛ ✛ ✛ ✛ ✛ ✛ ✜ ✜ ✜ ✜ ✜ ✜ ✢ ✢ ✢ ✢ ✢ ✢ ✣ ✣ ✣ ✣ ✣ ✣ ✤ ✤ ✤ ✤ ✤ ✤ ✥ ✥ ✥ ✥ ✦ ✦ ✦ ✦ ✧ ✧ ✧ ✧ ★ ★ ★ ★ ✩ ✩ ✩ ✩ ✪ ✪ ✪ ✪ ✫ ✫ ✫ ✫ ✬ ✬ ✬ ✬ ✭ ✭ ✭ ✭ ✭ ✭ ✮ ✮ ✮ ✮ ✮ ✮ ✯ ✯ ✯ ✰ ✰ ✰ ✱ ✱ ✱ ✱ ✱ ✱ ✲ ✲ ✲ ✲ ✲ ✲ ✳ ✳ ✳ ✳ ✴ ✴ ✴ ✴ ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹ ✹ ✹ ✹ ✹ ✺ ✺ ✺ ✺ ✺ ✺ ✻ ✻ ✻ ✼ ✼ ✼ ✽ ✽ ✽ ✽ ✽ ✽ ✾ ✾ ✾ ✾ ✿ ✿ ✿ ✿ ✿ ✿ ❀ ❀ ❀ ❀ ❀ ❀ ❁ ❁ ❁ ❁ ❁ ❁ ❂ ❂ ❂ ❂ ❂ ❂ ❃ ❃ ❃ ❃ ❃ ❃ ❄ ❄ ❄ ❄ ❅ ❅ ❅ ❆ ❆ ❆ ❇ ❇ ❇ ❇ ❇ ❇ ❈ ❈ ❈ ❈ ❈ ❈ ❉ ❉ ❉ ❉ ❊ ❊ ❊ ❊ ❋ ❋ ❋ ❋ ❋ ❋ • • • • • • ❍ ❍ ❍ ❍ ■ ■ ■ ■ ❏ ❏ ❏ ❏ ❑ ❑ ❑ ❑ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ◆ ◆ ◆ ◆ ❖ ❖ ❖ ❖ ❖ ❖ ◗ ◗ ◗ ◗ ◗ ◗ | | | | ❙ ❙ ❙ ❙
Figure 8: A conjectural view of stochastic classification for GCMs, using the concept of random attractors. Each point in red represents a GCM in which stochastic parametrizations are switched off, while each colored area represents a cluster of stochastically equivalent GCMs for a given level of the noise.

in many field of physics in general [START_REF] Feigenbaum | Quasiperiodicity in dissipative systems: A renormalization group analysis[END_REF][START_REF] Bak | One-dimensional Ising model and the complete devil's staircase[END_REF][START_REF] Bak | The devil's staircase[END_REF] and in some El-Niño/Southern-Oscillation (ENSO) models in particular [START_REF] F.-F. Jin | El Niño on the Devil's Staircase: Annual subharmonic steps to chaos[END_REF][START_REF] F.-F. Jin | El Niño/Southern Oscillation and the annual cycle: Subharmonic frequency locking and aperiodicity[END_REF][START_REF] Tziperman | El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator[END_REF][START_REF] Tziperman | Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos[END_REF][START_REF] Saunders | A Boolean delay equation model of ENSO variability[END_REF][START_REF] Ghil | A delay differential model of ENSO variability: Parametric instability and the distribution of extremes[END_REF]. Studying the effect of the noise on those two properties has, therefore, physical and mathematical, as well as climatological relevance.

Many physical and biological systems exhibit interference effects due to competing periodicities. One such effect is mode locking, which is due to nonlinear interaction between an "internal" frequency ω i of the system and an "external" frequency ω e . In the ENSO case, the external periodicity is the seasonal cycle. A simple model for systems with two competing periodicities is the well-known Arnol'd family of circle maps

x n+1 = F τ,ǫ (x n ) := x n + τ -ǫ sin(2πx n ) mod 1, (3.2)
where basically τ := ω i /ω e and ǫ parameterizes the magnitude of nonlinear effects; the map (3.2) is often called the standard circle map [START_REF] Arnol | Geometrical Methods in the Theory of Differential Equations[END_REF]. These maps also represent frequency locking near a bifurcation of Neimark-Sacker type (e.g. [START_REF] Yu | Elements of Applied Bifurcation Theory[END_REF], p. 434); here the parameter τ is typically interpreted as the novel (internal) frequency involved in the bifurcation and ǫ corresponds to the nonlinearity near the bifurcation. This kind of two-frequency behavior gives rise to a characteristic pattern, in the plane of ǫ vs. τ , called Arnol'd tongues. This pattern is computed numerically and shown in Fig. 9 for the family defined by (3.2), together with a cross-section at a fixed value of ǫ. This crosssection exhibits the so-called Devil's staircase, with "steps" on which the rotation number [START_REF] Poincaré | Sur les équations de la dynamique et le problème des trois corps[END_REF] is constant within each Arnol'd tongue; the rotation number measures the average rotation per iterate of (3.2).

For zero ǫ, two types of phenomena occur: either τ is rational and in this case the dynamics is periodic with period q, where τ = p/q, or τ is irrational and the iterates {x n } fill the whole circle densely. As ǫ increases, an Arnol'd tongue of increasing width grows out of each τ = p/q on the abscissa ǫ = 0. It follows that, in this very simple case, such an Arnol'd tongue corresponds to hyperbolic dynamics that is robust to perturbations, as verified by linearizing the map at the periodic point; the rotation number is then rational and equal to p/q.

The set of all these tongues is dense within the whole circle map family, while the Lebesgue measure of this set, at given ǫ, tends to zero as ǫ goes to zero. On the contrary, if a point in the (τ, ǫ)-plane does not belong to an Arnol'd tongue, the rotation number for those parameter values is irrational and the dynamics is nonhyperbolic; the latter fact follows, for instance, from a theorem of Denjoy [START_REF] Denjoy | Sur les courbes définies par les équations différentielles à la surface du tore[END_REF] showing that such dynamics is smoothly equivalent to an irrational rotation. The probability to observe nonhyperbolic dynamics tends therewith to unity as ǫ goes to zero. One has, therefore, a countably infinite number of distinct topological classes, namely the Arnol'd tongues p/q, and an uncoutably infinite number of maps with irrational rotation numbers.

What happens when noise is added in Eq. (3.2)? We consider here the case of additive forcing by a noise process obtained via sampling at each iterate n a random variable with uniform density and intensity σ. Experiments with colored, rather than white noise and multiplicative, rather than additive noise led to the same qualitative results. The results for additive white noise are shown in Fig. 10 for three different levels of noise intensity σ.

As expected, only the largest tongues survive the presence of the noise; in particular, there is only a finite number of surviving tongues, shown in red in Fig. 10. Within such a surviving tongue, the random attractor A(ω) is a random periodic cycle of period q (not shown). In the blue region outside the Arnol'd tongues, the random attractor is a fixed but random point A(ω) = {a(ω)}: if one starts a numerical simulation for a fixed realization of the noise ω 0 , all initial data x converge to the same fixed point a 0 , say; see Fig. 11 in the appendix.

This striking result tells us that noise destroys the irrational character of the dynamics; see the appendix: fractal structures, like the Devil's staircase in Fig. 9, are likely to disappear in the presence of noise. Of course the generality and scope of such a conclusion need considerable further study, numerical as well as rigorously mathematical.

As shown in the lower panel of Fig. 10, there is also a direct relationship between the random dynamics and the support of the PDF on the circle. For a given noise level, this support can either be the union of a finite number of disjoint intervals (red and blue curves) or it can fill the whole circle (black curve). The random attractor is, accordingly, either a random periodic orbit, with the disjoint intervals being visited in succession, or a random fixed point; this PDF behavior characterizes the level of the noise needed to destroy a given tongue. An exact definition of random fixed point and random periodic orbit with period q is given in the appendix. work of RDS theory and rigorous studies on random circle maps. This theoretical analysis helps clarify the interaction between noise and nonlinear dynamics in the context of the classification problem we are interested in and it is provided in Appendix A.

Concluding remarks

We recall that Section 2 dealt with the natural, interannual and interdecadal variability of the ocean's wind-driven circulation. The oceans' internal variability is an important source of uncertainty in past-climate reconstructions and future-climate projections [START_REF]Natural Climate Variability on Decade-to-Century Time Scales[END_REF][START_REF] Ghil | Hilbert problems for the geosciences in the 21st century[END_REF][START_REF] Dijkstra | Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach[END_REF][START_REF] Dijkstra | Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño[END_REF]. In Section 3 and Appendix A, we dealt more generally with the problem of structural instability as a possible cause for the stubborn tendency of the range of uncertainties in climate change projections to increase, rather than diminish over the last three decades [START_REF] Charney | Carbon Dioxide and Climate: A Scientific Assessment[END_REF][START_REF]Climate Change, The IPCC Scientific Assessment[END_REF][START_REF] Houghton | Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC)[END_REF][START_REF] Solomon | Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the IPCC[END_REF]. We summarize here the main results of the two sections in succession, and outline several open problems.

The wind-driven double-gyre circulation dominates the near-surface flow in the oceans' mid-latitude basins. Particular attention was paid to the North Atlantic and North Pacific, traversed by the best-known oceanic jets, namely the Gulf Stream and the Kuroshio Extension (see Fig. 2). The wind-driven circulation exhibits very rich internal dynamics and multiscale behavior associated with turbulent mesoscales (see Fig. 3). Aside from the intrinsic interest of this problem in physical oceanography, these major oceanic currents help regulate the climate of the adjacent continents, while their low-frequency variability affects past, present and furture global climate.

Thanks in part to the systematic use of dynamical systems theory, a comprehensive understanding of simple, barotropic, quasi-geostrophic (QG) models of the double-gyre circulation has been achieved over the last two decades, and was reviewed in Section 2 here. In particular, the importance of symmetry-breaking and homoclinic bifurcations (see Fig. 4) in explaining the observed low-frequency variability has been validated across a wide hierarchy of models, including models with much more comprehensive physical formulation, more realistic geometry, and greater resolution in the horizontal and vertical [START_REF] Dijkstra | Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach[END_REF][START_REF] Dijkstra | Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño[END_REF].

The next challenge in physical oceanography, as well as in fluid mechanics in general, is to reconcile the points of view of dynamical systems theory and statistical mechanics in describing the interaction between the largest scales of motion and geostrophic mesoscale turbulence, which is fully captured in baroclinic QG models. We emphasize that the complexity of these models of the double-gyre circulation is intermediate between high-end GCMs and simple "toy" models; these models offer, therefore, an ideal laboratory to test our ideas. In particular, stochastic parametrizations of the rectification process, absent in barotropic QG models, could be studied using some of the concepts and tools from RDS theory presented here. Note that the RDS approach has already been used in the context of stochastic partial differential equations, in particular for showing the existence of random attractors, as well as stable, unstable and inertial manifolds. Thus RDS concepts and tools are not restricted to finite-dimensional systems [START_REF] Caraballo | A stochatic pitchfork bifurcation in a reaction-diffusion equation[END_REF][START_REF] Duan | Smooth stable and unstable manifolds for stochastic evolutionary equations[END_REF][START_REF] Duan | Invariant manifolds for stochastic partial differential equations[END_REF].

In Section 3, we have addressed the range-of-uncertainty problem for IPCC-class GCM simulations (see Fig. 1) by considering them as stochastically perturbed dynamical systems. This approach is consonant with recent interest for stochastic parametrizations in the highend modeling-and-simulation community [START_REF] Lin | Influence of a stochastic moist convective parameterization on tropical climate variability[END_REF][START_REF] Lin | Considerations for stochastic convective parameterization[END_REF][START_REF] Lin | Toward stochastic deep convective parameterization in general circulation models[END_REF][START_REF] Palmer | The prediction of uncertainty in weather and climate forecasting[END_REF]73,[START_REF] Stevens | Stochastic effects in the representation of stratocumulus-topped mixed layers[END_REF]. We have emphasizedbased on rigorous mathematical results from the dynamical systems literature over the last few decades -that, in the absence of stochastic ingredients, GCMs as well as simpler models, on the lower rungs of the modeling hierarchy [START_REF] Ghil | Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy[END_REF], are bound to differ from each other in their results.

This sensitivity follows from the fact that deterministic, hyperbolic systems -for which the stable and unstable manifolds are in general position, i.e. roughly speaking satisfy Axiom A plus strong transversality assumptions -are essentially the only structurally stable ones, at least in the C 1 case [START_REF] Robbin | A structural stability theorem[END_REF][START_REF] Robinson | Structural stability of C 1 diffeomorphisms[END_REF][START_REF] Mañé | A proof of the C 1 -stability conjecture[END_REF][START_REF] Palis | A global perspective for non-conservative dynamics[END_REF]. Thus, because hyperbolic systems are not dense in the set of smooth deterministic ones [START_REF] Smale | Structurally stable systems are not dense[END_REF], we are led to conclude that the topological, structuralstability approach does not guarantee deterministic-model robustness, in spite of its many valuable insights so far. Related issues for GCM modeling were emphasized recently by Held [START_REF] Held | The gap between simulation and understanding in climate modeling[END_REF] and McWilliams [START_REF] Mcwilliams | Irreducible imprecision in atmospheric and oceanic simulations[END_REF] in a speculative mode.

We have proposed in this paper to go one step further, by considering the difficult question of model robustness in the presence of stochastic terms; such term could represent either parametrizations of unresolved processes in GCMs or stochastic components of natural or anthropogenic forcing, such as volcanic eruptions or fluctuations in greenhouse gas or aerosol emissions. Despite the obvious gap between idealized models and high-end simulations, we have addressed the problem here in the light of random dynamical systems (RDS) theory [START_REF] Arnold | Random Dynamical Systems[END_REF].

In this framework, we have considered a robustness criterion that could replace structural stability, through the concept of stochastic conjugacy (see Figs. 6 and7). We have shown, for a stochastically perturbed Arnol'd family of circle maps, that noise can enhance model robustness. More precisely, this circle map family exhibits structurally stable, as well as structurally unstable behavior. When noise is added, the entire family exhibits stochastic structural stability, based on the stochastic-conjugacy concept, even in those regions of parameter space where deterministic structural instability occurs for vanishing noise (see Figs. 9 and 10).

Clearly the hope that noise can smooth the very highly structured pattern of distinct behavior types for climate models, across the full hierarchy, has to be tempered by a number of caveats. First, serious questions remain at the fundamental, mathematical level about the behavior of nonhyperbolic chaotic attractors in the presence of noise [START_REF] Anishchenko | Effect of noise on the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors[END_REF][START_REF] Anishchenko | Influence of noise on statistical properties of nonhyperbolic attractors[END_REF][START_REF] Anishchenko | Peculiarities of the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors in the presence of noise[END_REF]. Likewise, the case of nonergodic noise driving is being actively studied [START_REF] Arnold | Normal forms for random differential equations[END_REF][START_REF] Arnold | Normal forms for stochastic differential equations[END_REF][START_REF] Li | Sternberg theorems for random dynamical systems[END_REF].

Second, the presence of certain manifestations of a Devil's staircase has been documented across the full hierarchy of ENSO models [START_REF] Ghil | Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy[END_REF][START_REF] F.-F. Jin | El Niño on the Devil's Staircase: Annual subharmonic steps to chaos[END_REF][START_REF] F.-F. Jin | El Niño/Southern Oscillation and the annual cycle: Subharmonic frequency locking and aperiodicity[END_REF][START_REF] Tziperman | El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator[END_REF][START_REF] Tziperman | Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos[END_REF][START_REF] Saunders | A Boolean delay equation model of ENSO variability[END_REF][START_REF] Ghil | A delay differential model of ENSO variability: Parametric instability and the distribution of extremes[END_REF], as well as in certain observations [START_REF] Ghil | Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy[END_REF][START_REF] Ghil | A delay differential model of ENSO variability: Parametric instability and the distribution of extremes[END_REF]. Interestingly, both GCMs and observations only exhibit a few, broad steps of the staircase, such as 4 : 1 = 4 yr, 4 : 2 = 2 yr, and 4 : 3 ∼ = 16 months. Does this result actually support the idea that nature and its detailed models always provide sufficient noise to achieve considerable smoothing of the much finer structure apparent in simpler models? Be that as it may, we need a much better understanding of how different types of noise -additive and multiplicative, white and colored -act across even a partial hierarchy of models, say from the simplest ones, like those studied in Section 3, to the intermediate ones considered in Section 2.

Third, a closer connection needs to be established between the nature of a stochastic parametrization and its effects on the model's behavior in phase-parameter space. As we saw in the appendix, not all types of noise are equal with respect to these effects. We are thus left with a rich, and hopefully fruitful, set of questions, which we expect to pursue in future work. Acknowledgements: It is a pleasure to thank the organizers and participants of the Conference on the "Euler Equations: 250 Years On" and, more than all, Uriel Frisch, for a very stimulating and altogether pleasant experience. We are grateful to our colleagues I.M. Held, J. C. McWilliams, J. D. Neelin and I. Zaliapin for many useful discussions and their continuing interest in the questions studied here. This study was supported by the DOE grant DE-FG02-07ER64439 from the Climate Change Prediction Program, and by the European Commission's No. 12975 (NEST) project "Extreme Events: Causes and Consequences (E2-C2)."

A Coarse-graining of the circle map family

We provide here a more rigorous justification of the numerical results obtained in Section 3.2.2 on the topological classification of the family of Arnol'd circle maps in the presence of noise. Consider the following random family of diffeomorphisms:

F τ,ǫ;ω (x) := x + τ + σω -ǫ sin(2πx) mod 1, (A.1)
for x ∈ S 1 , ǫ a real parameter in (0, 1), and ω a random parameter distributed in the compact interval I = [-1/2, 1/2] with fixed distribution ν and noise intensity σ. We denote by F τ,ǫ the corresponding deterministic family of diffeomorphisms when the noise is switched off, σ = 0. In order to work in the RDS framework, we need to specify the metric dynamical system modeling the noise. We retain here the choice of considering the interval σI as the base for the probability space Ω and defining the flow θ simply as the operator that maps the point ω into its successor in a sequence of realizations of the noise. One could also have used an irrational rotation on Ω for instance; in both cases, ergodicity is ensured.

For the sake of simplifying notation, we omit for a moment the dependence on τ and ǫ. Using the standard notation for discrete time T = Z, we define a map φ :

T × Ω × S 1 → S 1 , (n, ω, x) → φ(n, ω)x, such that φ(n, ω) :=    F θ n-1 ω • • • • • F ω , n ≥ 1, Id S 1 , n = 0, F -1 θ n ω • • • • • F -1 θ -1 ω , n ≤ -1. (A.2)
Theorem 2 For a random diffeomorphism F τ,ǫ;ω of the circle S 1 , with a stationary measure m supported on a union E of q disjoint intervals, the corresponding skew-product Θ restricted to E has precisely one attracting random periodic orbit and one repelling random periodic orbit.

Attraction in the preceding theorem means that lim n→∞ |F n ω (x) -F n ω (a(ω))| = 0, for a set of initial data (x, ω) ∈ S 1 × Ω with positive λ × ν-measure, in the case of a random attracting fixed point; here λ is Lebesgue measure on S 1 and the extension to a random periodic orbit is obvious.

Using these two theorems and rigorous results on random point attractors [START_REF] Crauel | A uniformly exponential attractor which is not a pullback attractor[END_REF], we can show that (i) if the support of the stationary measure is the whole circle (black curve in Fig. 10), then there exists one random fixed point which is pullback attracting; and (ii) if the support consists of q disjoint intervals, then the random attractor is a random periodic orbit of period q (red and blue curves).

We illustrate this remarkable property in Fig. 11 in the case of a random fixed point, for given ǫ and τ . The Lyapunov exponent for the three distinct trajectories shown in the figure is strictly negative and the trajectories are exponentially attracted to the single random fixed point a(ω), the realization of the driving system θ(ω) being the same for all the trajectories. Kaijser [START_REF] Kaijser | On stochastic perturbations of iterations of circle maps[END_REF] provided rigorous results on this type of synchronization phenomenon, but in a totally different conceptual setting. Interestingly, as the noise intensity increases the Lyapunov exponent becomes more negative, so that the synchronization is even faster. Of course, one needs to consider a fixed realization ω to see this effect. Having explained how the connectedness of the PDF support at different noise levels is related to the nature of the random attractor, we now turn to an explanation of the "disappearance" of the smaller steps in the Devil's staircase, as the noise level increases. To do so, we consider the Lyapunov spectrum of an RDS, which still relies on the celebrated multiplicative ergodic theorem (MET) of Oseledets [START_REF] Oseledets | A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems[END_REF]. To state an MET for RDS on manifolds, we differentiate φ(n, ω) at x ∈ S 1 , and obtain a linear mapping

T φ(n, ω, x) : T x M → T φ(n,ω)x M, (A.7)
where T φ is a continuous linear cocycle on the tangent bundle T M of the manifold M over the skew-product flow Θ. If φ possesses an ergodic invariant measure such that the required integrability condition for applying the MET is verified with respect to this measure, then the MET holds for flow φ over M [START_REF] Arnold | Trends and open problems in the theory of random dynamical systems[END_REF].

Because of Theorem 1 here, we can apply the MET to our problem and conclude that a unique Lyapunov exponent exists for the linearization of each diffeomorphism belonging to our family of random diffeomorphisms, and that this exponent is independent of the realization of the noise. We show next how the Lyapunov spectrum could be used to investigate the stochastic equivalence classes of a given RDS family, along with the study of its driving system θ. We develop this last aspect of the classification problem in the context of linear hyperbolic cocycles.

N.D. Cong [START_REF] Cong | Topological classification of linear hyperbolic cocycles[END_REF] has shown that, even in the linear context, the main difference with the deterministic case is that this classification depends strongly on the properties of θ, which is directly linked to the system noise and its modeling. For instance, if θ is an irrational rotation on S 1 , one can construct infinitely many classes of hyperbolic cocycles which are not pairwise topologically equivalent, playing essentially with the orientations of the cocyles, i.e. reversing between clockwise and anticlockwise rotation on S 1 . As we shall see, such difficulties can be avoided in the case of noisy Arnol'd tongues, especially for additive noise. Related issues still form an active research area in RDS theory; see [START_REF] Arnold | Trends and open problems in the theory of random dynamical systems[END_REF] for a brief survey.

A key ingredient for the linear classification is the notion of coboundary, which we recall herewith.

Definition 2 A measurable set K ⊂ Ω is called a coboundary if there exists a set H ∈ F such that K = H△θH, where H△θH denotes the symmetric difference of H and θH.

Let now A and B be two linear random maps on R d , and denote by deg A(ω) and deg B(ω) the degrees of the maps A(ω) and B(ω) with respect to a chosen random orientation. These degrees are just the sign of the determinant of the corresponding random matrices, and take their values in the set {-1, 1}; see [START_REF] Cong | Topological Dynamics of Random Dynamical Systems[END_REF][START_REF] Cong | Topological classification of linear hyperbolic cocycles[END_REF] for details. Consider the two linear hyperbolic cocycles Φ A and Φ B , associated with the maps A and B. Let us introduce the following subset of Ω:

C AB = {ω ∈ Ω|deg A(ω) • deg B(ω) = -1}, (A.8)
that is the set of all ω ∈ Ω for which the degrees of the two linear maps A(ω) and B(ω) differ.

The main theorem for the classification of our family of random diffeomorphisms of S 1 follows.

Theorem 3 [START_REF] Cong | Topological classification of linear hyperbolic cocycles[END_REF] Two one-dimensional linear hyperbolic cocycles Φ A and Φ B are conjugate if and only if the following conditions hold: Before applying this rigorous result, let us explain first heuristically how a Devil's staircase step corresponding to a rational rotation number can be "destroyed" by a sufficiently intense noise. Consider the period-1 locked state in the deterministic setting. At the beginning of this step, a pair of fixed points is created, one stable and the other unstable. As the bifurcation parameter is increased, these two points move away from each other, until they are π radians apart. Increasing the parameter further causes the fixed points to continue moving along, until they finally meet again and are annihilated in a saddle-node bifurcation, thus signaling the end of the locking interval.

When noise is added, we have to distinguish between a "strongly locked" regime, where the stable and unstable fixed points are nearly π radians apart, and a "weakly locked" regime, where these two fixed points are "close" to each other. In both regimes the relaxation time in the vicinity of the stable point represents an important time scale of the problem. In the strongly locked regime, this is the only time scale of interest. In the weakly locked regime, though, the process of escaping over the unstable fixed point is nonnegligible and the associated escape time becomes the second time scale of interest. From these heuristic considerations it follows that the distinction between strong and weak locking depends on the strength of the external noise.

If we consider period-T locked states, with T ≥ 2, the same kind of reasoning can be applied to the stable and unstable T -cycle. We conclude therefore, for a fixed ǫ > 0, that the shortest Devil's staircase steps are the least robust, while the widest are the most robust.

The fact that a locked case becomes unlocked when noise is growing implies in particular that the rotation number ρ τ,ǫ becomes irrational for a sufficiently high noise level. According to Theorem 2.1 of [START_REF] Kaijser | On stochastic perturbations of iterations of circle maps[END_REF], the Lyapunov exponent is strictly negative in this case almost surely. Moreover, by reinterpreting other results of Kaijser [START_REF] Kaijser | On stochastic perturbations of iterations of circle maps[END_REF] in our RDS framework, we can show that the random attractor is in fact a random fixed point; this, in turn, allows us to conclude that the corresponding linearized cocycle at this random fixed point is hyperbolic. Next, by using the Hartman-Grobman theorem for RDS [START_REF] Wanner | Linearization of random dynamical systems[END_REF][START_REF] Coayla-Teran | Random versions of Hartman-Grobman theorems[END_REF][START_REF] Coayla-Teran | Hartman-Grobman theorems along hyperbolic stationary trajectories[END_REF], we can conjugate the nonlinear cocycle with its linearization; in fact, Theorem 3.1 of [START_REF] Coayla-Teran | Random versions of Hartman-Grobman theorems[END_REF] says that this conjugacy is global.

Consider now two linearized cocycles Φ A and Φ B , at one and the same or two distinct random fixed points of the family of random diffeomorphisms, for the same noise intensity, and denote by A(ω) and B(ω) the random linear parts of the cocycles Φ A and Φ B respectively; it follows from our model of noisy circle maps that C AB is empty. Indeed, the noise being additive, the random orientation is preserved for different parameter values. But θ is assumed to be ergodic, and so we have that Ω△θΩ is empty and, therefore, C AB is a coboundary. Therewith, Theorem 3 can be applied to obtain the desired result for the problem considered here: with an appropriate amount of noise, two deterministic diffeomorphisms that are not topologically equivalent can fall into the same topological stochastic class! The numerical tests of Section 3.2 are entirely in agreement with this assertion.

Note that the set C AB could differ from a coboundary, if the noise occurred additively in the phase of the nonlinear term, for instance. Here we see the importance of noise modeling in obtaining the conjectural view of Figure 8 for a family of dynamical systems in general. It follows, in particular, that the exact nature of the stochastic parametrizations in a family of GCMs does matter. It's not enough to follow the trend and implement such parametrizations: one should test that they do improve the proximity, in an appropriate sense, between climate simulations in the family.

Figure 3 :

 3 Figure 3: A satellite image of the sea surface temperature (SST) over the northwestern North Atlantic, together with a sketch of the associated double-gyre circulation. An idealized view of the amount of potential vorticity injected into the ocean circulation by the trade winds, westerlies and polar easterlies is shown to the right.

23 Figure 4 :

 234 Figure 4: Generic bifurcation diagram for the barotropic QG model of the double-gyre problem: the asymmetry of the solution is plotted versus the intensity of the wind stress τ .The streamfunction field is plotted for a steady-state solution associated with each of the three branches; positive values in red and negative ones in blue (after[START_REF] Simonnet | Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation[END_REF]).

Figure 5 :

 5 Figure5: Two-parameter plane, with the wind-stress intensity τ vs. the eddy-viscosity coefficient ν: the curves indicate the locations of supercritical and subcritical pitchfork bifurcations. Each band is associated with a different wavenumber and timescale (from[START_REF] Simonnet | Quantization of the low-frequency variability of the double-gyre circulation[END_REF]).

Figure 9 : 3 Figure 10 :

 9310 Figure 9: Arnol'd tongues for the family of diffeomorphisms of the circle; units for τ and ǫ are 5 • 10 -4 and 10 -4 respectively. Devil's staircase in the cross-section to the right.Our numerical results are further supported by theoretical investigations using the frame-

Figure 11 :

 11 Figure 11: Synchronization by additive noise: three distinct trajectories (in blue, red and black) of x n+1 = F τ,ǫ;ω (x n ), with F τ,ǫ;ω given by (A.1); the three trajectories start from three initial points on S 1 , but are driven by the same realization ω of the noise, and thus converge to the same random fixed point a(ω), which is moving with time. The parameters are ǫ = 0.5, τ = 0.283 and σ = 0.3, and the corresponding Lyapunov exponent is λ ≃ -0.0104.

  (i) sign λ A = sign λ B , and (ii) the associated set C AB is a coboundary; here λ A and λ B indicate the Lyapunov exponents of Φ A and Φ B , respectively.

  illustrates this situation: two

		0.22
		0.2
		0.18
		0.16
	τ
	Wind stress	0.1 0.12 0.14
		0.08
		0.06
		0.04
		0.02
		0.4	0.6	0.8	1 Lateral friction ν	1.2	1.4	1.6 11 x 10

One can prove easily that φ thus defined satisfies the cocycle property and that it is in fact a C ∞ RDS on S 1 over θ.

The pair of mappings Θ := (θ, φ) is called the corresponding skew-product. This skewproduct defines a flow on Ω × S 1 by the relation:

A stationary measure m on S 1 under the random diffeomorphism F τ,ǫ;ω yields a Θ-invariant measure µ := m × ν, i.e. Θ n µ = µ; explicitly,

Let us recall the following important proposition [START_REF] Zmarrou | Bifurcations of stationary measures of random diffeomorphisms[END_REF] concerning the stationary measures obtained from the random family {F τ,ǫ;ω }.

Theorem 1

The random circle diffeomorphism F τ,ǫ;ω has a unique stationary measure m τ,ǫ . The support of m τ,ǫ consists either of q mutually disjoint intervals or of the entire circle S 1 . The density function φ τ,ǫ is in C ∞ (S 1 ) and depends C ∞ on τ . The invariant measure µ is ergodic. If the support of m is connected, then it is mixing and so is µ.

Mixing for m means that, for any bounded function f : S 1 → R, and for an arbitrary initial point x 0 ∈ S 1 , E(f (φ(n, ω)x 0 ) tends to S 1 f (x)m(dx) as n → +∞; see [START_REF] Kuksin | On random attractors for mixing-type systems[END_REF] for more on random attractors and mixing.

Central to the study of deterministic diffeomorphisms of the circle is the rotation number, which measures the average rotation per iterate of F τ,ǫ . In the presence of noise, one can still define a rotation number for F τ,ǫ;ω , namely

where F denotes the lift of a map F , acting on S 1 modulo 1, to a map acting on R. For a fixed τ and a fixed ǫ, we can then show that ρ τ,ǫ;ω exists for ν-almost all ω and is a constant; this constant ρ τ ;ω is independent of x and ω [START_REF] Zmarrou | Bifurcations of stationary measures of random diffeomorphisms[END_REF]. Furthermore, τ → ρ τ,ǫ is C ∞ for each ǫ, which is not true in the deterministic case with σ = 0; see again [START_REF] Zmarrou | Bifurcations of stationary measures of random diffeomorphisms[END_REF]. Theorem 1 has a natural geometric counterpart in terms of random attractors, as confirmed through our numerical study; see again Fig. 10. More precisely, we introduce also the following definitions of random fixed point and random periodic orbit; these definitions differ somewhat from those given in [START_REF] Zmarrou | Bifurcations of stationary measures of random diffeomorphisms[END_REF].

for ν-almost all ω ∈ Ω, i.e. such that Ω × a(Ω) is an invariant set for the flow given by the skew-product Θ. A random periodic orbit of period q is likewise an invariant set with cardinality q in fibers S 1 × {ω} for ν-almost all ω.

With these definitions, the following results of [START_REF] Zmarrou | Bifurcations of stationary measures of random diffeomorphisms[END_REF] still hold.