January  2017, 37(1): 189-228. doi: 10.3934/dcds.2017008

The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem

1. 

Geosciences Department and Laboratoire de Météorologie Dynamique (CNRS and IPSL), École Normale Supérieure, F-75231 Paris Cedex 05, France

2. 

Department of Atmospheric & Oceanic Sciences, University of California, Los Angeles, CA 90095-1565, USA

Received  March 2016 Revised  May 2016 Published  November 2016

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

Citation: Michael Ghil. The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 189-228. doi: 10.3934/dcds.2017008
References:
[1]

M. R. Allen, Do-it-yourself climate prediction, Nature, 401 (1999), p642. doi: 10.1038/44266. Google Scholar

[2]

A. A. Andronov and L. S. Pontryagin, Systémes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-250. Google Scholar

[3]

V. S. AnishchenkoT. E. VadivasovaA. S. KopeikinJ. Kurths and G. I. Strelkova, Effect of noise on the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors, Phys. Rev. Lett., 87 (2001), 054101. doi: 10.1103/PhysRevLett.87.054101. Google Scholar

[4]

V. S. AnishchenkoT. E. VadivasovaA. S. KopeikinG. I. Strelkova and J. Kurths, Influence of noise on statistical properties of nonhyperbolic attractors, Phys. Rev. E, 62 (2000), p7886. doi: 10.1103/PhysRevE.62.7886. Google Scholar

[5]

V. S. AnishchenkoT. E. VadivasovaA. S. KopeikinG. I. Strelkova and J. Kurths, Peculiarities of the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors in the presence of noise, Phys. Rev. E, 65 (2002), 036206. doi: 10.1103/PhysRevE.65.036206. Google Scholar

[6]

V. AraujoM. PacificoR. Pujal and M. Viana, Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485. doi: 10.1090/S0002-9947-08-04595-9. Google Scholar

[7]

L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar

[8]

L. Arnold, Trends and open problems in the theory of random dynamical systems, in: Probability Towards 2000, L. Accardi, C. C. Heyde (Eds.), Springer Lecture Notes in Statistics, 128 (1998), 34-46. doi: 10.1007/978-1-4612-2224-8_2. Google Scholar

[9]

L. Arnold and K. Xu, Normal forms for random differential equations, J. Diff. Eq., 116 (1995), 484-503. doi: 10.1006/jdeq.1995.1045. Google Scholar

[10]

L. Arnold and P. Imkeller, Normal forms for stochastic differential equations, Prob. Theory Relat. Fields, 110 (1998), 559-588. doi: 10.1007/s004400050159. Google Scholar

[11]

V. I. Arnol'd, Geometrical Methods in the Theory of Differential Equations, Springer, 1983,334 pp. Google Scholar

[12]

P. Bak, The devil's staircase, Physics Today, 39 (1986), 38-45. doi: 10.1063/1.881047. Google Scholar

[13]

P. Bak and R. Bruinsma, One-dimensional Ising model and the complete devil's staircase, Phys. Rev. Lett., 49 (1982), 249-251. doi: 10.1103/PhysRevLett.49.249. Google Scholar

[14]

J. J. Barsugli and D. S. Battisti, The basic effects of atmosphere-ocean thermal coupling on midlatitude variability, J. Atmos. Sci., 5 (1998), 477-493. doi: 10.1175/1520-0469(1998)055<0477:TBEOAO>2.0.CO;2. Google Scholar

[15]

D. R. Bell, Degenerate Stochastic Differential Equations and Hypoellipticity, Longman, Harlow, 1995. Google Scholar

[16]

A. Berger and S. Siegmund, On the gap between random dynamical systems and continuous skew products, J. Dyn. Diff. Eq., 15 (2003), 237-279. doi: 10.1023/B:JODY.0000009736.39445.c4. Google Scholar

[17]

P. BerloffA. Hogg and W. Dewar, The turbulent oscillator: A mechanism of low-frequency variability of the wind-driven ocean gyres, J. Phys. Oceanogr., J. Phys. Oceanogr., 37 (2007), 2363-2386. doi: 10.1175/JPO3118.1. Google Scholar

[18]

K. BhattacharyaM. Ghil and I. L. Vulis, Internal variability of an energy-balance model with delayed albedo effects, J. Atmos. Sci., 39 (1982), 1747-1773. doi: 10.1175/1520-0469. Google Scholar

[19]

T. BódaiG. Károlyi and T. Tél, A chaotically driven model climate: Extreme events and snapshot attractors, Nonlin. Processes Geophys, 18 (2011), 573-580. Google Scholar

[20]

T. BódaiV. LucariniF. Lunkeit and R. Boschi, Global instability in the Ghil-Sellers model, Clim. Dyn., 44 (2015), 3361-3381. Google Scholar

[21]

T. Bogenschütz and Z. S. Kowalski, A condition for mixing of skew products, Aequationes Math., 59 (2000), 222-234. doi: 10.1007/s000100050122. Google Scholar

[22]

F. Bouchet and J. Sommeria, Emergence of intense jets and Jupiter's great red spot as maximum entropy structures, J. Fluid. Mech., 464 (2002), 165-207. doi: 10.1017/S0022112002008789. Google Scholar

[23]

S. BrachetF. CodronY. FeliksM. GhilH. Le Treut and E. Simonnet, Atmospheric circulations induced by a mid-latitude SST front: A GCM study, J. Clim., 25 (2012), 1847-1853. doi: 10.1175/JCLI-D-11-00329.1. Google Scholar

[24]

A. BraccoJ. D. NeelinH. LuoJ. C. McWilliams and J. E. Meyerson, High-dimensional decision dilemmas in climate models, Geosci. Model Dev., 6 (2013), 1673-1687. doi: 10.5194/gmdd-6-2731-2013. Google Scholar

[25]

A. N. CarvalhoJ. A. Langa and J. C. Robinson, Lower semicontinuity of attractors for non-autonomous dynamical systems, Ergod. Th.. Dyn. Syst., 29 (2009), 1765-1780. doi: 10.1017/S0143385708000850. Google Scholar

[26]

A. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar

[27]

P. Cessi and G. R. Ierley, Symmetry-breaking multiple equilibria in quasigeostrophic, wind-driven flows, J. Phys. Oceanogr, 25 (1995), 1196-1205. Google Scholar

[28]

K. I. ChangK. IdeM. Ghil and C.-C. A. Lai, Transition to aperiodic variability in a wind-driven double-gyre circulation model, J. Phys. Oceanogr, 31 (2001), 1260-1286. Google Scholar

[29]

C. P. Chang, M. Ghil, M. Latif and J. M. Wallace (Eds.), Climate Change: Multidecadal and Beyond, World Scientific Publ. Co. /Imperial College Press, 2015,388 pp.Google Scholar

[30]

P. ChangL. JiH. Li and M. Flugel, Chaotic dynamics versus stochastic processes in El Niño-Southern Oscillation in coupled ocean-atmosphere models, Physica D, 98 (1996), 301-320. Google Scholar

[31]

Y. ChaoM. Ghil and J. C. McWilliams, Pacific interdecadal variability in this century's sea surface temperatures, Geophys. Res. Lett., 27 (), 2261-2264. Google Scholar

[32] J. G. Charney, Carbon Dioxide and Climate: A Scientific Assessment, National Academies Press, Washington, D.C, 1979. Google Scholar
[33]

J. G. Charney and D. M. Straus, Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, orographically forced, planetary wave systems, J. Atmos. Sci., 37 (1980), 1157-1176. Google Scholar

[34]

M. Chavez, M. Ghil and J. Urrutia Fucugauchi (Eds.), Extreme Events: Observations, Modeling and Economics, Geophysical Monograph 214, American Geophysical Union & Wiley, 2015,438 pp.Google Scholar

[35]

M. D. ChekrounD. Kondrashov and M. Ghil, Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation, Proc. Natl. Acad. Sci USA, 108 (2011), 11766-11771. Google Scholar

[36]

M. D. ChekrounE. Simonnet and M. Ghil, Stochastic climate dynamics: Random attractors and time-dependent invariant measures, Physica D, 240 (2011), 1685-1700. doi: 10.1016/j.physd.2011.06.005. Google Scholar

[37]

M. D. ChekrounJ. D. NeelinD. KondrashovJ. C. McWilliams and M. Ghil, Rough parameter dependence in climate models: The role of Ruelle-Pollicott resonances, Proc. Natl. Acad. Sci. USA, 111 (2014), 1684-1690. doi: 10.1073/pnas.1321816111. Google Scholar

[38]

M. D. ChekrounM. GhilH. Liu and S. Wang, Low-dimensional Galerkin approximations of nonlinear delay differential equations, Discr. Cont. Dyn. Syst., 36 (2016), 4133-4177. doi: 10.3934/dcds.2016.36.4133. Google Scholar

[39]

M. D. Chekroun, H. Liu and S. Wang, Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs Ⅰ, Springer Briefs in Mathematics, Springer, 2015. doi: 10.1007/978-3-319-12496-4. Google Scholar

[40]

M. D. Chekroun, M. Ghil and J. D. Neelin, Invariant measures on climatic pullback attractors, in preparation, 2016.Google Scholar

[41]

E. A. Coayla-Teran and P. R. C. Ruffino, Random versions of Hartman-Grobman theorems, Preprint IMECC, UNICAMP, No. 27/01 (2001).Google Scholar

[42]

P. Collet and C. Tresser, Ergodic theory and continuity of the Bowen-Ruelle measure for geometrical flows, Fyzika, 20 (1988), 33-48. Google Scholar

[43] N. D. Cong, Topological Dynamics of Random Dynamical Systems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1997. Google Scholar
[44]

H. Crauel, White noise eliminates instablity, Arch. Math., 75 (2000), 472-480. doi: 10.1007/s000130050532. Google Scholar

[45]

H. Crauel, A uniformly exponential attractor which is not a pullback attractor, Arch. Math, 78 (2002), 329-336. doi: 10.1007/s00013-002-8254-9. Google Scholar

[46]

H. Crauel, Random Probability Measures on Polish Spaces, Stochastic Monographs, vol. 11, Taylor & Francis, 2002. Google Scholar

[47]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Scuola Normale Superiore Pisa 148 (1992), Technical Report.Google Scholar

[48]

H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation, J. Dyn. Diff. Eqn., 10 (1998), 259-274. doi: 10.1023/A:1022665916629. Google Scholar

[49]

E. D. Da Costa and A. C. Colin de Verdière, The 7.7 year North Atlantic oscillation, Q. J. R. Meteorol. Soc., 128 (2002), 797-817. doi: 10.1256/0035900021643692. Google Scholar

[50]

A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pure Appl., 11 (1932), 333-375. Google Scholar

[51]

M. Dorfle and R. Graham, Probability density of the Lorenz model, Phys. Rev. A, 27 (1983), 1096-1105. doi: 10.1103/PhysRevA.27.1096. Google Scholar

[52]

H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño (2nd ed. ), Springer, 2005,532 pp. doi: 10.1007/978-94-015-9450-9. Google Scholar

[53]

H. A. Dijkstra, Nonlinear Climate Dynamics, Cambridge Univ. Press, 2013,367 pp. doi: 10.1017/CBO9781139034135. Google Scholar

[54]

H. A. Dijkstra and M. Ghil, Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach, Rev. Geophys., 43 (2005), RG3002. doi: 10.1029/2002RG000122. Google Scholar

[55]

H. A. Dijkstra and C. A. Katsman, Temporal variability of the wind-driven quasi-geostrophic double gyre ocean circulation: basic bifurcation diagrams, Geophys. Astrophys. Fluid Dyn., 85 (1997), 195-232. doi: 10.1080/03091929708208989. Google Scholar

[56]

R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Theor. Prob. Appl., 15 (1970), 458-486. doi: 10.1137/1115049. Google Scholar

[57]

G. DrótosT. Bódai and T. Tél, Probabilistic concepts in a changing climate: A snapshot attractor picture, J. Clim., 28 (2015), 3275-3288. Google Scholar

[58]

M. 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.) de l'aqueduc de Fréjus, Archeoscience, 30 (2006), 163-171. Google Scholar

[59]

J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617. Google Scholar

[60]

B. F. Farrel and P. J. Ioannou, Structural stability of turbulent jets, J. Atmos. Sci., 60 (2003), 2101-2118. doi: 10.1175/1520-0469(2003)060<2101:SSOTJ>2.0.CO;2. Google Scholar

[61]

M. J. FeigenbaumL. P. Kadanoff and S. J. Shenker, Quasiperiodicity in dissipative systems: A renormalization group analysis, Physica D, 5 (1982), 370-386. doi: 10.1016/0167-2789(82)90030-6. Google Scholar

[62]

Y. FeliksM. Ghil and E. Simonnet, Low-frequency variability in the mid-latitude atmosphere induced by an oceanic thermal front, J. Atmos. Sci., 61 (2004), 961-981. Google Scholar

[63]

Y. FeliksM. Ghil and E. Simonnet, Low-frequency variability in the mid-latitude baroclinic atmosphere induced by an oceanic thermal front, J. Atmos. Sci., 64 (2007), 97-116. Google Scholar

[64]

Y. FeliksM. Ghil and A. W. Robertson, Oscillatory climate modes in the Eastern Mediterranean and their synchronization with the north Atlantic Oscillation, J. Clim., 23 (2010), 4060-4079. doi: 10.1175/2010JCLI3181.1. Google Scholar

[65]

Y. FeliksM. Ghil and A. W. Robertson, The atmospheric circulation over the North Atlantic as induced by the SST field, J. Clim., 24 (2011), 522-542. doi: 10.1175/2010JCLI3859.1. Google Scholar

[66]

E. Galanti and E. Tziperman, ENSO's phase locking to the seasonal cycle in the fast-SST, fast-wave, and mixed-mode regimes, J. Atmos. Sci., 57 (2000), 2936-2950. doi: 10.1175/1520-0469(2000)057<2936:ESPLTT>2.0.CO;2. Google Scholar

[67]

M. Ghil, Steady-State Solutions of a Diffusive Energy-Balance Climate Model and Their Stability, Report IMM-410, Courant Institute of Mathematical Sciences, New York University, New York, 74 pp. , 1975; available in the Classic Reprint Series of Förlag Forgotten Books, http://www.bokus.com/bok/9781332200214/steady-state-solutions-of-a-diffusive-energy-balance-climate-model\-and-their-stability-classic-reprint/Google Scholar

[68]

M. Ghil, Climate stability for a Sellers-type model, J. Atmos. Sci., 33 (1976), 3-20. doi: 10.1175/1520-0469(1976)033<0003:CSFAST>2.0.CO;2. Google Scholar

[69]

M. Ghil, Cryothermodynamics: The chaotic dynamics of paleoclimate, Physica D, 77 (1994), 130-159. doi: 10.1016/0167-2789(94)90131-7. Google Scholar

[70]

M. Ghil, Hilbert problems for the geosciences in the 21st century, Nonlin. Proc. Geophys, 8 (2001), 211-211. doi: 10.5194/npg-8-211-2001. Google Scholar

[71]

M. Ghil, A mathematical theory of climate sensitivity or, How to deal with both anthropogenic forcing and natural variability?, in Climate Change: Multidecadal and Beyond, C. P. Chang, M. Ghil, M. Latif and J. M. Wallace (Eds.), World Scientific Publ. Co. /Imperial College Press, 2015, 31-51.Google Scholar

[72]

M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, Springer-Verlag, Berlin/Heidelberg/New York, 1987,512 pp. doi: 10.1007/978-1-4612-1052-8. Google Scholar

[73]

M. Ghil and R. Vautard, Interdecadal oscillations and the warming trend in global temperature time series, Nature, 350 (1991), 324-327. doi: 10.1038/350324a0. Google Scholar

[74]

M. Ghil and N. Jiang, Recent forecast skill for the El Niño/Southern Oscillation, Geophys. Res. Lett., 25 (1998), 171-174. Google Scholar

[75]

M. Ghil and A. W. Robertson, Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy, in: General Circulation Model Development: Past, Present and Future, D. Randall (Ed. ), Academic Press, San Diego, 70 (2000), 285--325. doi: 10.1016/S0074-6142(00)80058-3. Google Scholar

[76]

M. Ghil and A. W. Robertson, "Waves" vs "particles" in the atmosphere's phase space: A pathway to long-range forecasting?, Proc. Natl. Acad. Sci. USA, 99 (2002), 2493-2500. doi: 10.1073/pnas.012580899. Google Scholar

[77]

M. GhilY. Feliks and L. Sushama, Baroclinic and barotropic aspects of the wind-driven ocean circulation, Physica D, 167 (2002), 1-35. doi: 10.1016/S0167-2789(02)00392-5. Google Scholar

[78]

M. GhilR. M. AllenM. D. DettingerK. IdeD. KondrashovM. E. MannA. RobertsonA. SaundersY. TianF. Varadi and P. Yiou, Advanced spectral methods for climatic time series, Rev. Geophys., 40 (2002), 3.1-3.41. doi: 10.1029/2000RG000092. Google Scholar

[79]

M. GhilM. D. Chekroun and E. Simonnet, Climate dynamics and fluid mechanics: Natural variability and related uncertainties, Physica D, 237 (2008), 2111-2126. doi: 10.1016/j.physd.2008.03.036. Google Scholar

[80]

M. GhilI. Zaliapin and S. Thompson, A delay differential model of ENSO variability: Parametric instability and the distribution of extremes, Nonlin. Processes Geophys, 15 (2008), 417-433. doi: 10.5194/npg-15-417-2008. Google Scholar

[81]

M. GhilP. YiouS. HallegatteB. D. MalamudP. NaveauA. SolovievP. FriederichsV. Keilis-BorokD. KondrashovV. KossobokovO. MestreC. NicolisH. RustP. ShebalinM. VracA. Witt and I. Zaliapin, Extreme events: Dynamics, statistics and prediction, Nonlin. Processes Geophys, 18 (2011), 295-350. doi: 10.5194/npg-18-295-2011. Google Scholar

[82]

M. Ghil and I. Zaliapin, Understanding ENSO variability and its extrema: A delay differential equation approach, Ch. 6 in Extreme Events: Observations, Modeling and Economics, M. Chavez, M. Ghil and J. Urrutia-Fucugauchi (Eds.), Geophysical Monograph, American Geophysical Union & Wiley, 214 (2015), 63--78.Google Scholar

[83]

A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, 1982,662 pp.Google Scholar

[84]

C. GrebogiH. KantzA. PrasadY. C. Lai and E Sinde, Unexpected robustness-against-noise of a class of nonhyperbolic chaotic attractors, Phys. Rev. E, 65 (2002), 026209, 8 pp. doi: 10.1103/PhysRevE.65.026209. Google Scholar

[85]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (2nd ed. ), Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2. Google Scholar

[86]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Publ. Math. I.H.E.S., 50 (1979), 59-72. Google Scholar

[87]

I. M. Held, The gap between simulation and understanding in climate modeling, Bull. American. Meteorol. Soc., 86 (2005), 1609-1614. doi: 10.1175/BAMS-86-11-1609. Google Scholar

[88]

R. Hillerbrand and M. Ghil, Anthropogenic climate change: Scientific uncertainties and moral dilemmas, in this volume, Physica D, 2008.Google Scholar

[89]

J. T. Houghton, G. J. Jenkins and J. J. Ephraums (Eds.), Climate Change, The IPCC Scientific Assessment, Cambridge Univ. Press, Cambridge, MA, 1991,365 pp.Google Scholar

[90]

J. T. Houghton, Y. Ding, D. J. Griggs, M. Noguer, P. J. van der Linden, X. Dai, K. Maskell and C. A. Johnson (Eds.), Climate Change 2001: The Scientific Basis. Contribution of Working Group Ⅰ to the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC), Cambridge University Press, Cambridge, U. K. , 2001,944 pp.Google Scholar

[91]

S. Jiang, F. -F. Jin and M. Ghil, The nonlinear behavior of western boundary currents in a wind-driven, double-gyre, shallow-water model in: Ninth Conf. Atmos. & Oceanic Waves and Stability (San Antonio, TX), American Meterorological Society, Boston, Mass. , 1993, pp. 64-67.Google Scholar

[92]

S. JiangF.-F. Jin and M. Ghil, Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J. Phys. Oceanogr, 25 (1995), 764-786. doi: 10.1175/1520-0485(1995)025<0764:MEPAAS>2.0.CO;2. Google Scholar

[93]

F.-F. JinJ. D. Neelin and M. Ghil, El Niño on the Devil's Staircase: Annual subharmonic steps to chaos, Science, 264 (1994), 70-72. Google Scholar

[94]

F.-F. JinJ. D. Neelin and M. Ghil, El Niño/Southern Oscillation and the annual cycle: Subharmonic frequency locking and aperiodicity, Physica D, 98 (1996), 442-465. Google Scholar

[95]

T. Jung, T. N. Palmer and G. J. Shutts, Geophys. Res. Lett. , 32 (2005), Art. No. L23811.Google Scholar

[96]

T. Kaijser, On stochastic perturbations of iterations of circle maps, Physica D, 68 (1993), 201-231. doi: 10.1016/0167-2789(93)90081-B. Google Scholar

[97] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge Univ. Press, Cambridge/London, UK, 2003. doi: 10.1017/CBO9780511802270. Google Scholar
[98]

J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Differential Equations and Approximations of Fixed Points, H. -O. Peitgen and H. -O. Walter (Eds.), Lecture Notes in Mathematics, (Springer, Berlin), 730 (1979), 204-227. Google Scholar

[99]

A. Katok and B. Haselblatt, Introduction to the Modern Theroy of Dynamical Systems, Cambridge Univ. Press, Encycl. Math. Appl. , 54 1995. doi: 10.1017/CBO9780511809187. Google Scholar

[100]

C. A. KatsmanH. A. Dijkstra and S. S. Drijfhout, The rectification of the wind-driven ocean circulation due to its instabilities, J. Mar. Res., 56 (1998), 559-587. Google Scholar

[101]

Y. Kifer, Ergodic Theory of Random Perturbations, Birkhäuser, 1988.Google Scholar

[102]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. Google Scholar

[103]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, vol. 176, American Mathematical Society, 2011. doi: 10.1090/surv/176. Google Scholar

[104]

J. J. Kohn, Pseudo-differential operators and hypoellipticity, Proc. Amer. Math. Soc. Symp. Pure Math., 23 (1973), 61-69. Google Scholar

[105]

D. KondrashovY. Feliks and M. Ghil, Oscillatory modes of extended Nile River records (A.D. 622-1922), Geophys. Res. Lett., 32 (2005), L10702. doi: 10.1029/2004GL022156. Google Scholar

[106]

K. KondrashovM. D. ChekrounA. W. Robertson and M. Ghil, Low-order stochastic model and "past-noise forecasting" of the Madden-Julian oscillation, Geophys. Res. Lett., 40 (2013), 5303-5310. doi: 10.1002/grl.50991. Google Scholar

[107]

D. KondrashovM. D. Chekroun and M. Ghil, Data-driven non-Markovian closure models, Physica D, 297 (2015), 33-55. doi: 10.1016/j.physd.2014.12.005. Google Scholar

[108]

S. KravtsovP. BerloffW. K. DewarM. Ghil and J. C. McWilliams, Dynamical origin of low-frequency variability in a highly nonlinear mid-latitude coupled model, J. Climate, 19 (2007), 6391-6408. Google Scholar

[109]

S. Kuksin and A. Shirikyan, On random attractors for mixing-type systems, Funct. Anal. Appl., 38 (2004), 34-46. doi: 10.1023/B:FAIA.0000024865.78811.11. Google Scholar

[110]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York (3rd ed. ), 2004. doi: 10.1007/978-1-4757-3978-7. Google Scholar

[111]

J. A. LangaJ. C. Robinson and A. Suarez, Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity, 15 (2002), 887-903. doi: 10.1088/0951-7715/15/3/322. Google Scholar

[112]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Applied Mathematical Sciences, vol. 97, Springer-Verlag, 1994. doi: 10.1007/978-1-4612-4286-4. Google Scholar

[113]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Prob. Theory Related Fields, 80 (1988), 217-240. doi: 10.1007/BF00356103. Google Scholar

[114]

W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988. doi: 10.1002/cpa.20083. Google Scholar

[115]

J. W. B. Lin and J. D. Neelin, Influence of a stochastic moist convective parameterization on tropical climate variability, Geophys. Res. Lett., 27 (2000), 3691-3694. doi: 10.1029/2000GL011964. Google Scholar

[116]

J. W. B. Lin and J. D. Neelin, Considerations for stochastic convective parameterization, J. Atmos. Sci., 59 (2002), 959-975. doi: 10.1175/1520-0469(2002)059<0959:CFSCP>2.0.CO;2. Google Scholar

[117]

J. W. B. Lin and J. D. Neelin, Toward stochastic deep convective parameterization in general circulation models, Geophys. Res. Lett., 30 (2003), p1162. doi: 10.1029/2002GL016203. Google Scholar

[118]

P. C. Loikith and J. D. Neelin, Short-tailed temperature distributions over North America and implications for future changes in extremes, Geophys. Res. Lett., 42 (2015), 8577-8585. doi: 10.1002/2015GL065602. Google Scholar

[119]

E. N. Lorenz, Deterministic nonperiodic flow, The Theory of Chaotic Attractors, (2004), 25-36. doi: 10.1007/978-0-387-21830-4_2. Google Scholar

[120] E. N. Lorenz, The Essence of Chaos, Univ. of Washington Press, 1993. doi: 10.4324/9780203214589. Google Scholar
[121]

V. Lucarini and S. Sarno, A statistical mechanical approach for the computation of the climatic response to general forcings, Nonlin. Processes Geophys, 18 (2011), 7-28. doi: 10.5194/npg-18-7-2011. Google Scholar

[122]

V. LucariniR. BlenderC. HerbertF. RagoneS. Pascale and J. Wouters, Mathematical and physical ideas for climate science, Rev. Geophys, 52 (2014), 809-859. doi: 10.1002/2013RG000446. Google Scholar

[123]

V. LucariniF. Ragone and F. Lunkeit, Predicting climate change using response theory: Global averages and spatial patterns, Journal of Statistical Physics, (2016), 1-29. doi: 10.1007/s10955-016-1506-z. Google Scholar

[124]

R. A. Madden and P. R. Julian, Observations of the 40--50-day tropical oscillations -a review, Mon. Wea. Rev., 122 (1994), 814-837. doi: 10.1175/1520-0493(1994)122<0814:OOTDTO>2.0.CO;2. Google Scholar

[125]

A. Majda and X. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge Univ. Press, 551 pp. , 2006. doi: 10.1017/CBO9780511616778. Google Scholar

[126]

R. Mañé, A proof of the $ C^1$-stability conjecture, Publ. Math, I.H.E.S., 66 (1988), 161-210. Google Scholar

[127]

N. J. MantuaS. HareY. ZhangJ. M. Wallace and R. C. Francis, A Pacific interdecadal climate oscillation with impacts on salmon production, Bull. Am. Meteor. Soc., 78 (1997), 1069-1079. doi: 10.1175/1520-0477(1997)078<1069:APICOW>2.0.CO;2. Google Scholar

[128]

J. C. McWilliams, Irreducible imprecision in atmospheric and oceanic simulations, Proc. Natl. Acad. Sci. USA, 104 (2007), 8709-8713. doi: 10.1073/pnas.0702971104. Google Scholar

[129]

J. C. McWilliams, Fundamentals of Geophysical Fluid Dynamics, 2nd ed. , Cambridge Univ. Press, 272 pp. , 2011.Google Scholar

[130]

S. P. Meacham, Low-frequency variability in the wind-driven circulation, J. Phys. Oceanogr, 30 (2000), 269-293. doi: 10.1175/1520-0485(2000)030<0269:LFVITW>2.0.CO;2. Google Scholar

[131]

G. A. Meehl, Decadal climate variability and the early-2000s hiatus, in Understanding the Earth's Climate Warming Hiatus: Putting the Pieces Together, D. Menemenlis and J. Sprintall (Eds.), US CLIVAR Variations, US CLIVAR Project Office, 13 (2015), 1-6,Google Scholar

[132]

S. MinobeA. Kuwano-YoshidaN. KomoriS.-P. Xie and R. J. Small, Influence of the Gulf Stream on the troposphere, Nature, 452 (2008), 206-209. doi: 10.1038/nature06690. Google Scholar

[133]

J. F. B. Mitchell, Can we believe predictions of climate change?, Q. J. Roy. Meteorol. Soc. (Part A), 130 (2004), 2341-2360. doi: 10.1256/qj.04.74. Google Scholar

[134]

A. K. MittalS. Dwivedi and R. S. Yadav, Probability distribution for the number of cycles between successive regime transitions for the Lorenz model, Physica D, 233 (2007), 14-20. doi: 10.1016/j.physd.2007.06.014. Google Scholar

[135]

V. MoronR. Vautard and M. Ghil, Trends, interdecadal and interannual oscillations in global sea-surface temperatures, Clim. Dyn., 14 (1998), 545-569. doi: 10.1007/s003820050241. Google Scholar

[136]

N. T. Nadiga and B. P. Luce, Global bifurcation of Shilnikov type in a double-gyre ocean model, J. Phys. Oceanogr, 31 (2001), 2669-2690. doi: 10.1175/1520-0485(2001)031<2669:GBOSTI>2.0.CO;2. Google Scholar

[137]

National Research Council, Natural Climate Variability on Decade-to-Century Time Scales, D. G. Martinson, K. Bryan, M. Ghil et al. (Eds.), National Academies Press, Washington, D. C. , 1995,630 pp.Google Scholar

[138]

J. D. NeelinD. S. BattistiA. C. HirstF.-F. JinY. WakataT. Yamagata and S. Zebiak, ENSO Theory, J. Geophys. Res., 103 (1998), 14261-14290. doi: 10.1029/97JC03424. Google Scholar

[139]

S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. I.H.E.S., 50 (1979), 101-151. Google Scholar

[140]

N. Ohtomo, Exponential characteristics of power spectral densities caused by chaotic phenomena, J. Phys. Soc. Japan, 64 (1995), 1104-1113. Google Scholar

[141]

V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231. Google Scholar

[142]

J. Palis, A global perspective for non-conservative dynamics, Ann. I.H. Poincaré, 22 (2005), 485-507. doi: 10.1016/j.anihpc.2005.01.001. Google Scholar

[143]

T. N. Palmer, The prediction of uncertainty in weather and climate forecasting, Rep. Prog. Phys., 63 (2000), 71-116. Google Scholar

[144]

J. Pedlosky, Geophysical Fluid Dynamics (2nd ed. ), Springer-Verlag, 1987,710 pp.Google Scholar

[145]

J. Pedlosky, Ocean Circulation Theory, Springer, New York, 1996. doi: 10.1007/978-3-662-03204-6. Google Scholar

[146]

M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-110. doi: 10.1016/0040-9383(65)90018-2. Google Scholar

[147]

S. Pierini, Low-frequency variability, coherence resonance, and phase selection in a low-order model of the wind-driven ocean circulation, J. Phys. Oceanogr, 41 (2011), 1585-1604. doi: 10.1175/JPO-D-10-05018.1. Google Scholar

[148]

S. Pierini, Stochastic tipping points in climate dynamics, Phys. Rev. E, 85 (2012), 027101. doi: 10.1103/PhysRevE.85.027101. Google Scholar

[149]

S. Pierini, Ensemble simulations and pullback attractors of a periodically forced double-gyre sysyem, J. Phys. Oceanogr, 44 (2014), 3245-3254. Google Scholar

[150]

S. PieriniM. Ghil and M. D. Chekroun, Exploring the pullback attractors of a low-order quasigeostrophic ocean model: The deterministic case, J. Clim., in press, (2015). doi: 10.1175/JCLI-D-15-0848.1. Google Scholar

[151]

G. PlautM. Ghil and R. Vautard, Interannual and interdecadal variability in 335 Years of Central England temperatures, Science, 268 (1995), 710-713. doi: 10.1126/science.268.5211.710. Google Scholar

[152]

H. Poincaré, Sur les équations de la dynamique et le problème des trois corps, Acta Math., 13 (1890), 1-270. Google Scholar

[153]

V. D. PopeM. GallaniP. R. Rowntree and R. A. Stratton, The impact of new physical parameterisations in the Hadley Centre climate model -HadAM3, Clim. Dyn., 16 (2000), 123-146. Google Scholar

[154]

F. W. Primeau, Multiple equilibria and low-frequency variability of the wind-driven ocean circulation, J. Phys. Oceanogr, 32 (2002), 2236-2256. doi: 10.1175/1520-0485(2002)032<2236:MEALFV>2.0.CO;2. Google Scholar

[155]

M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Springer, 2007. Google Scholar

[156]

B. B. Reinhold and R. P. Pierrehumbert, Dynamics of weather regimes: Quasi-stationary waves and blocking, Mon. Wea. Rev., 110 (1982), 1105-1145. Google Scholar

[157]

J. Robbin, A structural stability theorem, Annals Math., 94 (1971), 447-493. doi: 10.2307/1970766. Google Scholar

[158]

R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows, J. Fluid. Mech., 229 (1991), 291-310. doi: 10.1017/S0022112091003038. Google Scholar

[159]

C. Robinson, Structural stability of $C^1$ diffeomorphisms, J. Diff. Equations, 22 (1976), 28-73. doi: 10.1016/0022-0396(76)90004-8. Google Scholar

[160]

T. W. Ruff and J. D. Neelin, Long tails in regional surface temperature probability distributions with implications for extremes under global warming, Geophys. Res. Lett., 39 (2012). doi: 10.1029/2011GL050610. Google Scholar

[161]

A. Saunders and M. Ghil, A Boolean delay equation model of ENSO variability, Physica D, 160 (2001), 54-78. doi: 10.1016/S0167-2789(01)00331-1. Google Scholar

[162]

S. H. Schneider and R. E. Dickinson, Climate modeling, Rev. Geophys. Space Phys., 12 (1974), 447-493. Google Scholar

[163]

G. Sell, Non-autonomous differential equations and dynamical systems, Trans. Amer. Math. Soc., 127 (1967), 241-283. Google Scholar

[164]

V. A. SheremetG. R. Ierley and V. M. Kamenkovitch, Eigenanalysis of the two-dimensional wind-driven ocean circulation problem, J. Mar. Res., 55 (1997), 57-92. doi: 10.1357/0022240973224463. Google Scholar

[165]

E. Simonnet, On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains, Physica D, 237 (2008), 2539-2552. doi: 10.1016/j.physd.2008.04.008. Google Scholar

[166]

E. Simonnet and H. A. Dijkstra, Spontaneous generation of low-frequency modes of variability in the wind-driven ocean circulation, J. Phys. Oceanogr, 32 (2002), 1747-1762. doi: 10.1175/1520-0485(2002)032<1747:SGOLFM>2.0.CO;2. Google Scholar

[167]

E. Simonnet, R. Temam, S. Wang, M. Ghil and K. Ide, Successive bifurcations in a shallow-water ocean model, in: 16th Intl. Conf. Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, Springer-Verlag, 515 (1998), 225-230. doi: 10.1007/BFb0106588. Google Scholar

[168]

E. SimonnetM. GhilK. IdeR. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation, Part Ⅰ: Steady-state solutions, J. Phys. Oceanogr, 33 (2003), 712-728. doi: 10.1175/1520-0485(2003)33<712:LVISMO>2.0.CO;2. Google Scholar

[169]

E. SimonnetM. GhilK. IdeR. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part Ⅱ: Time-dependent solutions, Oceanogr, 33 (2003), 729-752. doi: 10.1175/1520-0485(2003)33<729:LVISMO>2.0.CO;2. Google Scholar

[170]

E. SimonnetM. Ghil and H. A. Dijkstra, Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation, J. Mar. Res., 63 (2005), 931-956. doi: 10.1357/002224005774464210. Google Scholar

[171]

E. Simonnet, H. A. Dijkstra and M. Ghil, Bifurcation analysis of ocean, atmosphere and climate models, in Computational Methods for the Ocean and the Atmosphere, R. Temam and J. J. Tribbia, Eds. , North-Holland, 14 (2009), 187--229. doi: 10.1016/S1570-8659(08)00203-2. Google Scholar

[172]

E. Simonnet, Quantization of the low-frequency variability of the double-gyre circulation, J. Phys. Oceanogr, 35 (2005), 2268-2290. doi: 10.1175/JPO2806.1. Google Scholar

[173]

Y. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64. Google Scholar

[174]

S. Smale, Structurally stable systems are not dense, American J. Math., 88 (1966), 491-496. doi: 10.2307/2373203. Google Scholar

[175]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

[176]

R. J. SmallS. P. DeSzoekeS. P. XieL. O'NeillH. SeoQ. Song and P. Cornillon, Air-sea interaction over ocean fronts and eddies, Dyn. Atmos. Oceans, 45 (2008), 274-319. doi: 10.1016/j.dynatmoce.2008.01.001. Google Scholar

[177]

C. Soize, The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, World Scientic Publishing Co. , 1994. doi: 10.1142/9789814354110. Google Scholar

[178]

S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor and H. L. Miller (Eds.), Climate Change 2007: The Physical Science Basis. Contribution of Working Group Ⅰ to the Fourth Assessment Report of the IPCC, Cambridge University Press, Cambridge, UK and New York, NY, USA, 2007.Google Scholar

[179]

S. SpeichH. A. Dijkstra and M. Ghil, Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation, Nonlin, Processes Geophys, 2 (1995), 241-268. Google Scholar

[180]

D. A. Stainforth, Uncertainty in predictions of the climate response to rising levels of greenhouse gases, Nature, 433 (2005), 403-406. Google Scholar

[181]

B. Stevens, Y. Zhang and M. Ghil, Stochastic effects in the representation of stratocumulus-topped mixed layers, Proc. ECMWF Workshop on Representation of Sub-grid Processes Using Stochastic-Dynamic Models, 6-8 June 2005, Shinfield Park, Reading, UK, pp. 79-90.Google Scholar

[182]

T. F. Stocker, D. Qin, G. K. Plattner, M. Tignor, S. K. Allen, J. Boschung, A. Nauels, Y. Xia, B. Bex and B. M. Midgley, (Eds.), Climate Change 2013. The Physical Science Basis: Contribution of Working Group Ⅰ to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, UK and New York, NY, USA, 2013.Google Scholar

[183]

H. Stommel, Thermohaline convection with two stable regimes of flow, Tellus, 13 (1961), 224-230. Google Scholar

[184] H. Stommel, The Gulf Stream: A Physical and Dynamical Description, 2nd ed, Cambridge Univ. Press, London, 1965. Google Scholar
[185]

L. SushamaM. Ghil and K. Ide, Spatio-temporal variability in a mid-latitude ocean basin subject to periodic wind forcing, Atmosphere-Ocean, 45 (2007), 227-250. doi: 10.3137/ao.450404. Google Scholar

[186]

H. U. Sverdrup, Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific, Proc. Natl. Acad. Sci. USA, 33 (1947), 318-326. doi: 10.1073/pnas.33.11.318. Google Scholar

[187]

H. U. Sverdrup, M. W. Johnson and R. H. Fleming, The oceans: Their physics, chemistry and general biology, Prentice-Hall, New York, 1942, available at http://ark.cdlib.org/ark:/13030/kt167nb66r/Google Scholar

[188] T. Tél and M. Gruiz, Chaotic Dynamics, Cambridge University Press, 2006. doi: 10.1017/CBO9780511803277. Google Scholar
[189]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 2nd Ed. , 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar

[190]

P. D. Thompson, Numerical Weather Analysis and Prediction, Macmillan, New York, 1961,170 pp. Google Scholar

[191]

L. N. TrefethenA. TrefethenS. C. Reddy and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261 (1993), 578-584. doi: 10.1126/science.261.5121.578. Google Scholar

[192]

W. Tucker, Lorenz attractor exists, C. R. Acad. Sci. Paris, 328 (1999), 1197-1202. doi: 10.1016/S0764-4442(99)80439-X. Google Scholar

[193]

E. TzipermanL. StoneM. Cane and H. Jarosh, El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator, Science, 264 (1994), 72-74. Google Scholar

[194]

E. TzipermanM. A. Cane and S. E. Zebiak, Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos, J. Atmos. Sci., 50 (1995), 293-306. doi: 10.1175/1520-0469(1995)052<0293:IALTTS>2.0.CO;2. Google Scholar

[195]

G. Vallis, Amospheric and Oceanic Fluid Dynamics, Cambridge University Press, UK, 745 pp. , 2006.Google Scholar

[196]

S. Vannitsem, Dynamics and predictability of a low-order wind-driven ocean--atmosphere coupled model, Clim. Dyn., 42 (2014), 1981-1998. doi: 10.1007/s00382-013-1815-8. Google Scholar

[197]

S. Vannitsem and L. De Cruz, A 24-variable low-order coupled ocean-atmosphere model: OA-QG-WS v2, Geoscientific Model Development, 7 (2014), 649-662. Google Scholar

[198]

S. VannitsemJ. DemaeyerL. De Cruz and M. Ghil, Low-frequency variability and heat transport in a low-order nonlinear coupled ocean-atmosphere model, Physica D, 309 (2015), 71-85. doi: 10.1016/j.physd.2015.07.006. Google Scholar

[199]

T. Wanner, Linearization of random dynamical systems, in: C. K. R. T. Jones, U. Kirchgraber, and H. O. Walther (Eds), Dynamics Reported, Springer, Berlin, Heidelberg, New York, 4 (1995), 203-269. Google Scholar

[200]

R. F. Williams, The structure of Lorenz attractors, Publ. Math. I.H.E.S., 50 (1979), 73-99. Google Scholar

[201]

C. Wunsch, The interpretation of short climate records, with comments on the North Atlantic and Southern Oscillations, Bull. Am. Meteorol. Soc., 80 (1999), 245-255. doi: 10.1175/1520-0477(1999)080<0245:TIOSCR>2.0.CO;2. Google Scholar

[202]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717. Google Scholar

show all references

References:
[1]

M. R. Allen, Do-it-yourself climate prediction, Nature, 401 (1999), p642. doi: 10.1038/44266. Google Scholar

[2]

A. A. Andronov and L. S. Pontryagin, Systémes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-250. Google Scholar

[3]

V. S. AnishchenkoT. E. VadivasovaA. S. KopeikinJ. Kurths and G. I. Strelkova, Effect of noise on the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors, Phys. Rev. Lett., 87 (2001), 054101. doi: 10.1103/PhysRevLett.87.054101. Google Scholar

[4]

V. S. AnishchenkoT. E. VadivasovaA. S. KopeikinG. I. Strelkova and J. Kurths, Influence of noise on statistical properties of nonhyperbolic attractors, Phys. Rev. E, 62 (2000), p7886. doi: 10.1103/PhysRevE.62.7886. Google Scholar

[5]

V. S. AnishchenkoT. E. VadivasovaA. S. KopeikinG. I. Strelkova and J. Kurths, Peculiarities of the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors in the presence of noise, Phys. Rev. E, 65 (2002), 036206. doi: 10.1103/PhysRevE.65.036206. Google Scholar

[6]

V. AraujoM. PacificoR. Pujal and M. Viana, Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485. doi: 10.1090/S0002-9947-08-04595-9. Google Scholar

[7]

L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar

[8]

L. Arnold, Trends and open problems in the theory of random dynamical systems, in: Probability Towards 2000, L. Accardi, C. C. Heyde (Eds.), Springer Lecture Notes in Statistics, 128 (1998), 34-46. doi: 10.1007/978-1-4612-2224-8_2. Google Scholar

[9]

L. Arnold and K. Xu, Normal forms for random differential equations, J. Diff. Eq., 116 (1995), 484-503. doi: 10.1006/jdeq.1995.1045. Google Scholar

[10]

L. Arnold and P. Imkeller, Normal forms for stochastic differential equations, Prob. Theory Relat. Fields, 110 (1998), 559-588. doi: 10.1007/s004400050159. Google Scholar

[11]

V. I. Arnol'd, Geometrical Methods in the Theory of Differential Equations, Springer, 1983,334 pp. Google Scholar

[12]

P. Bak, The devil's staircase, Physics Today, 39 (1986), 38-45. doi: 10.1063/1.881047. Google Scholar

[13]

P. Bak and R. Bruinsma, One-dimensional Ising model and the complete devil's staircase, Phys. Rev. Lett., 49 (1982), 249-251. doi: 10.1103/PhysRevLett.49.249. Google Scholar

[14]

J. J. Barsugli and D. S. Battisti, The basic effects of atmosphere-ocean thermal coupling on midlatitude variability, J. Atmos. Sci., 5 (1998), 477-493. doi: 10.1175/1520-0469(1998)055<0477:TBEOAO>2.0.CO;2. Google Scholar

[15]

D. R. Bell, Degenerate Stochastic Differential Equations and Hypoellipticity, Longman, Harlow, 1995. Google Scholar

[16]

A. Berger and S. Siegmund, On the gap between random dynamical systems and continuous skew products, J. Dyn. Diff. Eq., 15 (2003), 237-279. doi: 10.1023/B:JODY.0000009736.39445.c4. Google Scholar

[17]

P. BerloffA. Hogg and W. Dewar, The turbulent oscillator: A mechanism of low-frequency variability of the wind-driven ocean gyres, J. Phys. Oceanogr., J. Phys. Oceanogr., 37 (2007), 2363-2386. doi: 10.1175/JPO3118.1. Google Scholar

[18]

K. BhattacharyaM. Ghil and I. L. Vulis, Internal variability of an energy-balance model with delayed albedo effects, J. Atmos. Sci., 39 (1982), 1747-1773. doi: 10.1175/1520-0469. Google Scholar

[19]

T. BódaiG. Károlyi and T. Tél, A chaotically driven model climate: Extreme events and snapshot attractors, Nonlin. Processes Geophys, 18 (2011), 573-580. Google Scholar

[20]

T. BódaiV. LucariniF. Lunkeit and R. Boschi, Global instability in the Ghil-Sellers model, Clim. Dyn., 44 (2015), 3361-3381. Google Scholar

[21]

T. Bogenschütz and Z. S. Kowalski, A condition for mixing of skew products, Aequationes Math., 59 (2000), 222-234. doi: 10.1007/s000100050122. Google Scholar

[22]

F. Bouchet and J. Sommeria, Emergence of intense jets and Jupiter's great red spot as maximum entropy structures, J. Fluid. Mech., 464 (2002), 165-207. doi: 10.1017/S0022112002008789. Google Scholar

[23]

S. BrachetF. CodronY. FeliksM. GhilH. Le Treut and E. Simonnet, Atmospheric circulations induced by a mid-latitude SST front: A GCM study, J. Clim., 25 (2012), 1847-1853. doi: 10.1175/JCLI-D-11-00329.1. Google Scholar

[24]

A. BraccoJ. D. NeelinH. LuoJ. C. McWilliams and J. E. Meyerson, High-dimensional decision dilemmas in climate models, Geosci. Model Dev., 6 (2013), 1673-1687. doi: 10.5194/gmdd-6-2731-2013. Google Scholar

[25]

A. N. CarvalhoJ. A. Langa and J. C. Robinson, Lower semicontinuity of attractors for non-autonomous dynamical systems, Ergod. Th.. Dyn. Syst., 29 (2009), 1765-1780. doi: 10.1017/S0143385708000850. Google Scholar

[26]

A. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar

[27]

P. Cessi and G. R. Ierley, Symmetry-breaking multiple equilibria in quasigeostrophic, wind-driven flows, J. Phys. Oceanogr, 25 (1995), 1196-1205. Google Scholar

[28]

K. I. ChangK. IdeM. Ghil and C.-C. A. Lai, Transition to aperiodic variability in a wind-driven double-gyre circulation model, J. Phys. Oceanogr, 31 (2001), 1260-1286. Google Scholar

[29]

C. P. Chang, M. Ghil, M. Latif and J. M. Wallace (Eds.), Climate Change: Multidecadal and Beyond, World Scientific Publ. Co. /Imperial College Press, 2015,388 pp.Google Scholar

[30]

P. ChangL. JiH. Li and M. Flugel, Chaotic dynamics versus stochastic processes in El Niño-Southern Oscillation in coupled ocean-atmosphere models, Physica D, 98 (1996), 301-320. Google Scholar

[31]

Y. ChaoM. Ghil and J. C. McWilliams, Pacific interdecadal variability in this century's sea surface temperatures, Geophys. Res. Lett., 27 (), 2261-2264. Google Scholar

[32] J. G. Charney, Carbon Dioxide and Climate: A Scientific Assessment, National Academies Press, Washington, D.C, 1979. Google Scholar
[33]

J. G. Charney and D. M. Straus, Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, orographically forced, planetary wave systems, J. Atmos. Sci., 37 (1980), 1157-1176. Google Scholar

[34]

M. Chavez, M. Ghil and J. Urrutia Fucugauchi (Eds.), Extreme Events: Observations, Modeling and Economics, Geophysical Monograph 214, American Geophysical Union & Wiley, 2015,438 pp.Google Scholar

[35]

M. D. ChekrounD. Kondrashov and M. Ghil, Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation, Proc. Natl. Acad. Sci USA, 108 (2011), 11766-11771. Google Scholar

[36]

M. D. ChekrounE. Simonnet and M. Ghil, Stochastic climate dynamics: Random attractors and time-dependent invariant measures, Physica D, 240 (2011), 1685-1700. doi: 10.1016/j.physd.2011.06.005. Google Scholar

[37]

M. D. ChekrounJ. D. NeelinD. KondrashovJ. C. McWilliams and M. Ghil, Rough parameter dependence in climate models: The role of Ruelle-Pollicott resonances, Proc. Natl. Acad. Sci. USA, 111 (2014), 1684-1690. doi: 10.1073/pnas.1321816111. Google Scholar

[38]

M. D. ChekrounM. GhilH. Liu and S. Wang, Low-dimensional Galerkin approximations of nonlinear delay differential equations, Discr. Cont. Dyn. Syst., 36 (2016), 4133-4177. doi: 10.3934/dcds.2016.36.4133. Google Scholar

[39]

M. D. Chekroun, H. Liu and S. Wang, Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs Ⅰ, Springer Briefs in Mathematics, Springer, 2015. doi: 10.1007/978-3-319-12496-4. Google Scholar

[40]

M. D. Chekroun, M. Ghil and J. D. Neelin, Invariant measures on climatic pullback attractors, in preparation, 2016.Google Scholar

[41]

E. A. Coayla-Teran and P. R. C. Ruffino, Random versions of Hartman-Grobman theorems, Preprint IMECC, UNICAMP, No. 27/01 (2001).Google Scholar

[42]

P. Collet and C. Tresser, Ergodic theory and continuity of the Bowen-Ruelle measure for geometrical flows, Fyzika, 20 (1988), 33-48. Google Scholar

[43] N. D. Cong, Topological Dynamics of Random Dynamical Systems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1997. Google Scholar
[44]

H. Crauel, White noise eliminates instablity, Arch. Math., 75 (2000), 472-480. doi: 10.1007/s000130050532. Google Scholar

[45]

H. Crauel, A uniformly exponential attractor which is not a pullback attractor, Arch. Math, 78 (2002), 329-336. doi: 10.1007/s00013-002-8254-9. Google Scholar

[46]

H. Crauel, Random Probability Measures on Polish Spaces, Stochastic Monographs, vol. 11, Taylor & Francis, 2002. Google Scholar

[47]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Scuola Normale Superiore Pisa 148 (1992), Technical Report.Google Scholar

[48]

H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation, J. Dyn. Diff. Eqn., 10 (1998), 259-274. doi: 10.1023/A:1022665916629. Google Scholar

[49]

E. D. Da Costa and A. C. Colin de Verdière, The 7.7 year North Atlantic oscillation, Q. J. R. Meteorol. Soc., 128 (2002), 797-817. doi: 10.1256/0035900021643692. Google Scholar

[50]

A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pure Appl., 11 (1932), 333-375. Google Scholar

[51]

M. Dorfle and R. Graham, Probability density of the Lorenz model, Phys. Rev. A, 27 (1983), 1096-1105. doi: 10.1103/PhysRevA.27.1096. Google Scholar

[52]

H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño (2nd ed. ), Springer, 2005,532 pp. doi: 10.1007/978-94-015-9450-9. Google Scholar

[53]

H. A. Dijkstra, Nonlinear Climate Dynamics, Cambridge Univ. Press, 2013,367 pp. doi: 10.1017/CBO9781139034135. Google Scholar

[54]

H. A. Dijkstra and M. Ghil, Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach, Rev. Geophys., 43 (2005), RG3002. doi: 10.1029/2002RG000122. Google Scholar

[55]

H. A. Dijkstra and C. A. Katsman, Temporal variability of the wind-driven quasi-geostrophic double gyre ocean circulation: basic bifurcation diagrams, Geophys. Astrophys. Fluid Dyn., 85 (1997), 195-232. doi: 10.1080/03091929708208989. Google Scholar

[56]

R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Theor. Prob. Appl., 15 (1970), 458-486. doi: 10.1137/1115049. Google Scholar

[57]

G. DrótosT. Bódai and T. Tél, Probabilistic concepts in a changing climate: A snapshot attractor picture, J. Clim., 28 (2015), 3275-3288. Google Scholar

[58]

M. 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.) de l'aqueduc de Fréjus, Archeoscience, 30 (2006), 163-171. Google Scholar

[59]

J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617. Google Scholar

[60]

B. F. Farrel and P. J. Ioannou, Structural stability of turbulent jets, J. Atmos. Sci., 60 (2003), 2101-2118. doi: 10.1175/1520-0469(2003)060<2101:SSOTJ>2.0.CO;2. Google Scholar

[61]

M. J. FeigenbaumL. P. Kadanoff and S. J. Shenker, Quasiperiodicity in dissipative systems: A renormalization group analysis, Physica D, 5 (1982), 370-386. doi: 10.1016/0167-2789(82)90030-6. Google Scholar

[62]

Y. FeliksM. Ghil and E. Simonnet, Low-frequency variability in the mid-latitude atmosphere induced by an oceanic thermal front, J. Atmos. Sci., 61 (2004), 961-981. Google Scholar

[63]

Y. FeliksM. Ghil and E. Simonnet, Low-frequency variability in the mid-latitude baroclinic atmosphere induced by an oceanic thermal front, J. Atmos. Sci., 64 (2007), 97-116. Google Scholar

[64]

Y. FeliksM. Ghil and A. W. Robertson, Oscillatory climate modes in the Eastern Mediterranean and their synchronization with the north Atlantic Oscillation, J. Clim., 23 (2010), 4060-4079. doi: 10.1175/2010JCLI3181.1. Google Scholar

[65]

Y. FeliksM. Ghil and A. W. Robertson, The atmospheric circulation over the North Atlantic as induced by the SST field, J. Clim., 24 (2011), 522-542. doi: 10.1175/2010JCLI3859.1. Google Scholar

[66]

E. Galanti and E. Tziperman, ENSO's phase locking to the seasonal cycle in the fast-SST, fast-wave, and mixed-mode regimes, J. Atmos. Sci., 57 (2000), 2936-2950. doi: 10.1175/1520-0469(2000)057<2936:ESPLTT>2.0.CO;2. Google Scholar

[67]

M. Ghil, Steady-State Solutions of a Diffusive Energy-Balance Climate Model and Their Stability, Report IMM-410, Courant Institute of Mathematical Sciences, New York University, New York, 74 pp. , 1975; available in the Classic Reprint Series of Förlag Forgotten Books, http://www.bokus.com/bok/9781332200214/steady-state-solutions-of-a-diffusive-energy-balance-climate-model\-and-their-stability-classic-reprint/Google Scholar

[68]

M. Ghil, Climate stability for a Sellers-type model, J. Atmos. Sci., 33 (1976), 3-20. doi: 10.1175/1520-0469(1976)033<0003:CSFAST>2.0.CO;2. Google Scholar

[69]

M. Ghil, Cryothermodynamics: The chaotic dynamics of paleoclimate, Physica D, 77 (1994), 130-159. doi: 10.1016/0167-2789(94)90131-7. Google Scholar

[70]

M. Ghil, Hilbert problems for the geosciences in the 21st century, Nonlin. Proc. Geophys, 8 (2001), 211-211. doi: 10.5194/npg-8-211-2001. Google Scholar

[71]

M. Ghil, A mathematical theory of climate sensitivity or, How to deal with both anthropogenic forcing and natural variability?, in Climate Change: Multidecadal and Beyond, C. P. Chang, M. Ghil, M. Latif and J. M. Wallace (Eds.), World Scientific Publ. Co. /Imperial College Press, 2015, 31-51.Google Scholar

[72]

M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, Springer-Verlag, Berlin/Heidelberg/New York, 1987,512 pp. doi: 10.1007/978-1-4612-1052-8. Google Scholar

[73]

M. Ghil and R. Vautard, Interdecadal oscillations and the warming trend in global temperature time series, Nature, 350 (1991), 324-327. doi: 10.1038/350324a0. Google Scholar

[74]

M. Ghil and N. Jiang, Recent forecast skill for the El Niño/Southern Oscillation, Geophys. Res. Lett., 25 (1998), 171-174. Google Scholar

[75]

M. Ghil and A. W. Robertson, Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy, in: General Circulation Model Development: Past, Present and Future, D. Randall (Ed. ), Academic Press, San Diego, 70 (2000), 285--325. doi: 10.1016/S0074-6142(00)80058-3. Google Scholar

[76]

M. Ghil and A. W. Robertson, "Waves" vs "particles" in the atmosphere's phase space: A pathway to long-range forecasting?, Proc. Natl. Acad. Sci. USA, 99 (2002), 2493-2500. doi: 10.1073/pnas.012580899. Google Scholar

[77]

M. GhilY. Feliks and L. Sushama, Baroclinic and barotropic aspects of the wind-driven ocean circulation, Physica D, 167 (2002), 1-35. doi: 10.1016/S0167-2789(02)00392-5. Google Scholar

[78]

M. GhilR. M. AllenM. D. DettingerK. IdeD. KondrashovM. E. MannA. RobertsonA. SaundersY. TianF. Varadi and P. Yiou, Advanced spectral methods for climatic time series, Rev. Geophys., 40 (2002), 3.1-3.41. doi: 10.1029/2000RG000092. Google Scholar

[79]

M. GhilM. D. Chekroun and E. Simonnet, Climate dynamics and fluid mechanics: Natural variability and related uncertainties, Physica D, 237 (2008), 2111-2126. doi: 10.1016/j.physd.2008.03.036. Google Scholar

[80]

M. GhilI. Zaliapin and S. Thompson, A delay differential model of ENSO variability: Parametric instability and the distribution of extremes, Nonlin. Processes Geophys, 15 (2008), 417-433. doi: 10.5194/npg-15-417-2008. Google Scholar

[81]

M. GhilP. YiouS. HallegatteB. D. MalamudP. NaveauA. SolovievP. FriederichsV. Keilis-BorokD. KondrashovV. KossobokovO. MestreC. NicolisH. RustP. ShebalinM. VracA. Witt and I. Zaliapin, Extreme events: Dynamics, statistics and prediction, Nonlin. Processes Geophys, 18 (2011), 295-350. doi: 10.5194/npg-18-295-2011. Google Scholar

[82]

M. Ghil and I. Zaliapin, Understanding ENSO variability and its extrema: A delay differential equation approach, Ch. 6 in Extreme Events: Observations, Modeling and Economics, M. Chavez, M. Ghil and J. Urrutia-Fucugauchi (Eds.), Geophysical Monograph, American Geophysical Union & Wiley, 214 (2015), 63--78.Google Scholar

[83]

A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, 1982,662 pp.Google Scholar

[84]

C. GrebogiH. KantzA. PrasadY. C. Lai and E Sinde, Unexpected robustness-against-noise of a class of nonhyperbolic chaotic attractors, Phys. Rev. E, 65 (2002), 026209, 8 pp. doi: 10.1103/PhysRevE.65.026209. Google Scholar

[85]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (2nd ed. ), Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2. Google Scholar

[86]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Publ. Math. I.H.E.S., 50 (1979), 59-72. Google Scholar

[87]

I. M. Held, The gap between simulation and understanding in climate modeling, Bull. American. Meteorol. Soc., 86 (2005), 1609-1614. doi: 10.1175/BAMS-86-11-1609. Google Scholar

[88]

R. Hillerbrand and M. Ghil, Anthropogenic climate change: Scientific uncertainties and moral dilemmas, in this volume, Physica D, 2008.Google Scholar

[89]

J. T. Houghton, G. J. Jenkins and J. J. Ephraums (Eds.), Climate Change, The IPCC Scientific Assessment, Cambridge Univ. Press, Cambridge, MA, 1991,365 pp.Google Scholar

[90]

J. T. Houghton, Y. Ding, D. J. Griggs, M. Noguer, P. J. van der Linden, X. Dai, K. Maskell and C. A. Johnson (Eds.), Climate Change 2001: The Scientific Basis. Contribution of Working Group Ⅰ to the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC), Cambridge University Press, Cambridge, U. K. , 2001,944 pp.Google Scholar

[91]

S. Jiang, F. -F. Jin and M. Ghil, The nonlinear behavior of western boundary currents in a wind-driven, double-gyre, shallow-water model in: Ninth Conf. Atmos. & Oceanic Waves and Stability (San Antonio, TX), American Meterorological Society, Boston, Mass. , 1993, pp. 64-67.Google Scholar

[92]

S. JiangF.-F. Jin and M. Ghil, Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J. Phys. Oceanogr, 25 (1995), 764-786. doi: 10.1175/1520-0485(1995)025<0764:MEPAAS>2.0.CO;2. Google Scholar

[93]

F.-F. JinJ. D. Neelin and M. Ghil, El Niño on the Devil's Staircase: Annual subharmonic steps to chaos, Science, 264 (1994), 70-72. Google Scholar

[94]

F.-F. JinJ. D. Neelin and M. Ghil, El Niño/Southern Oscillation and the annual cycle: Subharmonic frequency locking and aperiodicity, Physica D, 98 (1996), 442-465. Google Scholar

[95]

T. Jung, T. N. Palmer and G. J. Shutts, Geophys. Res. Lett. , 32 (2005), Art. No. L23811.Google Scholar

[96]

T. Kaijser, On stochastic perturbations of iterations of circle maps, Physica D, 68 (1993), 201-231. doi: 10.1016/0167-2789(93)90081-B. Google Scholar

[97] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge Univ. Press, Cambridge/London, UK, 2003. doi: 10.1017/CBO9780511802270. Google Scholar
[98]

J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Differential Equations and Approximations of Fixed Points, H. -O. Peitgen and H. -O. Walter (Eds.), Lecture Notes in Mathematics, (Springer, Berlin), 730 (1979), 204-227. Google Scholar

[99]

A. Katok and B. Haselblatt, Introduction to the Modern Theroy of Dynamical Systems, Cambridge Univ. Press, Encycl. Math. Appl. , 54 1995. doi: 10.1017/CBO9780511809187. Google Scholar

[100]

C. A. KatsmanH. A. Dijkstra and S. S. Drijfhout, The rectification of the wind-driven ocean circulation due to its instabilities, J. Mar. Res., 56 (1998), 559-587. Google Scholar

[101]

Y. Kifer, Ergodic Theory of Random Perturbations, Birkhäuser, 1988.Google Scholar

[102]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. Google Scholar

[103]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, vol. 176, American Mathematical Society, 2011. doi: 10.1090/surv/176. Google Scholar

[104]

J. J. Kohn, Pseudo-differential operators and hypoellipticity, Proc. Amer. Math. Soc. Symp. Pure Math., 23 (1973), 61-69. Google Scholar

[105]

D. KondrashovY. Feliks and M. Ghil, Oscillatory modes of extended Nile River records (A.D. 622-1922), Geophys. Res. Lett., 32 (2005), L10702. doi: 10.1029/2004GL022156. Google Scholar

[106]

K. KondrashovM. D. ChekrounA. W. Robertson and M. Ghil, Low-order stochastic model and "past-noise forecasting" of the Madden-Julian oscillation, Geophys. Res. Lett., 40 (2013), 5303-5310. doi: 10.1002/grl.50991. Google Scholar

[107]

D. KondrashovM. D. Chekroun and M. Ghil, Data-driven non-Markovian closure models, Physica D, 297 (2015), 33-55. doi: 10.1016/j.physd.2014.12.005. Google Scholar

[108]

S. KravtsovP. BerloffW. K. DewarM. Ghil and J. C. McWilliams, Dynamical origin of low-frequency variability in a highly nonlinear mid-latitude coupled model, J. Climate, 19 (2007), 6391-6408. Google Scholar

[109]

S. Kuksin and A. Shirikyan, On random attractors for mixing-type systems, Funct. Anal. Appl., 38 (2004), 34-46. doi: 10.1023/B:FAIA.0000024865.78811.11. Google Scholar

[110]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York (3rd ed. ), 2004. doi: 10.1007/978-1-4757-3978-7. Google Scholar

[111]

J. A. LangaJ. C. Robinson and A. Suarez, Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity, 15 (2002), 887-903. doi: 10.1088/0951-7715/15/3/322. Google Scholar

[112]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Applied Mathematical Sciences, vol. 97, Springer-Verlag, 1994. doi: 10.1007/978-1-4612-4286-4. Google Scholar

[113]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Prob. Theory Related Fields, 80 (1988), 217-240. doi: 10.1007/BF00356103. Google Scholar

[114]

W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988. doi: 10.1002/cpa.20083. Google Scholar

[115]

J. W. B. Lin and J. D. Neelin, Influence of a stochastic moist convective parameterization on tropical climate variability, Geophys. Res. Lett., 27 (2000), 3691-3694. doi: 10.1029/2000GL011964. Google Scholar

[116]

J. W. B. Lin and J. D. Neelin, Considerations for stochastic convective parameterization, J. Atmos. Sci., 59 (2002), 959-975. doi: 10.1175/1520-0469(2002)059<0959:CFSCP>2.0.CO;2. Google Scholar

[117]

J. W. B. Lin and J. D. Neelin, Toward stochastic deep convective parameterization in general circulation models, Geophys. Res. Lett., 30 (2003), p1162. doi: 10.1029/2002GL016203. Google Scholar

[118]

P. C. Loikith and J. D. Neelin, Short-tailed temperature distributions over North America and implications for future changes in extremes, Geophys. Res. Lett., 42 (2015), 8577-8585. doi: 10.1002/2015GL065602. Google Scholar

[119]

E. N. Lorenz, Deterministic nonperiodic flow, The Theory of Chaotic Attractors, (2004), 25-36. doi: 10.1007/978-0-387-21830-4_2. Google Scholar

[120] E. N. Lorenz, The Essence of Chaos, Univ. of Washington Press, 1993. doi: 10.4324/9780203214589. Google Scholar
[121]

V. Lucarini and S. Sarno, A statistical mechanical approach for the computation of the climatic response to general forcings, Nonlin. Processes Geophys, 18 (2011), 7-28. doi: 10.5194/npg-18-7-2011. Google Scholar

[122]

V. LucariniR. BlenderC. HerbertF. RagoneS. Pascale and J. Wouters, Mathematical and physical ideas for climate science, Rev. Geophys, 52 (2014), 809-859. doi: 10.1002/2013RG000446. Google Scholar

[123]

V. LucariniF. Ragone and F. Lunkeit, Predicting climate change using response theory: Global averages and spatial patterns, Journal of Statistical Physics, (2016), 1-29. doi: 10.1007/s10955-016-1506-z. Google Scholar

[124]

R. A. Madden and P. R. Julian, Observations of the 40--50-day tropical oscillations -a review, Mon. Wea. Rev., 122 (1994), 814-837. doi: 10.1175/1520-0493(1994)122<0814:OOTDTO>2.0.CO;2. Google Scholar

[125]

A. Majda and X. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge Univ. Press, 551 pp. , 2006. doi: 10.1017/CBO9780511616778. Google Scholar

[126]

R. Mañé, A proof of the $ C^1$-stability conjecture, Publ. Math, I.H.E.S., 66 (1988), 161-210. Google Scholar

[127]

N. J. MantuaS. HareY. ZhangJ. M. Wallace and R. C. Francis, A Pacific interdecadal climate oscillation with impacts on salmon production, Bull. Am. Meteor. Soc., 78 (1997), 1069-1079. doi: 10.1175/1520-0477(1997)078<1069:APICOW>2.0.CO;2. Google Scholar

[128]

J. C. McWilliams, Irreducible imprecision in atmospheric and oceanic simulations, Proc. Natl. Acad. Sci. USA, 104 (2007), 8709-8713. doi: 10.1073/pnas.0702971104. Google Scholar

[129]

J. C. McWilliams, Fundamentals of Geophysical Fluid Dynamics, 2nd ed. , Cambridge Univ. Press, 272 pp. , 2011.Google Scholar

[130]

S. P. Meacham, Low-frequency variability in the wind-driven circulation, J. Phys. Oceanogr, 30 (2000), 269-293. doi: 10.1175/1520-0485(2000)030<0269:LFVITW>2.0.CO;2. Google Scholar

[131]

G. A. Meehl, Decadal climate variability and the early-2000s hiatus, in Understanding the Earth's Climate Warming Hiatus: Putting the Pieces Together, D. Menemenlis and J. Sprintall (Eds.), US CLIVAR Variations, US CLIVAR Project Office, 13 (2015), 1-6,Google Scholar

[132]

S. MinobeA. Kuwano-YoshidaN. KomoriS.-P. Xie and R. J. Small, Influence of the Gulf Stream on the troposphere, Nature, 452 (2008), 206-209. doi: 10.1038/nature06690. Google Scholar

[133]

J. F. B. Mitchell, Can we believe predictions of climate change?, Q. J. Roy. Meteorol. Soc. (Part A), 130 (2004), 2341-2360. doi: 10.1256/qj.04.74. Google Scholar

[134]

A. K. MittalS. Dwivedi and R. S. Yadav, Probability distribution for the number of cycles between successive regime transitions for the Lorenz model, Physica D, 233 (2007), 14-20. doi: 10.1016/j.physd.2007.06.014. Google Scholar

[135]

V. MoronR. Vautard and M. Ghil, Trends, interdecadal and interannual oscillations in global sea-surface temperatures, Clim. Dyn., 14 (1998), 545-569. doi: 10.1007/s003820050241. Google Scholar

[136]

N. T. Nadiga and B. P. Luce, Global bifurcation of Shilnikov type in a double-gyre ocean model, J. Phys. Oceanogr, 31 (2001), 2669-2690. doi: 10.1175/1520-0485(2001)031<2669:GBOSTI>2.0.CO;2. Google Scholar

[137]

National Research Council, Natural Climate Variability on Decade-to-Century Time Scales, D. G. Martinson, K. Bryan, M. Ghil et al. (Eds.), National Academies Press, Washington, D. C. , 1995,630 pp.Google Scholar

[138]

J. D. NeelinD. S. BattistiA. C. HirstF.-F. JinY. WakataT. Yamagata and S. Zebiak, ENSO Theory, J. Geophys. Res., 103 (1998), 14261-14290. doi: 10.1029/97JC03424. Google Scholar

[139]

S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. I.H.E.S., 50 (1979), 101-151. Google Scholar

[140]

N. Ohtomo, Exponential characteristics of power spectral densities caused by chaotic phenomena, J. Phys. Soc. Japan, 64 (1995), 1104-1113. Google Scholar

[141]

V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231. Google Scholar

[142]

J. Palis, A global perspective for non-conservative dynamics, Ann. I.H. Poincaré, 22 (2005), 485-507. doi: 10.1016/j.anihpc.2005.01.001. Google Scholar

[143]

T. N. Palmer, The prediction of uncertainty in weather and climate forecasting, Rep. Prog. Phys., 63 (2000), 71-116. Google Scholar

[144]

J. Pedlosky, Geophysical Fluid Dynamics (2nd ed. ), Springer-Verlag, 1987,710 pp.Google Scholar

[145]

J. Pedlosky, Ocean Circulation Theory, Springer, New York, 1996. doi: 10.1007/978-3-662-03204-6. Google Scholar

[146]

M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-110. doi: 10.1016/0040-9383(65)90018-2. Google Scholar

[147]

S. Pierini, Low-frequency variability, coherence resonance, and phase selection in a low-order model of the wind-driven ocean circulation, J. Phys. Oceanogr, 41 (2011), 1585-1604. doi: 10.1175/JPO-D-10-05018.1. Google Scholar

[148]

S. Pierini, Stochastic tipping points in climate dynamics, Phys. Rev. E, 85 (2012), 027101. doi: 10.1103/PhysRevE.85.027101. Google Scholar

[149]

S. Pierini, Ensemble simulations and pullback attractors of a periodically forced double-gyre sysyem, J. Phys. Oceanogr, 44 (2014), 3245-3254. Google Scholar

[150]

S. PieriniM. Ghil and M. D. Chekroun, Exploring the pullback attractors of a low-order quasigeostrophic ocean model: The deterministic case, J. Clim., in press, (2015). doi: 10.1175/JCLI-D-15-0848.1. Google Scholar

[151]

G. PlautM. Ghil and R. Vautard, Interannual and interdecadal variability in 335 Years of Central England temperatures, Science, 268 (1995), 710-713. doi: 10.1126/science.268.5211.710. Google Scholar

[152]

H. Poincaré, Sur les équations de la dynamique et le problème des trois corps, Acta Math., 13 (1890), 1-270. Google Scholar

[153]

V. D. PopeM. GallaniP. R. Rowntree and R. A. Stratton, The impact of new physical parameterisations in the Hadley Centre climate model -HadAM3, Clim. Dyn., 16 (2000), 123-146. Google Scholar

[154]

F. W. Primeau, Multiple equilibria and low-frequency variability of the wind-driven ocean circulation, J. Phys. Oceanogr, 32 (2002), 2236-2256. doi: 10.1175/1520-0485(2002)032<2236:MEALFV>2.0.CO;2. Google Scholar

[155]

M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Springer, 2007. Google Scholar

[156]

B. B. Reinhold and R. P. Pierrehumbert, Dynamics of weather regimes: Quasi-stationary waves and blocking, Mon. Wea. Rev., 110 (1982), 1105-1145. Google Scholar

[157]

J. Robbin, A structural stability theorem, Annals Math., 94 (1971), 447-493. doi: 10.2307/1970766. Google Scholar

[158]

R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows, J. Fluid. Mech., 229 (1991), 291-310. doi: 10.1017/S0022112091003038. Google Scholar

[159]

C. Robinson, Structural stability of $C^1$ diffeomorphisms, J. Diff. Equations, 22 (1976), 28-73. doi: 10.1016/0022-0396(76)90004-8. Google Scholar

[160]

T. W. Ruff and J. D. Neelin, Long tails in regional surface temperature probability distributions with implications for extremes under global warming, Geophys. Res. Lett., 39 (2012). doi: 10.1029/2011GL050610. Google Scholar

[161]

A. Saunders and M. Ghil, A Boolean delay equation model of ENSO variability, Physica D, 160 (2001), 54-78. doi: 10.1016/S0167-2789(01)00331-1. Google Scholar

[162]

S. H. Schneider and R. E. Dickinson, Climate modeling, Rev. Geophys. Space Phys., 12 (1974), 447-493. Google Scholar

[163]

G. Sell, Non-autonomous differential equations and dynamical systems, Trans. Amer. Math. Soc., 127 (1967), 241-283. Google Scholar

[164]

V. A. SheremetG. R. Ierley and V. M. Kamenkovitch, Eigenanalysis of the two-dimensional wind-driven ocean circulation problem, J. Mar. Res., 55 (1997), 57-92. doi: 10.1357/0022240973224463. Google Scholar

[165]

E. Simonnet, On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains, Physica D, 237 (2008), 2539-2552. doi: 10.1016/j.physd.2008.04.008. Google Scholar

[166]

E. Simonnet and H. A. Dijkstra, Spontaneous generation of low-frequency modes of variability in the wind-driven ocean circulation, J. Phys. Oceanogr, 32 (2002), 1747-1762. doi: 10.1175/1520-0485(2002)032<1747:SGOLFM>2.0.CO;2. Google Scholar

[167]

E. Simonnet, R. Temam, S. Wang, M. Ghil and K. Ide, Successive bifurcations in a shallow-water ocean model, in: 16th Intl. Conf. Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, Springer-Verlag, 515 (1998), 225-230. doi: 10.1007/BFb0106588. Google Scholar

[168]

E. SimonnetM. GhilK. IdeR. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation, Part Ⅰ: Steady-state solutions, J. Phys. Oceanogr, 33 (2003), 712-728. doi: 10.1175/1520-0485(2003)33<712:LVISMO>2.0.CO;2. Google Scholar

[169]

E. SimonnetM. GhilK. IdeR. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part Ⅱ: Time-dependent solutions, Oceanogr, 33 (2003), 729-752. doi: 10.1175/1520-0485(2003)33<729:LVISMO>2.0.CO;2. Google Scholar

[170]

E. SimonnetM. Ghil and H. A. Dijkstra, Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation, J. Mar. Res., 63 (2005), 931-956. doi: 10.1357/002224005774464210. Google Scholar

[171]

E. Simonnet, H. A. Dijkstra and M. Ghil, Bifurcation analysis of ocean, atmosphere and climate models, in Computational Methods for the Ocean and the Atmosphere, R. Temam and J. J. Tribbia, Eds. , North-Holland, 14 (2009), 187--229. doi: 10.1016/S1570-8659(08)00203-2. Google Scholar

[172]

E. Simonnet, Quantization of the low-frequency variability of the double-gyre circulation, J. Phys. Oceanogr, 35 (2005), 2268-2290. doi: 10.1175/JPO2806.1. Google Scholar

[173]

Y. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64. Google Scholar

[174]

S. Smale, Structurally stable systems are not dense, American J. Math., 88 (1966), 491-496. doi: 10.2307/2373203. Google Scholar

[175]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

[176]

R. J. SmallS. P. DeSzoekeS. P. XieL. O'NeillH. SeoQ. Song and P. Cornillon, Air-sea interaction over ocean fronts and eddies, Dyn. Atmos. Oceans, 45 (2008), 274-319. doi: 10.1016/j.dynatmoce.2008.01.001. Google Scholar

[177]

C. Soize, The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, World Scientic Publishing Co. , 1994. doi: 10.1142/9789814354110. Google Scholar

[178]

S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor and H. L. Miller (Eds.), Climate Change 2007: The Physical Science Basis. Contribution of Working Group Ⅰ to the Fourth Assessment Report of the IPCC, Cambridge University Press, Cambridge, UK and New York, NY, USA, 2007.Google Scholar

[179]

S. SpeichH. A. Dijkstra and M. Ghil, Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation, Nonlin, Processes Geophys, 2 (1995), 241-268. Google Scholar

[180]

D. A. Stainforth, Uncertainty in predictions of the climate response to rising levels of greenhouse gases, Nature, 433 (2005), 403-406. Google Scholar

[181]

B. Stevens, Y. Zhang and M. Ghil, Stochastic effects in the representation of stratocumulus-topped mixed layers, Proc. ECMWF Workshop on Representation of Sub-grid Processes Using Stochastic-Dynamic Models, 6-8 June 2005, Shinfield Park, Reading, UK, pp. 79-90.Google Scholar

[182]

T. F. Stocker, D. Qin, G. K. Plattner, M. Tignor, S. K. Allen, J. Boschung, A. Nauels, Y. Xia, B. Bex and B. M. Midgley, (Eds.), Climate Change 2013. The Physical Science Basis: Contribution of Working Group Ⅰ to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, UK and New York, NY, USA, 2013.Google Scholar

[183]

H. Stommel, Thermohaline convection with two stable regimes of flow, Tellus, 13 (1961), 224-230. Google Scholar

[184] H. Stommel, The Gulf Stream: A Physical and Dynamical Description, 2nd ed, Cambridge Univ. Press, London, 1965. Google Scholar
[185]

L. SushamaM. Ghil and K. Ide, Spatio-temporal variability in a mid-latitude ocean basin subject to periodic wind forcing, Atmosphere-Ocean, 45 (2007), 227-250. doi: 10.3137/ao.450404. Google Scholar

[186]

H. U. Sverdrup, Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific, Proc. Natl. Acad. Sci. USA, 33 (1947), 318-326. doi: 10.1073/pnas.33.11.318. Google Scholar

[187]

H. U. Sverdrup, M. W. Johnson and R. H. Fleming, The oceans: Their physics, chemistry and general biology, Prentice-Hall, New York, 1942, available at http://ark.cdlib.org/ark:/13030/kt167nb66r/Google Scholar

[188] T. Tél and M. Gruiz, Chaotic Dynamics, Cambridge University Press, 2006. doi: 10.1017/CBO9780511803277. Google Scholar
[189]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 2nd Ed. , 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar

[190]

P. D. Thompson, Numerical Weather Analysis and Prediction, Macmillan, New York, 1961,170 pp. Google Scholar

[191]

L. N. TrefethenA. TrefethenS. C. Reddy and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261 (1993), 578-584. doi: 10.1126/science.261.5121.578. Google Scholar

[192]

W. Tucker, Lorenz attractor exists, C. R. Acad. Sci. Paris, 328 (1999), 1197-1202. doi: 10.1016/S0764-4442(99)80439-X. Google Scholar

[193]

E. TzipermanL. StoneM. Cane and H. Jarosh, El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator, Science, 264 (1994), 72-74. Google Scholar

[194]

E. TzipermanM. A. Cane and S. E. Zebiak, Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos, J. Atmos. Sci., 50 (1995), 293-306. doi: 10.1175/1520-0469(1995)052<0293:IALTTS>2.0.CO;2. Google Scholar

[195]

G. Vallis, Amospheric and Oceanic Fluid Dynamics, Cambridge University Press, UK, 745 pp. , 2006.Google Scholar

[196]

S. Vannitsem, Dynamics and predictability of a low-order wind-driven ocean--atmosphere coupled model, Clim. Dyn., 42 (2014), 1981-1998. doi: 10.1007/s00382-013-1815-8. Google Scholar

[197]

S. Vannitsem and L. De Cruz, A 24-variable low-order coupled ocean-atmosphere model: OA-QG-WS v2, Geoscientific Model Development, 7 (2014), 649-662. Google Scholar

[198]

S. VannitsemJ. DemaeyerL. De Cruz and M. Ghil, Low-frequency variability and heat transport in a low-order nonlinear coupled ocean-atmosphere model, Physica D, 309 (2015), 71-85. doi: 10.1016/j.physd.2015.07.006. Google Scholar

[199]

T. Wanner, Linearization of random dynamical systems, in: C. K. R. T. Jones, U. Kirchgraber, and H. O. Walther (Eds), Dynamics Reported, Springer, Berlin, Heidelberg, New York, 4 (1995), 203-269. Google Scholar

[200]

R. F. Williams, The structure of Lorenz attractors, Publ. Math. I.H.E.S., 50 (1979), 73-99. Google Scholar

[201]

C. Wunsch, The interpretation of short climate records, with comments on the North Atlantic and Southern Oscillations, Bull. Am. Meteorol. Soc., 80 (1999), 245-255. doi: 10.1175/1520-0477(1999)080<0245:TIOSCR>2.0.CO;2. Google Scholar

[202]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717. Google Scholar

Figure 1.  A map of the main oceanic currents: warm currents in red and cold ones in blue. Reproduced from [79], with permission from Elsevier
Figure 2.  A satellite image of the sea surface temperature (SST) field over the northwestern North Atlantic (in false color, from the U.S. National Oceanic and Atmospheric Administration), together with a sketch of the associated double-gyre circulation (white arrows). A highly simplified and smoothed view of the amount of potential vorticity injected into the ocean circulation by the equatorial trade winds, the mid-latitudes' prevailing westerlies and the polar easterlies is shown in the sketch to the right. Reproduced from [79], with permission from Elsevier
Figure 3.  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 $\tau$. 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 [170]
Figure 4.  Pullback attractor (PBA) for a scalar linear equation, given by the solid black line. We wish to observe the PBA at times $t = t_1, t_2$, marked by dashed vertical lines, and consider the convergence to the straight line of orbits started at past times $s = s_1, s_2$; sample orbits for $s_1$ and $s_2$ are plotted in red and blue, respectively. Courtesy of M. D. Chekroun
Figure 5.  Snapshot of the Lorenz [119] model's random attractor ${\mathcal A}(\omega)$ and of the corresponding sample measure $\mu_\omega$, for a given, fixed realization $\omega$. The figure corresponds to projection onto the $(y,z)$ plane, i.e. $\int \mu_{\omega} (x,y,z)\mbox{d}x$. One billion initial points have been used in both panels and the pullback attractor is computed for $t = 40$. The parameter values are the classical ones -- $r=28$, $s=10$, and $b=8/3$, while the time step is $\Delta t = 5\cdot 10^{-3}$. The color bar to the right of each panel is on a log-scale and quantifies the probability to end up in a particular region of phase space. Both panels use the same noise realization $\omega$ but with noise intensity (a) $\sigma=0.3$ and (b) $\sigma=0.5$. Notice the interlaced filamentary structures between highly (yellow) and moderately (red) populated regions; these structures are much more complex in panel (b), where the noise is stronger. Weakly populated regions cover an important part of the random attractor and are, in turn, entangled with (almost) zero-probability regions (black). After [36]
Figure 6.  Four snapshots of the random attractor and sample measure supported on it, for the same parameter values as in Fig.5. The time interval $\Delta t$ between two successive snapshots -- moving from left to right and top to bottom -- is $\Delta t = 0.0875$. Note that the support of the sample measure may change quite abruptly, from time to time, cf. short video in [36,Supplementary Material] for details. Reproduced from [36], with permission from Elsevier
Figure 7.  Ensemble behavior of forced solutions of the double-gyre ocean model of [150]. (a) Time dependence of the total forcing $1+\epsilon f(t)$, for $\epsilon = 0.2$. (b, c) Evolution of $N = 644$ initial states emanating from the subset $\Gamma$ in the $(\Psi_1, \Psi_3)$-plane for (b) $\gamma=0.96$ and (c) $\gamma=1.1$. (b', c') Corresponding time series of $P_{\Psi_3}$. Reproduced from [150], with the permission of the American Meteorological Society
Figure 8.  Mean normalized distance $\sigma(\Psi_1,\Psi_3)$ for 15 000 trajectories of the double-gyre ocean model starting in the initial set $\Gamma$: (a) $\gamma=0.96$, and (b) $\gamma=1.1$. Reproduced from [150], with the permission of the American Meteorological Society
Figure 9.  Long-periodic orbits and slow manifold of the coupled ocean-atmosphere model of [198], in a three-dimensional (3-D) projection onto the leading modes $(\psi_{{\rm a},1}, \psi_{{\rm o},2}, T_{{\rm o},2})$ of the atmospheric and oceanic streamfunction fields and that of the oceanic temperature field. (a) Long-periodic orbits of the coupled model, for the friction parameter $d=1 \times 10^{-8}$ s$^{-1}$ and several values of $C_{\rm o}$ (see legend). (b) Long-periodic orbits and LFV-dominated ones, for the radiative-input parameter $C_{\rm o} = 300$ Wm$^{-2}$ and several values of $d$: $5 \times 10^{-9}$ s$^{-1}$ (red), $1 \times 10^{-8}$ s$^{-1}$ (green), $2 \times 10^{-8}$ s$^{-1}$ (dark blue), $3 \times10^{-8}$ s$^{-1}$ (magenta), and $8 \times 10^{-8}$ s$^{-1}$ (light blue). After [198]
Figure 10.  Low-frequency variability (LFV) of the coupled ocean-atmosphere model. Time series of geopotential height difference between locations ($\pi/n, \pi/4)$) and ($\pi/n, 3 \pi/4)$) of the model's nondimensional domain, for different values of meridional temperature gradient $C_{\rm o}$ and coupling coefficient $d$; this height difference plays the role of a North Atlantic Oscillation index in the model. (a) Chaotic but smooth trajectories living on a hypothetical slow attractor; and (b) strongly fluctuating trajectories that are not lying close to such a slow attractor. Reproduced from [198], with permission from Elsevier
Figure 11.  Climate sensitivity for (a) an equilibrium model; and (b, c) a nonequilibrium model. Given a jump in a parameter, such as CO$_2$ concentration, only the mean global temperature $\overline T$ changes in (a), while in (b) it is also the period, amplitude and phase of a purely periodic oscillation, such as the seasonal cycle or the intrinsic ENSO cycle. Finally, in panel (c), it is also the character of the oscillation, whether deterministic or stochastically perturbed, which may change. After [71]
Figure 12.  Time-dependent invariant measures of the [66] model; snapshots shown at three times $t$: (a) $t_1 = 19.23$ yr, (b) $t_2 = 20$ yr and (c) $t_3=20.833$ yr. After [40], with the authors' permision
Figure 13.  Random dynamical systems (RDS) viewed as a flow on the bundle $X \times \Omega$ = "dynamical space" $\times$ "probability space." For a given state $x$ and realization $\omega$, the RDS $\varphi$ is such that $\Theta(t)(x,\omega) = (\theta(t)\omega,\varphi(t,\omega)x)$ is a flow on the bundle. Reproduced from [79], with permission from Elsevier
Figure 14.  Schematic diagram of a random attractor ${\mathcal A}(\omega)$, where $\omega \in \Omega$ is a fixed realization of the noise. To be attracting, for every set $B$ of $X$ in a family $\mathfrak{B}$ of such sets, one must have $\lim_{t\to +\infty} {\text{dist}}(B(\theta(-t)\omega), {\mathcal A}(\omega)) = 0$ with $B(\theta(-t)\omega):=\varphi(t,\theta(-t)\omega) B$; to be invariant, one must have $\varphi(t,\omega) {\mathcal A}(\omega) = {\mathcal A}(\theta(t)\omega)$. This definition depends strongly on $\mathfrak{B}$; see [45] for more details. Reproduced from [79], with permission from Elsevier
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