July  2011, 16(1): 73-107. doi: 10.3934/dcdsb.2011.16.73

The dynamics of a low-order model for the Atlantic multidecadal oscillation

1. 

Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK Groningen, Netherlands, Netherlands

2. 

Institute for Marine and Atmospheric Research, Utrecht University, Princetonplein 5, 3584 CC Utrecht, Netherlands

3. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona

4. 

College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, North Park Road, EX4 4QF, Exeter

Received  February 2010 Revised  March 2010 Published  April 2011

Observational and model based studies provide ample evidence for the presence of multidecadal variability in the North Atlantic sea-surface temperature known as the Atlantic Multidecadal Oscillation (AMO). This variability is characterised by a multidecadal time scale, a westward propagation of temperature anomalies, and a phase difference between the anomalous meridional and zonal overturning circulations.
    We study the AMO in a low-order model obtained by projecting a model for thermally driven ocean flows onto a 27-dimensional function space. We study bifurcations of attractors by varying the equator-to-pole temperature gradient ($\Delta T$) and a damping parameter ($\gamma$).
    For $\Delta T = 20^\circ$C and $\gamma = 0$ the low-order model has a stable equilibrium corresponding to a steady ocean flow. By increasing $\gamma$ to 1 a supercritical Hopf bifurcation gives birth to a periodic attractor with the spatio-temporal signature of the AMO. Through a period doubling cascade this periodic orbit gives birth to Hénon-like strange attractors. Finally, we study the effects of annual modulation by introducing a time-periodic forcing. Then the AMO appears through a Hopf-Neĭmark-Sacker bifurcation. For $\Delta T = 24^\circ$C we detected at least 11 quasi-periodic doublings of the invariant torus. After these doublings we find quasi-periodic Hénon-like strange attractors.
Citation: Henk Broer, Henk Dijkstra, Carles Simó, Alef Sterk, Renato Vitolo. The dynamics of a low-order model for the Atlantic multidecadal oscillation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 73-107. doi: 10.3934/dcdsb.2011.16.73
References:
[1]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-periodic Motions in Families of Dynamical Systems, Order Amidst Chaos,'', Lecture Notes in Mathematics 1645, 1645 (1996). Google Scholar

[2]

H. W. Broer, G. B. Huitema, F. Takens and B. L. J. Braaksma, "Unfoldings and Bifurcations of Quasi-Periodic Tori,'', Memoirs Amer. Math. Soc., 83 (1990). Google Scholar

[3]

H. W. Broer and M. B. Sevryuk, KAM theory: quasi-periodicity in dynamical systems, in, (2010), 249. doi: 10.1016/S1874-575X(10)00314-0. Google Scholar

[4]

H. W. Broer and C. Simó, Hill's equation with quasi-periodic forcing: Resonance tongues, instability pockets and global phenomena,, Bol. Soc. Bras. Mat., 29 (1998), 253. doi: 10.1007/BF01237651. Google Scholar

[5]

H. W. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing,, Nonlinearity, 15 (2002), 1205. doi: 10.1088/0951-7715/15/4/312. Google Scholar

[6]

H. W. Broer, C. Simó and R. Vitolo, Quasi-periodic Hénon-like attractors in the Lorenz-84 climate model with seasonal forcing,, in, (2005), 601. Google Scholar

[7]

H. W. Broer, C. Simó and R. Vitolo, Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: analysis of a resonance 'bubble',, Physica D, 237 (2008), 1773. doi: 10.1016/j.physd.2008.01.026. Google Scholar

[8]

H. W. Broer, C. Simó and R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol$'$d resonance web,, Bull. Belgian Math. Soc. Simon Stevin, 15 (2008), 769. Google Scholar

[9]

H. W. Broer, C. Simó and R. Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus,, Discrete and Continuous Dynamical Systems B, 14 (2010), 871. doi: 10.3934/dcdsb.2010.14.871. Google Scholar

[10]

H. W. Broer and F. Takens, "Dynamical Systems and Chaos,'', Applied Mathematical Sciences 172, 172 (2011). Google Scholar

[11]

D. R. Cayan, Latent and sensible heat flux anomalies over the northern oceans: the connection to monthly atmospheric cirulation,, J. Climate, 5 (1992), 354. doi: 10.1175/1520-0442(1992)005<0354:LASHFA>2.0.CO;2. Google Scholar

[12]

F. Chen and M. Ghil, Interdecadal variability of the thermohaline circulation and high-latitude surface fluxes,, J. Physical Oceanography, 25 (1995), 2547. doi: 10.1175/1520-0485(1995)025<2547:IVOTTC>2.0.CO;2. Google Scholar

[13]

A. Chenciner, Bifurcations de points fixes elliptiques I. Courbes invariantes,, Inst. Hautes Études Sci. Publ. Math., 61 (1985), 67. Google Scholar

[14]

A. Chenciner, Bifurcations de points fixes elliptiques II. Orbites péridiques et ensembles de Cantor invariants,, Invent. Math., 80 (1985), 81. doi: 10.1007/BF01388549. Google Scholar

[15]

A. Chenciner, Bifurcations de points fixes elliptiques III. Orbites péridiquesde "petites'' périodes et élimination résonnante des couples de courbes invariantes,, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 5. Google Scholar

[16]

A. Colin de Verdière and T. Huck, Baroclinic instability: An oceanic wavemaker for interdecadal variability,, J. Physical Oceanography, 29 (1999), 893. doi: 10.1175/1520-0485(1999)029<0893:BIAOWF>2.0.CO;2. Google Scholar

[17]

T. L. Delworth and M. E. Mann, Observed and simulated multidecadal variability in the Northern Hemisphere,, Climate Dynamics, 16 (2000), 661. doi: 10.1007/s003820000075. Google Scholar

[18]

H. A. Dijkstra, "Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño,'' 2nd edition,, Springer, (2005). Google Scholar

[19]

H. A. Dijkstra, Interaction of SST modes in the North Atlantic ocean,, J. Physical Oceanography, 36 (2006), 286. doi: 10.1175/JPO2851.1. Google Scholar

[20]

H. A. Dijkstra, L. A. te Raa, M. Schmeits and J. Gerrits, On the physics of the Atlantic multidecadal Oscillation,, Ocean Dynanmics, 56 (2006), 36. doi: 10.1007/s10236-005-0043-0. Google Scholar

[21]

H. A. Dijkstra, L. M. Frankcombe and A. S. von der Heydt, A stochastic dynamical systems view of the Atlantic multidecadal Oscillation,, Phil. Trans. R. Soc. A, 366 (2008), 2545. doi: 10.1098/rsta.2008.0031. Google Scholar

[22]

E. J. Doedel and B. E. Oldeman, "AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations,", Concordia University, (2007). Google Scholar

[23]

D. Enfield, A. Mestas-Nuñez and P. Trimble, The Atlantic multidecadal oscillation and its relation to rainfall and river flows in the continental U.S.,, Geophys. Res. Let., 28 (2001), 2077. doi: 10.1029/2000GL012745. Google Scholar

[24]

L. M. Frankcombe, H. A. Dijkstra and A. S. von der Heydt, Noise induced multidecadal variability in the North Atlantic: Excitation of normal modes,, J. Physical Oceanography, 39 (2009), 220. doi: 10.1175/2008JPO3951.1. Google Scholar

[25]

S. B. Goldenberg, C. W. Landsea, A. M. Mestas-Nuñez and W. M. Gray, The recent increase in Atlantic hurricane activity: Causes and implications,, Science, 293 (2001), 474. doi: 10.1126/science.1060040. Google Scholar

[26]

R. J. Greatbatch and S. Zhang, An interdecadal oscillation in an idealized ocean basin forced by constant heat flux,, J. Climate, 8 (1995), 81. doi: 10.1175/1520-0442(1995)008<0081:AIOIAI>2.0.CO;2. Google Scholar

[27]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'', Springer, (1983). Google Scholar

[28]

T. Huck and G. K. Vallis, Linear stability analysis of the three-dimensional thermally-driven ocean circulation: Application to interdecadal oscillations,, Tellus A, 53 (2001), 526. doi: 10.1111/j.1600-0870.2001.00526.x. Google Scholar

[29]

\`A. Jorba, Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps,, Nonlinearity, 14 (2001), 943. doi: 10.1088/0951-7715/14/5/303. Google Scholar

[30]

À. Jorba and M. Zou, A software package for the numerical integration of ODEs by means of high-order Taylor methods,, Experimental Mathematics, 14 (2005), 99. doi: 10.1080/10586458.2005.10128904. Google Scholar

[31]

R. A. Kerr, A North Atlantic climate pacemaker for the centuries,, Science, 288 (2000), 1984. doi: 10.1126/science.288.5473.1984. Google Scholar

[32]

Y. Kushnir, Interdecadal variations in North Atlantic sea surface temperature and associated atmospheric conditions,, J. Climate, 7 (1994), 141. doi: 10.1175/1520-0442(1994)007<0141:IVINAS>2.0.CO;2. Google Scholar

[33]

Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3rd edition,, Springer-Verlag, (2004). Google Scholar

[34]

F. Kwasniok, The reduction of complex dynamical systems using principal interaction patterns,, Physica D, 92 (1996), 28. doi: 10.1016/0167-2789(95)00280-4. Google Scholar

[35]

A. I. Neishtadt, C. Simó and D. V. Treschev, On stability loss delay for a periodic trajectory,, in, 19 (1995), 253. Google Scholar

[36]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems,'', Cambridge University Press, (2001). Google Scholar

[37]

C. Simó, On the use of Lyapunov exponents to detect global properties of the dynamics,, in, (2005), 631. Google Scholar

[38]

C. Simó, On the analytical and numerical continuation of invariant manifolds,, in, (1990), 285. Google Scholar

[39]

A. E. Sterk, R. Vitolo, H. W. Broer, C. Simó and H. A. Dijkstra, New nonlinear mechanisms of midlatitude atmospheric low-frequency variability,, Physica D, 239 (2010), 702. doi: 10.1016/j.physd.2010.02.003. Google Scholar

[40]

L. A. te Raa and H. A. Dijkstra, Instability of the thermohaline ocean circulation on interdecadal timescales,, J. Physical Oceanography, 32 (2002), 138. doi: 10.1175/1520-0485(2002)032<0138:IOTTOC>2.0.CO;2. Google Scholar

[41]

L. A. te Raa and H. A. Dijkstra, Modes of internal thermohaline variability in a single-hemispheric ocean basin,, J. Marine Res., 61 (2003), 491. doi: 10.1357/002224003322384906. Google Scholar

[42]

L. A. te Raa, J. Gerrits and H. A. Dijkstra, Identification of the mechanisms of interdecadal variability in the North Atlantic Ocean,, J. Physical Oceanography, 34 (2004), 2792. doi: 10.1175/JPO2655.1. Google Scholar

[43]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition,, Springer-Verlag, (1997). Google Scholar

[44]

P. C. F. van der Vaart, H. M. Schuttelaars, D. Calvete and H. A. Dijkstra, Instability of time-dependent wind-driven ocean gyres,, Phys. Fluids, 14 (2002), 3601. doi: 10.1063/1.1503804. Google Scholar

[45]

R. Vitolo, H. W. Broer and C. Simó, Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms,, Nonlinearity, 23 (2010), 1919. doi: 10.1088/0951-7715/23/8/007. Google Scholar

show all references

References:
[1]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-periodic Motions in Families of Dynamical Systems, Order Amidst Chaos,'', Lecture Notes in Mathematics 1645, 1645 (1996). Google Scholar

[2]

H. W. Broer, G. B. Huitema, F. Takens and B. L. J. Braaksma, "Unfoldings and Bifurcations of Quasi-Periodic Tori,'', Memoirs Amer. Math. Soc., 83 (1990). Google Scholar

[3]

H. W. Broer and M. B. Sevryuk, KAM theory: quasi-periodicity in dynamical systems, in, (2010), 249. doi: 10.1016/S1874-575X(10)00314-0. Google Scholar

[4]

H. W. Broer and C. Simó, Hill's equation with quasi-periodic forcing: Resonance tongues, instability pockets and global phenomena,, Bol. Soc. Bras. Mat., 29 (1998), 253. doi: 10.1007/BF01237651. Google Scholar

[5]

H. W. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing,, Nonlinearity, 15 (2002), 1205. doi: 10.1088/0951-7715/15/4/312. Google Scholar

[6]

H. W. Broer, C. Simó and R. Vitolo, Quasi-periodic Hénon-like attractors in the Lorenz-84 climate model with seasonal forcing,, in, (2005), 601. Google Scholar

[7]

H. W. Broer, C. Simó and R. Vitolo, Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: analysis of a resonance 'bubble',, Physica D, 237 (2008), 1773. doi: 10.1016/j.physd.2008.01.026. Google Scholar

[8]

H. W. Broer, C. Simó and R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol$'$d resonance web,, Bull. Belgian Math. Soc. Simon Stevin, 15 (2008), 769. Google Scholar

[9]

H. W. Broer, C. Simó and R. Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus,, Discrete and Continuous Dynamical Systems B, 14 (2010), 871. doi: 10.3934/dcdsb.2010.14.871. Google Scholar

[10]

H. W. Broer and F. Takens, "Dynamical Systems and Chaos,'', Applied Mathematical Sciences 172, 172 (2011). Google Scholar

[11]

D. R. Cayan, Latent and sensible heat flux anomalies over the northern oceans: the connection to monthly atmospheric cirulation,, J. Climate, 5 (1992), 354. doi: 10.1175/1520-0442(1992)005<0354:LASHFA>2.0.CO;2. Google Scholar

[12]

F. Chen and M. Ghil, Interdecadal variability of the thermohaline circulation and high-latitude surface fluxes,, J. Physical Oceanography, 25 (1995), 2547. doi: 10.1175/1520-0485(1995)025<2547:IVOTTC>2.0.CO;2. Google Scholar

[13]

A. Chenciner, Bifurcations de points fixes elliptiques I. Courbes invariantes,, Inst. Hautes Études Sci. Publ. Math., 61 (1985), 67. Google Scholar

[14]

A. Chenciner, Bifurcations de points fixes elliptiques II. Orbites péridiques et ensembles de Cantor invariants,, Invent. Math., 80 (1985), 81. doi: 10.1007/BF01388549. Google Scholar

[15]

A. Chenciner, Bifurcations de points fixes elliptiques III. Orbites péridiquesde "petites'' périodes et élimination résonnante des couples de courbes invariantes,, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 5. Google Scholar

[16]

A. Colin de Verdière and T. Huck, Baroclinic instability: An oceanic wavemaker for interdecadal variability,, J. Physical Oceanography, 29 (1999), 893. doi: 10.1175/1520-0485(1999)029<0893:BIAOWF>2.0.CO;2. Google Scholar

[17]

T. L. Delworth and M. E. Mann, Observed and simulated multidecadal variability in the Northern Hemisphere,, Climate Dynamics, 16 (2000), 661. doi: 10.1007/s003820000075. Google Scholar

[18]

H. A. Dijkstra, "Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño,'' 2nd edition,, Springer, (2005). Google Scholar

[19]

H. A. Dijkstra, Interaction of SST modes in the North Atlantic ocean,, J. Physical Oceanography, 36 (2006), 286. doi: 10.1175/JPO2851.1. Google Scholar

[20]

H. A. Dijkstra, L. A. te Raa, M. Schmeits and J. Gerrits, On the physics of the Atlantic multidecadal Oscillation,, Ocean Dynanmics, 56 (2006), 36. doi: 10.1007/s10236-005-0043-0. Google Scholar

[21]

H. A. Dijkstra, L. M. Frankcombe and A. S. von der Heydt, A stochastic dynamical systems view of the Atlantic multidecadal Oscillation,, Phil. Trans. R. Soc. A, 366 (2008), 2545. doi: 10.1098/rsta.2008.0031. Google Scholar

[22]

E. J. Doedel and B. E. Oldeman, "AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations,", Concordia University, (2007). Google Scholar

[23]

D. Enfield, A. Mestas-Nuñez and P. Trimble, The Atlantic multidecadal oscillation and its relation to rainfall and river flows in the continental U.S.,, Geophys. Res. Let., 28 (2001), 2077. doi: 10.1029/2000GL012745. Google Scholar

[24]

L. M. Frankcombe, H. A. Dijkstra and A. S. von der Heydt, Noise induced multidecadal variability in the North Atlantic: Excitation of normal modes,, J. Physical Oceanography, 39 (2009), 220. doi: 10.1175/2008JPO3951.1. Google Scholar

[25]

S. B. Goldenberg, C. W. Landsea, A. M. Mestas-Nuñez and W. M. Gray, The recent increase in Atlantic hurricane activity: Causes and implications,, Science, 293 (2001), 474. doi: 10.1126/science.1060040. Google Scholar

[26]

R. J. Greatbatch and S. Zhang, An interdecadal oscillation in an idealized ocean basin forced by constant heat flux,, J. Climate, 8 (1995), 81. doi: 10.1175/1520-0442(1995)008<0081:AIOIAI>2.0.CO;2. Google Scholar

[27]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'', Springer, (1983). Google Scholar

[28]

T. Huck and G. K. Vallis, Linear stability analysis of the three-dimensional thermally-driven ocean circulation: Application to interdecadal oscillations,, Tellus A, 53 (2001), 526. doi: 10.1111/j.1600-0870.2001.00526.x. Google Scholar

[29]

\`A. Jorba, Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps,, Nonlinearity, 14 (2001), 943. doi: 10.1088/0951-7715/14/5/303. Google Scholar

[30]

À. Jorba and M. Zou, A software package for the numerical integration of ODEs by means of high-order Taylor methods,, Experimental Mathematics, 14 (2005), 99. doi: 10.1080/10586458.2005.10128904. Google Scholar

[31]

R. A. Kerr, A North Atlantic climate pacemaker for the centuries,, Science, 288 (2000), 1984. doi: 10.1126/science.288.5473.1984. Google Scholar

[32]

Y. Kushnir, Interdecadal variations in North Atlantic sea surface temperature and associated atmospheric conditions,, J. Climate, 7 (1994), 141. doi: 10.1175/1520-0442(1994)007<0141:IVINAS>2.0.CO;2. Google Scholar

[33]

Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3rd edition,, Springer-Verlag, (2004). Google Scholar

[34]

F. Kwasniok, The reduction of complex dynamical systems using principal interaction patterns,, Physica D, 92 (1996), 28. doi: 10.1016/0167-2789(95)00280-4. Google Scholar

[35]

A. I. Neishtadt, C. Simó and D. V. Treschev, On stability loss delay for a periodic trajectory,, in, 19 (1995), 253. Google Scholar

[36]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems,'', Cambridge University Press, (2001). Google Scholar

[37]

C. Simó, On the use of Lyapunov exponents to detect global properties of the dynamics,, in, (2005), 631. Google Scholar

[38]

C. Simó, On the analytical and numerical continuation of invariant manifolds,, in, (1990), 285. Google Scholar

[39]

A. E. Sterk, R. Vitolo, H. W. Broer, C. Simó and H. A. Dijkstra, New nonlinear mechanisms of midlatitude atmospheric low-frequency variability,, Physica D, 239 (2010), 702. doi: 10.1016/j.physd.2010.02.003. Google Scholar

[40]

L. A. te Raa and H. A. Dijkstra, Instability of the thermohaline ocean circulation on interdecadal timescales,, J. Physical Oceanography, 32 (2002), 138. doi: 10.1175/1520-0485(2002)032<0138:IOTTOC>2.0.CO;2. Google Scholar

[41]

L. A. te Raa and H. A. Dijkstra, Modes of internal thermohaline variability in a single-hemispheric ocean basin,, J. Marine Res., 61 (2003), 491. doi: 10.1357/002224003322384906. Google Scholar

[42]

L. A. te Raa, J. Gerrits and H. A. Dijkstra, Identification of the mechanisms of interdecadal variability in the North Atlantic Ocean,, J. Physical Oceanography, 34 (2004), 2792. doi: 10.1175/JPO2655.1. Google Scholar

[43]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition,, Springer-Verlag, (1997). Google Scholar

[44]

P. C. F. van der Vaart, H. M. Schuttelaars, D. Calvete and H. A. Dijkstra, Instability of time-dependent wind-driven ocean gyres,, Phys. Fluids, 14 (2002), 3601. doi: 10.1063/1.1503804. Google Scholar

[45]

R. Vitolo, H. W. Broer and C. Simó, Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms,, Nonlinearity, 23 (2010), 1919. doi: 10.1088/0951-7715/23/8/007. Google Scholar

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