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 and 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, Springer-Verlag, Berlin, 1996.

[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), viii+175 pp.

[3]

H. W. Broer and M. B. Sevryuk, KAM theory: quasi-periodicity in dynamical systems in "Handbook of Dynamical Systems, Volume 3'' (eds. H.W. Broer, B. Hasselblatt and F. Takens) North-Holland, (2010), 249-344. doi: 10.1016/S1874-575X(10)00314-0.

[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), 253-293. doi: 10.1007/BF01237651.

[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), 1205-1267. doi: 10.1088/0951-7715/15/4/312.

[6]

H. W. Broer, C. Simó and R. Vitolo, Quasi-periodic Hénon-like attractors in the Lorenz-84 climate model with seasonal forcing, in "Equadiff 2003, Proceedings International Conference on Differential Equations, Hasselt 2003'' (eds. F. Dumortier, H.W. Broer, J. Mahwin, A. Vanderbauwhede and S.M. Verduyn Lunel), World Scientific, (2005), 601-606.

[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), 1773-1799. doi: 10.1016/j.physd.2008.01.026.

[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), 769-787.

[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), 871-905. doi: 10.3934/dcdsb.2010.14.871.

[10]

H. W. Broer and F. Takens, "Dynamical Systems and Chaos,'' Applied Mathematical Sciences 172, Springer, 2011.

[11]

D. R. Cayan, Latent and sensible heat flux anomalies over the northern oceans: the connection to monthly atmospheric cirulation, J. Climate, 5 (1992), 354-369. doi: 10.1175/1520-0442(1992)005<0354:LASHFA>2.0.CO;2.

[12]

F. Chen and M. Ghil, Interdecadal variability of the thermohaline circulation and high-latitude surface fluxes, J. Physical Oceanography, 25 (1995), 2547-2568. doi: 10.1175/1520-0485(1995)025<2547:IVOTTC>2.0.CO;2.

[13]

A. Chenciner, Bifurcations de points fixes elliptiques I. Courbes invariantes, Inst. Hautes Études Sci. Publ. Math., 61 (1985), 67-127.

[14]

A. Chenciner, Bifurcations de points fixes elliptiques II. Orbites péridiques et ensembles de Cantor invariants, Invent. Math., 80 (1985), 81-106. doi: 10.1007/BF01388549.

[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), 5-91.

[16]

A. Colin de Verdière and T. Huck, Baroclinic instability: An oceanic wavemaker for interdecadal variability, J. Physical Oceanography, 29 (1999), 893-910. doi: 10.1175/1520-0485(1999)029<0893:BIAOWF>2.0.CO;2.

[17]

T. L. Delworth and M. E. Mann, Observed and simulated multidecadal variability in the Northern Hemisphere, Climate Dynamics, 16 (2000), 661-676. doi: 10.1007/s003820000075.

[18]

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

[19]

H. A. Dijkstra, Interaction of SST modes in the North Atlantic ocean, J. Physical Oceanography, 36 (2006), 286-299. doi: 10.1175/JPO2851.1.

[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), 36-50. doi: 10.1007/s10236-005-0043-0.

[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), 2545-2560. doi: 10.1098/rsta.2008.0031.

[22]

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

[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), 2077-2080. doi: 10.1029/2000GL012745.

[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), 220-233. doi: 10.1175/2008JPO3951.1.

[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), 474-479. doi: 10.1126/science.1060040.

[26]

R. J. Greatbatch and S. Zhang, An interdecadal oscillation in an idealized ocean basin forced by constant heat flux, J. Climate, 8 (1995), 81-91. doi: 10.1175/1520-0442(1995)008<0081:AIOIAI>2.0.CO;2.

[27]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'' Springer, New York, 1983.

[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), 526-545. doi: 10.1111/j.1600-0870.2001.00526.x.

[29]

\`A. Jorba, Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps, Nonlinearity, 14 (2001), 943-976. doi: 10.1088/0951-7715/14/5/303.

[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), 99-117. doi: 10.1080/10586458.2005.10128904.

[31]

R. A. Kerr, A North Atlantic climate pacemaker for the centuries, Science, 288 (2000), 1984-1985. doi: 10.1126/science.288.5473.1984.

[32]

Y. Kushnir, Interdecadal variations in North Atlantic sea surface temperature and associated atmospheric conditions, J. Climate, 7 (1994), 141-157. doi: 10.1175/1520-0442(1994)007<0141:IVINAS>2.0.CO;2.

[33]

Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3rd edition, Springer-Verlag, New York, 2004.

[34]

F. Kwasniok, The reduction of complex dynamical systems using principal interaction patterns, Physica D, 92 (1996), 28-60. doi: 10.1016/0167-2789(95)00280-4.

[35]

A. I. Neishtadt, C. Simó and D. V. Treschev, On stability loss delay for a periodic trajectory, in "Progress in Nonlinear Differential Equations and their Applications,'' 19, Birkhäuser-Verlag, (1995), 253-278.

[36]

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

[37]

C. Simó, On the use of Lyapunov exponents to detect global properties of the dynamics, in "Equadiff 2003, Proceedings International Conference on Differential Equations, Hasselt 2003'' (eds. F. Dumortier, H.W. Broer, J. Mahwin, A. Vanderbauwhede and S.M. Verduyn Lunel), World Scientific, (2005), 631-636.

[38]

C. Simó, On the analytical and numerical continuation of invariant manifolds, in "Modern Methods in Celestial Mechanics'' (eds. D. Benest and C. Froeschlé), Éditions Frontières, (1990), 285-330.

[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), 702-718. doi: 10.1016/j.physd.2010.02.003.

[40]

L. A. te Raa and H. A. Dijkstra, Instability of the thermohaline ocean circulation on interdecadal timescales, J. Physical Oceanography, 32 (2002), 138-160. doi: 10.1175/1520-0485(2002)032<0138:IOTTOC>2.0.CO;2.

[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), 491-516. doi: 10.1357/002224003322384906.

[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), 2792-2807. doi: 10.1175/JPO2655.1.

[43]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition, Springer-Verlag, New York, 1997.

[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), 3601-3615. doi: 10.1063/1.1503804.

[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), 1919-1947. doi: 10.1088/0951-7715/23/8/007.

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, Springer-Verlag, Berlin, 1996.

[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), viii+175 pp.

[3]

H. W. Broer and M. B. Sevryuk, KAM theory: quasi-periodicity in dynamical systems in "Handbook of Dynamical Systems, Volume 3'' (eds. H.W. Broer, B. Hasselblatt and F. Takens) North-Holland, (2010), 249-344. doi: 10.1016/S1874-575X(10)00314-0.

[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), 253-293. doi: 10.1007/BF01237651.

[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), 1205-1267. doi: 10.1088/0951-7715/15/4/312.

[6]

H. W. Broer, C. Simó and R. Vitolo, Quasi-periodic Hénon-like attractors in the Lorenz-84 climate model with seasonal forcing, in "Equadiff 2003, Proceedings International Conference on Differential Equations, Hasselt 2003'' (eds. F. Dumortier, H.W. Broer, J. Mahwin, A. Vanderbauwhede and S.M. Verduyn Lunel), World Scientific, (2005), 601-606.

[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), 1773-1799. doi: 10.1016/j.physd.2008.01.026.

[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), 769-787.

[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), 871-905. doi: 10.3934/dcdsb.2010.14.871.

[10]

H. W. Broer and F. Takens, "Dynamical Systems and Chaos,'' Applied Mathematical Sciences 172, Springer, 2011.

[11]

D. R. Cayan, Latent and sensible heat flux anomalies over the northern oceans: the connection to monthly atmospheric cirulation, J. Climate, 5 (1992), 354-369. doi: 10.1175/1520-0442(1992)005<0354:LASHFA>2.0.CO;2.

[12]

F. Chen and M. Ghil, Interdecadal variability of the thermohaline circulation and high-latitude surface fluxes, J. Physical Oceanography, 25 (1995), 2547-2568. doi: 10.1175/1520-0485(1995)025<2547:IVOTTC>2.0.CO;2.

[13]

A. Chenciner, Bifurcations de points fixes elliptiques I. Courbes invariantes, Inst. Hautes Études Sci. Publ. Math., 61 (1985), 67-127.

[14]

A. Chenciner, Bifurcations de points fixes elliptiques II. Orbites péridiques et ensembles de Cantor invariants, Invent. Math., 80 (1985), 81-106. doi: 10.1007/BF01388549.

[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), 5-91.

[16]

A. Colin de Verdière and T. Huck, Baroclinic instability: An oceanic wavemaker for interdecadal variability, J. Physical Oceanography, 29 (1999), 893-910. doi: 10.1175/1520-0485(1999)029<0893:BIAOWF>2.0.CO;2.

[17]

T. L. Delworth and M. E. Mann, Observed and simulated multidecadal variability in the Northern Hemisphere, Climate Dynamics, 16 (2000), 661-676. doi: 10.1007/s003820000075.

[18]

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

[19]

H. A. Dijkstra, Interaction of SST modes in the North Atlantic ocean, J. Physical Oceanography, 36 (2006), 286-299. doi: 10.1175/JPO2851.1.

[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), 36-50. doi: 10.1007/s10236-005-0043-0.

[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), 2545-2560. doi: 10.1098/rsta.2008.0031.

[22]

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

[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), 2077-2080. doi: 10.1029/2000GL012745.

[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), 220-233. doi: 10.1175/2008JPO3951.1.

[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), 474-479. doi: 10.1126/science.1060040.

[26]

R. J. Greatbatch and S. Zhang, An interdecadal oscillation in an idealized ocean basin forced by constant heat flux, J. Climate, 8 (1995), 81-91. doi: 10.1175/1520-0442(1995)008<0081:AIOIAI>2.0.CO;2.

[27]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'' Springer, New York, 1983.

[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), 526-545. doi: 10.1111/j.1600-0870.2001.00526.x.

[29]

\`A. Jorba, Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps, Nonlinearity, 14 (2001), 943-976. doi: 10.1088/0951-7715/14/5/303.

[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), 99-117. doi: 10.1080/10586458.2005.10128904.

[31]

R. A. Kerr, A North Atlantic climate pacemaker for the centuries, Science, 288 (2000), 1984-1985. doi: 10.1126/science.288.5473.1984.

[32]

Y. Kushnir, Interdecadal variations in North Atlantic sea surface temperature and associated atmospheric conditions, J. Climate, 7 (1994), 141-157. doi: 10.1175/1520-0442(1994)007<0141:IVINAS>2.0.CO;2.

[33]

Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3rd edition, Springer-Verlag, New York, 2004.

[34]

F. Kwasniok, The reduction of complex dynamical systems using principal interaction patterns, Physica D, 92 (1996), 28-60. doi: 10.1016/0167-2789(95)00280-4.

[35]

A. I. Neishtadt, C. Simó and D. V. Treschev, On stability loss delay for a periodic trajectory, in "Progress in Nonlinear Differential Equations and their Applications,'' 19, Birkhäuser-Verlag, (1995), 253-278.

[36]

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

[37]

C. Simó, On the use of Lyapunov exponents to detect global properties of the dynamics, in "Equadiff 2003, Proceedings International Conference on Differential Equations, Hasselt 2003'' (eds. F. Dumortier, H.W. Broer, J. Mahwin, A. Vanderbauwhede and S.M. Verduyn Lunel), World Scientific, (2005), 631-636.

[38]

C. Simó, On the analytical and numerical continuation of invariant manifolds, in "Modern Methods in Celestial Mechanics'' (eds. D. Benest and C. Froeschlé), Éditions Frontières, (1990), 285-330.

[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), 702-718. doi: 10.1016/j.physd.2010.02.003.

[40]

L. A. te Raa and H. A. Dijkstra, Instability of the thermohaline ocean circulation on interdecadal timescales, J. Physical Oceanography, 32 (2002), 138-160. doi: 10.1175/1520-0485(2002)032<0138:IOTTOC>2.0.CO;2.

[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), 491-516. doi: 10.1357/002224003322384906.

[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), 2792-2807. doi: 10.1175/JPO2655.1.

[43]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition, Springer-Verlag, New York, 1997.

[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), 3601-3615. doi: 10.1063/1.1503804.

[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), 1919-1947. doi: 10.1088/0951-7715/23/8/007.

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