Low-frequency climate variability: a dynamical systems a approach

Low-frequency climate variability: a dynamical systems a approach Henk Broer Rijksuniversiteit Groningen http://www.math.rug.nl/ broer 19 October 2006 Page 1 of 40

1. Variability at low-frequencies Making a long story short: Power spectrum of a time series related to mean solid-body rotation of atmosphere relative to Earth, from spectral atmospheric model (James & James 1989). Similar spectra typical for atmosphere/ocean data and models often at decadal-to-multidecadal scale. Page 2 of 40 Example: Atlantic Multidecadal Oscillation Can such phenomena be studied by dynamical system techniques?

Dynamical systems perspective Deterministic modelling plays central role in: weather prediction (Numerical Weather Prediction Models) and in climate analysis (General Circulation Models) These are intense research fields large scale computational efforts (Earth Simulator) huge financial commitments (Kyoto, etc.). Weather unpredictability and climate statistics: related to turbulence and chaos (Lorenz 1963, Ruelle-Takens 1971). Basic idea: identify regimes with stationary solutions; explain low-frequency variability and regime transitions by intermittency due to homo- and heteroclinicity Page 3 of 40

Dynamical systems principles Statespace with deterministic evolution (NB: Three-body problem has 3 6 = 18D state space) Stationary state point attractor Periodic state circle attractor Multiperiodic state torus Strange attractor (chaos & unpredictability, fractal structure) Search for persistent / robust phenomena, often also universal (i.e., context independent). Example: Transition upon variation of parameters like Hopf- or period doubling bifurcation (think of Feigenbaum cascade). Page 4 of 40

Towards low dimensional models Start with first principle PDE models: Navier-Stokes plus mass and energy conservation laws infinite dimensional dynamics in function space I. Conceptual reduction to finite dimension: invariant / inertial manifold containing attractors (Ruelle & Takens, Mallet-Paret, Temam, etc.). II. Analytical / computational approximation: Galerkin like truncations (e.g., Lorenz 1963, 1984). I and II not so easy to marry: a mathematical challenge... Search is for reduced models by projection of PDE s onto their attractors, leading to finite (low) dimensional ODE s Page 5 of 40

Example: Torus and Lorenz attractor Lorenz attractor (1963, from a convection model) is first strange attractor; only proven so by Tucker (1999). x2 Z Page 6 of 40 x1 3D torus admits strange attractor X

Example: towards reduced models Two-Layer Quasi-Geostrophic Model on circumpolar annulus domain. Ansätze: Hydrostatic atmosphere near geostrophic equilibrium, in two layers, with a Coriolis force that is linear in latitude, parametrizations... 180 o W 150 o W 150 o E 120 o W 120 o E L y 90 o W 90 o E Page 7 of 40 67 o 30 N 45 o N 60 o W 22 o 30 N 60 o E 30 o W y^ ^ x 30 o E From: Lucarini, Speranza & Vitolo (2006). 0 o

Minimal models: Lorenz-84 Fourier expansion + Galerkin truncation + reduction to 3D manifold Lorenz-84 ODE: ẋ = ax y 2 z 2 + af, ẏ = y + xy bxz + G, ż = z + bxy + xz. E.N. Lorenz (1984,1990): simplest model for atmospheric dynamics at midlatitudes. Shil nikov, Nicolis& Nicolis (1995): comprehensive bifurcation analysis. Page 8 of 40 Pielke & Zheng (1994): low-frequency variability induced by seasonal forcing. L. van Veen (2003): derivation from two-layer quasi-geostrophic equations, usage in low-order coupled atmosphere/ocean models. Broer, Simó & Vitolo (2002): influence on dynamics of seasonal variation of (F, G), new types of strange attractors.

Variables and parameters in Lorenz-84 x: strength of westerly wind current y, z: sine and cosine phases of travelling waves F, G: thermal forcings Lorenz-84 is a toy -model only for detecting qualitative aspects in atmospheric baroclinic jet at mid-latitudes. Page 9 of 40

2. Bifurcations in Lorenz-84 Message: Lorenz 84 has rich dynamics. Codimension 2 organizing centers in (F, G) parameter plane: - Hopf-saddle-node bifo of equilibria. - Cusp of equilibria. - 1:2 strong resonance bifo of periodic orbits. - 1:1 (Bogdanov-Takens) bifo of periodic orbits. Page 10 of 40 Homoclinic bifurcations: Shil nikov tangencies. Several codimension 1 bifos of equilibria (Hopf, saddle-node) and of periodic orbits (Hopf, saddle-node, period doubling). Shil nikov, Nicolis & Nicolis (1995).

Page 11 of 40 Figure: Van Veen (2003).

Page 12 of 40 Figure: Van Veen (2003).

Shil nikov-like strange attractors Page 13 of 40 Parameter values are s.t. a Shil nikov bifurcation takes place in 6D ODE, similar attractors also in Lorenz-84. Blue: 4-times period-doubled periodic orbit. Red: Shil nikov homoclinic orbit. Green: strange attractor. Figure: Van Veen (2003).

Shil nikov bifurcation Page 14 of 40 Sketch of Shil nikov bifurcation (codimension 1) associated to chaos

Two broken sphere scenario s Page 15 of 40 Broer & Vegter (1984)

Conclusions I (see Appendices) 1. Hopf-saddle-node and Shil nikov bifurcations occur in several, very different low-order models of atmospheric circulation. 2. Rich dynamical phenonena are related to this: (a) Transversal heteroclinic orbit; (b) Chaos and strange attractors; (c) Intermittency (and regime transiton ) near both polar equilibria. 3. Traces of homo- and heteroclinic behaviour are found in reduced models of large-scale atmospheric flow (of dimensions 6D to 30D), where low-frequency variability has been observed (Charney & DeVore (1979), Branstator & Opsteegh (1989), De Swart (1989), James et al. (1994), Crommelin et al. (2004). Page 16 of 40

Conclusions II (see Appendices) 1. Homo- and heteroclinicity: a possible explanation for (a) low-frequency variability (b) regime transitions (heteroclinic intermittency) (c) bimodality (observed in data) 2. What is the analogue of Shil nikov in more complex, higher dimensional models such as PDE s or the General Circulation Models? 3. Can traces of homoclinic behaviour also be found in climatic data (i.e., in observations)? Page 17 of 40

Bibliography 1. E.N. Lorenz: Energy and numerical weather prediction, Tellus 12(4) (1960), 364 373. 2. E.N. Lorenz: Irregularity: a fundamental property of the atmosphere, Tellus 36A (1984), 98 100. 3. E.N. Lorenz: Can chaos and intransitivity lead to interannual variability? Tellus 42A (1990), 378 389. 4. E.N. Lorenz: Regimes in simple systems, J. Atmos. Sci. 63 (2006), 2056 2073. 5. I.N. James, P.M. James: Ultra-low-frequency variability in a simple atmospheric circulation model, Nature 342 (1989), 53-55. 6. R. Pielke, X. Zeng: Long-term variability of climate, J. Atmos. Sci. 51 (1994), 155-159. Page 18 of 40 7. A. Shil nikov, G. Nicolis, C. Nicolis: Bifurcation and predictability analysis of a low-order atmospheric circulation model, Int.J.Bifur.Chaos 5(6) (1995), 1701 1711.

Bibliography (continued) 8. D.T. Crommelin: Homoclinic Dynamics: A Scenario for Atmospheric Ultralow-Frequency Variability, J. Atmos. Sci. 59(9) (2002), 1533 1549 9. D.T. Crommelin: Regime transitions and heteroclinic connections in a barotropic atmosphere J. Atmos. Sci., 60(2) (2003), 229 246. 10. D.T. Crommelin, J.D. Opsteegh, F. Verhulst: A mechanism for atmospheric regime behaviour, J. Atmos. Sci. 61(12) (2004), 1406 1419. 11. L. van Veen: Baroclinic flow and the Lorenz-84 model, Int.J.Bifur.Chaos 13 (2003), 2117 2139. 12. L. van Veen, T. Opsteegh and F. Verhulst: Active and passive ocean regimes in a low-order climate model, Tellus 53A (2001), 616 628. Page 19 of 40 13. L. van Veen: Overturning and wind driven circulation in a low-order ocean-atmosphere model, Dyn.Atmos.Oceans 37 (2003), 197 221.

Bibliography (continued) 14. H.W. Broer, C. Simó, R. Vitolo: Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing, Nonlinearity 15(4) (2002), 1205-1267. 15. H.W. Broer, R. Vitolo: Dynamical systems modeling of low-frequency variability in low-order atmospheric models. Submitted. 16. H. E. De Swart: Analysis of a six-component atmospheric spectral model: Chaos, predictability and vacillation. Physica D 36 (1989), 222 234. Page 20 of 40 17. H. Itoh, and M. Kimoto: Multiple attractors and chaotic itinerancy in a quasigeostrophic model with realistic topography: Implications for weather regimes and low-frequency variability, J. Atmos. Sci. 53 (1996), 22172231.

Bibliography (continued) 18. V. Lucarini, A. Speranza, R. Vitolo: Physical and Mathematical Properties of a Quasi-Geostrophic Model of Intermediate Complexity of the Mid-Latitudes Atmospheric Circulation, preprint PASEF: www.unicam.it/matinf/pasef (2006). 19. P. Glendinning: Differential equations with bifocal homoclinic orbits, IJBC 7 (1997), 27-37. 20. G. Branstator, and J. D. Opsteegh: Free solutions of the barotropic vorticity equation, J. Atmos. Sci. 46 (1989), 1799 1814. 21. P. M. James, K. Fraedrich, and I. N. James: Wave-zonal-flow interaction and ultra-low-frequency variability in a simplified global circulation model, Quart. J. Roy. Meteor. Soc. 120 (1994), 1045 2013. 1067. Page 21 of 40 22. M. Viana: What s new on Lorenz strange attractors, The Mathematical Intelligencer 22 (3) (2000) 6-19.

Appendices Summary and connection of the Appendices A, B, C and D to main text: 1. There really exists something like low frequency variability. 2. Dynamical systems concepts play a key role in current activities, which are of societal relevance (weather forecasts, climate analysis). 3. Understanding of low frequency (regimes, regime transitions) in terms of Shil nikov, homo- and heteroclinic intermittency. 4. First step: from infinite (PDE) to finite (ODE) (Lorenz-84, Charney, DeVore & De Swart) by means of filtering and spectral projection. 5. Small models: we understand something of Shil nikov, Hopf-saddlenode, etc. 6. How does all of this look when returning from small to finite and to infinite? What is the role of 3. in the PDE s of the GCM? Do we find traces of this in NATURE? Page 22 of 40

A. Equations of 2LQG Lorenz, 1960: t 2 ψ = J(ψ, 2 ψ + f) J(τ, 2 τ) C 2 (ψ τ), t 2 τ = J(τ, 2 ψ + f) J(ψ, 2 τ) + C 2 (ψ τ)+ t θ = J(ψ, θ) + σ 2 χ 1 h N (θ θ ). + (f χ 1 ) 2C 2 τ, Model for atmospheric dynamics at mid-latitudes (Lorenz 1960). Coordinates (x, y) [0, L x ] [0, L y ] horizontally (x periodic). Pressure p vertically: discretized at two levels! Page 23 of 40 Unknowns: ψ, τ, θ, χ 1.

Unknowns in 2LQG model ψ = 1 2 (Ψ 3 + Ψ 1 ) τ = 1 2 (Ψ 3 Ψ 1 ) barotropic streamfunction baroclinic streamfunction θ = 1 2 (Θ 3 + Θ 1 ) mean potential temperature σ = 1 2 (Θ 3 Θ 1 ) static stability (fixed) Ψ 1,3 : streamfunction at lower and upper layer, respectively Θ 1,3 : potential temperature at lower and upper layer, respectively Page 24 of 40 C: friction at layer interface C : friction at layer interface h N : Newtonian cooling f: Coriolis parameter: constant (f-plane approx.) θ = 1 T cos(πy/l 2 y): imposed temperature profile (halfcosine shape) T : north-south temperature gradient

Original variables (non-discretized model) Let v be velocity (wind) in initial 3D equations: v = v h + v v with horizontal velocity: v h = v r + v d, where v r = k Ψ divergence-free component of v h, v d = χ irrotational component of v h, v v = p t k vertical velocity. Ψ : streamfunction χ : velocity potential T : temperature Θ : potential temperature ( p cp T = Θ, where p s ) cp cv p s surface pressure, c p, c v : specific heat of dry air at constant pressure and volume Page 25 of 40

B. The seasonally driven Lorenz-84 model ẋ = ax y 2 z 2 + af (1 + ɛ cos(ωt)) ẏ = y + xy bxz + G(1 + ɛ cos(ωt)) (1) ż = z + bxy + xz. T = 2π/ω = 73: period of the forcing (a, b: constants) F, G, ɛ: control parameters Poincaré (time T ) map P F,G,ɛ : P F,G,ɛ : R 3 R 3 is diffeomorphism. Page 26 of 40 Problem setting: - Coherent inventory of dynamics of P F,G,ɛ depending on F, G, ɛ. H.W. Broer, C. Simó & R. Vitolo, Nonlinearity 15(4) (2002), 1205-1267.

Bifurcation diagram of fixed points of P F,G Page 27 of 40 ɛ = 0.5 fixed. H 2 : Hopf, H sub 1 : subcritical Hopf, SN 0 sub: saddle-node.

Page 28 of 40 Magnification of box A in Fig.1. Cusp C terminates two saddle-node curves. H 1 : Hopf. A 1 1:1 resonance tongue. Strange attractors in L 1.

Disappearance of HSN bifurcation point Page 29 of 40 ɛ = 0.01 ɛ = 0.5 Hopf and saddle-node bifurcation curves are no longer tangent for ɛ = 0.5. Several strong resonances interrupt Hopf bifurcation curve. Codimension 3 bifurcation between ɛ = 0.01 and 0.5?

Quasi-periodic bifos of invariant circles 2 1 0-1 z 1e-05 1e-15 1e-25 16 10 6 22 20 4 12 14 2 18 8 24 x -2 0.7 0.9 1.1 1.3 2 1 0 z -1 x -2 0.7 0.9 1.1 1.3 1e-35 1e-05 1e-15 1e-25 1e-35 0 0.1 0.2 0.3 0.4 0.5 13 1 3 5 7 9 11 19 23 0 0.1 0.2 0.3 0.4 0.5 17 15 21 Page 30 of 40 Left: projections on (x, z) of attractors. Right: power spectra. Top: G = 0.4872. Bottom: G = 0.4874 (F = 11).

Quasi-periodic strange attractors 2 0.01 1 1e-06 0 1e-10-1 -2 0.7 1e-14 0.9 1.1 1.3 1.5 0 0.1 0.2 0.3 0.4 1 2 0.01 3 0.5 JJ II J I 2 4 Page 31 of 40 1 6 0 S 1e-07 z -1-2 0.7 x 0.9 1.1 1.3 1e-12 1.5 0 0.1 0.2 0.3 0.4 0.5 Left: projections on (x, z) of attractors. Right: power spectra. Top: G = 0.497011. Bottom: G = 0.4972 (F = 11).

Hénon-like attractors Ansatz: Hénon-like strange attractor = closure of unstable manifold Page 32 of 40

C. Homoclinic dynamics and ultralow-frequency variability D. Crommelin: J. Atmos. Sci. 59 (2002). Ultralow-frequency: timescale beyond several months. Common physical explanations: 1. Associated to slow dynamical components: ice, oceans. 2. Seasonal variations of parameters (James & James, 1989) 3. Interaction of zonal (longitudinal) flow and baroclinic waves (James & James, 1994) Page 33 of 40 But what is mathematical structure? Connection with homoclinic dynamics? Present investigation: consider various models: 1. A General Circulation Model (GCM): NCAR CCM version 0B. 2. Empirical Orthogonal Projection (EOF) of a quasi-geostrophic model. 3. A 4D simplified model.

Page 34 of 40 Low-frequency variability in power spectra of EOF1 GCM (top), 30D EOF model (middle), 10D EOF (bottom)

Page 35 of 40 Traces of homoclinic dynamics in attractors of 4D model (2D projection) for four values of a control parameter. Last plot (h) is attracting periodic orbit: suggests bifocal homoclinic, two equilibria of saddle-focus type.

D. The 6D ODE model Charney and DeVore (1979), De Swart (1989). From barotropic vorticity equation (PDE), Galerkin projection ẋ 1 = γ 1 x 3 C(x 1 x 1), ẋ 2 = (α 1 x 1 β 1 )x 3 Cx 2 δ 1 x 4 x 6, ẋ 3 = (α 1 x 1 β 1 )x 2 γ 1 x 1 Cx 3 + δ 1 x 4 x 5, ẋ 4 = γ 2 x 6 C(x 4 x 4) + ε(x 2 x 6 x 3 x 5 ), ẋ 5 = (α 2 x 1 β 2 )x 6 Cx 5 δ 2 x 4 x 3, ẋ 6 = (α 2 x 1 β 2 )x 5 γ 2 x 4 Cx 6 + δ 2 x 4 x 2. Physical meaning of terms: α j : advection of waves by zonal (longitudinal) flow. β j : Coriolis force. γ j, γ j : topography. C: Newtonian damping to zonal profile (x 1, 0, 0, x 4, 0, 0). δ, ε: Fourier modes interactions due to nonlinearity. Page 36 of 40 Control parameters: x 1, γ, r (where x 4 = rx 1).

Bifurcations in Charney-DeVore-De Swart Crommelin, Opsteegh, Verhulst, 2004. Codimension 2 organizing centers in (F, G) parameter plane: - Hopf-saddle-node bifo of equilibria. - Cusp of equilibria. - 1:2 strong resonance bifo of periodic orbits. Homoclinic bifurcations: Shil nikov tangencies. Codimension 1 bifos of equilibria (Hopf, saddle-node) and of periodic orbits (Hopf, period doubling). Page 37 of 40 Very strong analogies with Lorenz-84!

Bifurcation analysis fh: Hopf-saddle-node c: cusp sn1,sn2: saddle-node pd: period doubling (of periodic orbits) Page 38 of 40

Homoclinic (Shil nikov) orbits Homoclinic orbits of an equilibrium eq1 occurring, from top to bottom, at different values of the parameters (x 1, r). Page 39 of 40

Bimodality, regimes, and heteroclinics Intermittency of saddle-node type occurs after equilibrium eq2 coalesces with another equilibrium eq3 (at sn2). Intermittent heteroclinic behaviour: orbits visit alternatively vicinity of eq1 and (formerly existing) eq2,eq3. Bimodality of probability distribution function (bottom right): near eq1 and near eq2,eq3. Page 40 of 40 Notice high speed in phase space along transitions between two regions.