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Low-frequency variability in climate models:

a dynamical systems perspective.

Henk Broer

Rijksuniversiteit Groningen

http://www.math.rug.nl/∼broer

23 May 2006

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Low-frequency atmospheric variability

Variability: nonzero spectral power.

Low-frequency: timescale of one week to a month.

Power spectrum of a time series

related to mean solid-body ro-

tation of atmosphere relative to

Earth (Y 10 vorticity coefficient av-

eraged wrt pressure), from multi-

level spectral hemispheric model

of atmosphere (James & James

1989).

“Red noise” spectrum.

Similar spectra are typical in atmosphere/ocean data and models

Can this be studied by dynamical system techniques?

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Towards “simple” atmospheric models

Search for simplified models, derived from general equations of motion.

Common technique: start from Navier-Stokes and use approximations,

expansions in small parameters, and “global balances”.

simplified models: quasi-geostrophic eqs., barotropic vorticity eq.

Further: suitable boundary conditions, Galerkin truncations.

low-order models: Lorenz-84 (and many more).

Qualitative representation of fundamental physical processes, no realism.

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The Lorenz-84 model: derivation

Lennaert van Veen, IJBC 13(8) (2003).

Starting point: two-layer quasi-geostrophic model (Lorenz 1960):

∂

∂t∇2ψ = −J(ψ,∇2ψ + f )− J(τ,∇2τ )− C∇2(ψ − τ ),

∂

∂t∇2τ = −J(τ,∇2ψ + f )− J(ψ,∇2τ ) + C∇2(ψ − τ )+

+∇ · (f∇χ1)− 2C ′∇2τ,

∂

∂tθ = −J(ψ, θ) + σ∇2χ1 − hN(θ − θ∗).

Coordinates (x, y) ∈ [0, Lx]× [0, Ly] horizontally.

Pressure p vertically: discretized at two levels!

Unknowns: ψ, τ , θ, χ1.

Obtained from Navier-Stokes (+ energy equation) by hydrostatic and quasi-

geostrophic balances.

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Two-layer quasi-geostrophic model

Obtained by discretization of pressure in two vertical layers:

Ψ3, Θ3, χ3

Ψ1, Θ1, χ1

Earth’s surface, p = p0

Bottom layer, p = 3

4p0

Boundary between layers, p = p0/2

Top layer, p = 1

4p0

Upper boundary, p = 0

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Domain of 2LQG

Model for atmospheric dynamics at mid-latitudes.

Domain: β-channel (f -plane): (x, y)-rectangle s.t. Coriolis f ∼ const.

150o W

120

o W

90oW

60 oW

30 oW

0o

30o E

60

o E

90o E

120 oE

150 oE

180oW

45oN

22o30’N

67o30’N

Ly

x

y ^ ^

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The Lorenz-84 model

Fourier expansion + Galerkin projection Lorenz-84 ODE:

x = −ax− y2 − z2 + aF,

y = −y + xy − bxz +G,

z = −z + bxy + xz.

E.N. Lorenz (1984,1990): simplest model for atmospheric dynamics at mid-

latitudes.

Shil′nikov, Nicolis, Nicolis (1995): comprehensive bifurcation analysis.

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, Simo, Vitolo (2002): influence of periodic forcing on (F,G), new types

of strange attractors.

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Bifurcations in Lorenz-84

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.

Homoclinic bifurcations: Shil′nikov tangencies.

Several codimension 1 bifos of equilibria (Hopf, saddle-node)

and of periodic orbits (Hopf, saddle-node, period doubling).

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Strange attractors in Lorenz-84

Parameter values are s.t. a Shil′nikov bifurcation takes place.

Blue: 4-times period-doubled periodic orbit.

Red: Shil′nikov homoclinic orbit.

Green: strange attractor.

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The periodically driven Lorenz-84 model

x = −ax− y2 − z2 + aF (1 + ε cos(ωt))

y = −y + xy − bxz +G(1 + ε cos(ωt))

z = −z + bxy + xz.

(1)

T = 2π/ω = 73: period of the forcing (a, b: constants)

F , G, ε: control parameters

Poincare map PF,G,ε (time T map of system (1)):

PF,G,ε : R3 → R3 is diffeomorphism.

Problem setting:

- Coherent inventory of dynamics of PF,G,ε depending on F , G, ε.

H.W. Broer, C. Simo, R. Vitolo, Nonlinearity 15(4) (2002), 1205-1267.

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Bifurcation diagram of fixed points of PF,G

Hopf curve H2 = ∂Q2. Hsub1 : subcritical Hopf. SN 0

sub: saddle-node.

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Magnification of box A in Fig.1.

Cusp C terminates two saddle-node curves.

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Disappearance of HSN bifurcation point

ε = 0.01 ε = 0.5

Hopf and saddle-node bifurcation curves are no longer tangent for ε = 0.5.

Several strong resonances interrupt Hopf bifurcation curve.

Higher codimension bifurcation between ε = 0.01 and 0.5?

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Quasi-periodic bifos of invariant circles

-2

-1

0

1

2

0.7 0.9 1.1 1.3

A

1e-35

1e-25

1e-15

1e-05

0 0.1 0.2 0.3 0.4 0.5

a24

6

810

1214

16

1820

22 24

-2

-1

0

1

2

0.7 0.9 1.1 1.3

B

1e-35

1e-25

1e-15

1e-05

0 0.1 0.2 0.3 0.4 0.5

b 13 5

79

1113

1517

19

2123

Left: projections on (x, z) of attractors. Right: power spectra.

Top: G = 0.4872. Bottom: G = 0.4874.

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Quasi-periodic strange attractors

-2

-1

0

1

2

0.7 0.9 1.1 1.3 1.5

d

1e-14

1e-10

1e-06

0.01

0 0.1 0.2 0.3 0.4 0.5

-2

-1

0

1

2

0.7 0.9 1.1 1.3 1.5

B

S

1e-12

1e-07

0.01

0 0.1 0.2 0.3 0.4 0.5

b12

3 4

6

Left: projections on (x, z) of attractors. Right: power spectra.

Top: G = 0.497011. Bottom: G = 0.4972.

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Homoclinic dynamics and

ultralow-frequency variability

D. Crommelin: J. Atmos. Sci. 59 (2002).

Ultralow-frequency: timescale beyond several months. Hypotheses:

1. Associated to slow dynamical components: ice, oceans.

2. Periodic variations of parameters.

3. Interaction of zonal flow and baroclinic waves.

What is mathematical structure? Connection with homoclinic dynamics?

Present investigation: consider various models:

1. General Circulation Model (GCM): NCAR CCM version 0B.

2. Empirical Orthogonal Projection (EOF) of a quasi-geostrophic model.

3. A 4D simplified model.

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Attractors of 4D model, projection on (x9, x10), for ω5 fixed at:

(a) 0.0353438 (d) 0.0323438 (e) 0.0313438 (h) 0.0290188

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Homoclinic orbits in atmospheric models?

Last plot (h) is attracting periodic orbit: suggests bifocal homoclinic, two

equilibria of focus type.

Two equilibria are found in 4D model, but not for original value of ω5.

Homoclinic orbit: not found! Is it there?

Main idea: mean state is equilibrium, system returns often near it.

Homoclinicity: model for recurrence of system to vicinity of mean state.

Existence of homoclinic orbit in a GCM? Very hard problem!

Perhaps, homoclinicity characterizes large-scale atmospheric flow.

Increasing dimension of phase space means introducing small spatial scales.

This provides “noise”, obscuring the homoclinic-like recurrence in realistic

atmospheric models.

However, homoclinic-like intermittency is found in other models:

James et al. (1994), Branstator and Opsteegh (1989).

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Atmospheric regimes and transitions

D. Crommelin, J.D. Opsteegh, F. Verhulst, J. Atmos. Sci. 61 (2004).

Regime: preferred flow pattern.

Evidence found in northern hemisphere atmospheric data.

Charney and DeVore (1979): regimes are coexisting equilibria of equations.

How to explain regime transitions?

Old idea: stable equilibria + stochastic (random) perturbation.

New idea: chaotic itinerancy.

Several attractor coexist for other parameter values.

By changing parameters, the attractors lose stability.

Then, typical orbits visit the remnants of the formerly existing attractors.

In other words: intermittency due to heteroclinic connections.

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A 6D model: derivation

Barotropic vorticity equation in a β-plane channel:

∂

∂t∇2ψ = −J(ψ,∇2ψ + f + γh)− C∇2(ψ − ψ∗).

(t, x, y) ∈ R× [0, 2π]× [0, πb] are time, longitude and latitude.

b = 2B/L ratio between zonal length L and meridional width B of channel.

Streamfunction ψ(t, x, y) is periodic in x.

f is Coriolis parameter, h is orography.

Newtonian relaxation towards profile ψ∗.

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The Charney-DeVore model

Galerkin projection

x1 = γ1x3 − C(x1 − x∗1),

x2 = −(α1x1 − β1)x3 − Cx2 − δ1x4x6,

x3 = (α1x1 − β1)x2 − γ1x1 − Cx3 + δ1x4x5,

x4 = γ2x6 − C(x4 − x∗4) + ε(x2x6 − x3x5),

x5 = −(α2x1 − β2)x6 − Cx5 − δ2x4x3,

x6 = (α2x1 − β2)x5 − γ2x4 − Cx6 + δ2x4x2.

Physical meaning of terms. αj: advection of waves by zonal flow.

βj: Coriolis force. γj, γj: topography.

C: Newtonian damping to zonal profile (x∗1, 0, 0, x∗4, 0, 0).

δ, ε: Fourier modes interactions due to nonlinearity.

Control parameters: x∗1, γ, r (where x∗4 = rx∗1).

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Hopf-saddle-node bifurcation

Third order normal form:

y = ρ1 + y2 + s|z|2

z = (ρ2 + iω1)z + (θ + iϑ)yz + y2z

y ∈ R, z ∈ C.

Unfolding case: s = 1, θ < 0.

sn: saddle-node

hb: Hopf

ns: Neimark-Sacker

hc: heteroclinic connection

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Sphere-like heteroclinic structure

Truncated normal form has invariant sphere with heteroclinic cycle. Inclu-

sion of generic higher order terms breaks both structures, and Shil′nikov

homoclinic bifurcations appear.

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Bifurcation analysis

fh: fold (saddle-node)-Hopf

c: cusp

sn1,sn2: saddle-node

pd: period doubling

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Homoclinic (Shil′nikov) orbits

Homoclinic orbits of the “zonal” equi-

librium eq1 occurring, from top to

bottom, at different values of the pa-

rameters (x∗1, r).

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Bimodality, regimes, and heteroclinics

Intermittency of saddle-node type occurs after eq2,eq3 coalesce at sn2.

Due to near-heteroclinic behaviour, orbits visit alternatively vicinity of eq1

and (formerly existing) eq2,eq3.

High speed in phase space

along transitions between

two regions implies

bimodality of probability

distribution function (bot-

tom right).

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Conclusions

1. Hopf-saddle-node bifurcations in very different low-order models of at-

mospheric circulation.

2. Homoclinic- and heteroclinic-like behaviour: an explanation for

(a) regime transitions (chaotic itinerancy)

(b) bimodality (observed in data)

3. What is relation with more complex models? Original PDEs?

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Bibliography

1. E.N. Lorenz: Irregularity: a fundamental property of the atmosphere,

Tellus 36A (1984), 98–100.

2. E.N. Lorenz: Can chaos and intransitivity lead to interannual variabil-

ity? Tellus 42A (1990), 378–389.

3. I.N. James, P.M. James: Ultra-low-frequency variability in a simple

atmospheric circulation model, Nature 342 (1989), 53-55.

4. R. Pielke, X. Zeng: Long-term variability of climate, J. Atmos. Sci.

51 (1994), 155-159.

5. A. Shilnikov, G. Nicolis, C. Nicolis: Bifurcation and predictability anal-

ysis of a low-order atmospheric circulation model, Int.J.Bifur.Chaos

5(6) (1995), 1701–1711.

6. H.W. Broer, C. Simo, R. Vitolo: Bifurcations and strange attractors in

the Lorenz-84 climate model with seasonal forcing, Nonlinearity 15(4)

(2002), 1205-1267.

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Bibliography (continued)

7. D.T. Crommelin: Homoclinic Dynamics: A Scenario for Atmospheric

Ultralow-Frequency Variability, J. Atmos. Sci. 59(9) (2002), 1533–

1549

8. D.T. Crommelin: Regime transitions and heteroclinic connections in a

barotropic atmosphere J. Atmos. Sci., 60(2) (2003), 229–246.

9. D.T. Crommelin, J.D. Opsteegh, F. Verhulst: A mechanism for atmo-

spheric regime behaviour, J. Atmos. Sci. 61(12) (2004), 1406–1419.

10. L. van Veen: Baroclinic flow and the Lorenz-84 model,

Int.J.Bifur.Chaos 13 (2003), 2117–2139.

11. L. van Veen, T. Opsteegh and F. Verhulst: Active and passive ocean

regimes in a low-order climate model, Tellus 53A (2001), 616–628.

12. L. van Veen: Overturning and wind driven circulation in a low-order

ocean-atmosphere model, Dyn.Atmos.Oceans 37 (2003), 197–221.