Winter Is Coming: A Dynamical SystemsApproach to Better El Nino Predictions

Andrew Roberts

Esther Widiasih, Chris Jones, Axel Timmerman

October 14, 2014

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Outline

• What is ENSO?

• Why do we care?

• Conceptual ENSO Models

• Dynamical Systems

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The Data

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2014 PredictionsAugust:

September: No ENSO this yearOctober 9: El Nin0 watch back on. Chances back over 65 % (NOAA).

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Physical Cartoon

From: NOAA 5/26

Jin’s Model - 1997

Recharge Oscillator Model:

dTE

dt= RTE + γhW − en(hW + bTE )3

dhWdt

= −rhW − αbTE .

Variables:

• TE temperature anomaly in E Pacific

• hW thermocline depth anomaly in W Pacific

Processes:

• Relaxation to mean state

• Upwelling

• Thermocline adjustment to wind-stress

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Results

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Supercharged Recharge Oscillator

dT1

dt= −α(T1 − Tr )− δµ(T2 − T1)2

dT2

dt= −α(T2 − Tr ) + ζµ(T2 − T1)[T2 − Tsub(T1,T2, h1)]

dh1dt

= −r[h1 −

bLµ

2β(T2 − T1)

]Modifications from Recharge Oscillator model:

• Variables are no longer anomalies

• Subscript 1 indicates West, 2 indicates East

• Includes advection

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Simulation vs. Data

From: Timmerman, Jin, and Abshagen – 2003 9/26

MMOs

• Timmerman, Jin, Abshagen (2003): Shilnikov-type mechansim(saddle-focus)

• Other methods require multiple time-scales!

Change of variables:

• S = T2 − T1

• T = T1 − Tr

• h = h1 − K

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New System

dS

dt= −αS + δµS2 + ζµS [S + T + Cf (S ,T , h)]

dT

dt= −αT − δµS2

dh

dt= −r

(h − K +

bLµ

2βS

)We want to analyze as a fast/slow system → need to non-dimensionalize!

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Dimensionless System

x ′ = ε(x2 − ax) + x

[x + y − nz + d − c

(x − x3

3

)]y ′ = −ε(ay + x2)

z ′ = m(k − z − x

2

)• x ∼ S

• y ∼ T

• z ∼ h

• ε = δ/ζ

• c = bLµ(Tr−Tr0)2h∗β

• a = αbLδβh∗

• m = rbLζβh∗

• n, k, d come from f

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Dimensionless System

x ′ = ε(x2 − ax) + x

[x + y − nz + d − c

(x − x3

3

)]y ′ = −ε(ay + x2)

z ′ = m(k − z − x

2

)• If ε is small (how small?), we have a fast/slow system.

• If m is also small, we have 1-fast, 2-slow OR 3 time-scale system.

• If m = O(1), we have a 2-fast, 1-slow system.

εx = ε(x2 − ax) + x

[x + y − nz + d − c

(x − x3

3

)]y = −(ay + x2)

εz = m(k − z − x

2

)13/26

Fast and Slow Dynamics

If we take the limit as ε→ 0, the two versions become: (1) The layerproblem

x ′ = x

[x + y − nz + d − c

(x − x3

3

)]z ′ = m

(k − z − x

2

)y ′ = 0

Or (2) The reduced problem

0 = x

[x + y − nz + d − c

(x − x3

3

)]0 = m

(k − z − x

2

)y = −(ay + x2)

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The Critical Manifold (ε = 0)

Figure: a = 0.4, c = 4, d = 3.69, k = 1,m = 0.26, n = 2.69

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The Critical Manifold (ε = 0)

Figure: a = 0.4, c = 4, d = 3.69, k = 1,m = 0.26, n = 2.69

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The Critical Manifold (ε = 0)

Figure: a = 0.4, c = 4, d = 3.69, k = 1,m = 0.26, n = 2.69

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The Critical Manifold (ε = 0)

Figure: a = 0.4, c = 4, d = 3.69, k = 1,m = 0.26, n = 2.69

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The Critical Manifold (ε = 0)

Figure: a = 0.4, c = 4, d = 3.69, k = 1,m = 0.26, n = 2.69

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The Critical Manifold (ε = 0)

Figure: a = 0.4, c = 4, d = 3.69, k = 1,m = 0.26, n = 2.69

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The Critical Manifold (ε = 0)

Figure: a = 0.4, c = 4, d = 3.69, k = 1,m = 0.26, n = 2.69

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The Critical Manifold (ε = 0)

Figure: a = 0.4, c = 4, d = 3.69, k = 1,m = 0.26, n = 2.69

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MMO Orbit: ε = 0.001

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MMO Orbit: ε = 0.1

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Re-dimensionalized Model Output

0 20 40 60 80 100 120 140 160 180 20022

23

24

25

26

27

28Cubic Approximation−Dimensionalized

T1

T2

0 20 40 60 80 100 120 140 160 180 20060

70

80

90

100

h

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Consequences in the US

From: NOAA26/26